Asymptotic analysis of high order solitons for the Hirota equation
AASYMPTOTIC ANALYSIS OF HIGH ORDER SOLITONS FOR THE HIROTA EQUATION
XIAOEN ZHANG AND LIMING LINGA
BSTRACT . In this paper, we mainly analyze the long-time asymptotics of high order soliton for the Hirota equa-tion. With the aid of Darboux transformation, we construct the exact high order soliton in a determinant form.Two different Riemann-Hilbert representations of Darboux matrices with high order soliton are given to establishthe relationships between inverse scattering method and Darboux transformation. The asymptotic analysis withsingle spectral parameter is derived through the formulas of determinant directly. Furthermore, the long-timeasymptotics with k spectral parameters is given by combining the iterated Darboux matrices and the result ofhigh order soliton with single spectral parameter, which discloses the structure of high order soliton clearly andis possible to be utilized in the optic experiments. Keywords:
Hirota equation, Asymptotic analysis, High order soliton
We are concerned with the following Hirota equation,(1) i q t + γ (cid:16) q xx + | q | q (cid:17) + i δ (cid:16) q xxx + | q | q x (cid:17) = γ = δ =
0, Eq.(1) can be reduced to the NLS equation [2, 3], which is a universal model that describes theevolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media. If γ = δ =
1, Eq. (1) will turn into the complex modified Korteweg-de Vries equation[4]. Due to the effect ofhigh order dispersion, Hirota equation (1) has better properties than NLS equation in some aspect. Froma physical view point, Eq.(1) can describe the ultra-short pulse propagation and the phenomenon of oceanwaves more precisely than NLS equation. From the mathematical view point, NLS equation has five kindsof symmetries, while Hirota equation just has four types, that is(2) ˜ q ( x , t ) = e i (cid:101) q ( x , t ) ,˜ q ( x , t ) = q ( x + (cid:101) , t ) ,˜ q ( x , t ) = q ( x , t + (cid:101) ) ,˜ q ( x , t ) = e (cid:101) − i γ (cid:32) ( e (cid:101) − ) δ x + γ ( e (cid:101) − ) ( e (cid:101) + ) δ t (cid:33) q (cid:32) e (cid:101) (cid:32) x − γ (cid:0) − e (cid:101) (cid:1) δ t (cid:33) , e (cid:101) t (cid:33) .The first three symmetries are consistent with the NLS equation. The fourth one corresponds to the Galieansymmetry, whose parameter δ is involved in the denominator and can not be equal to zero, so it can not de-generate to the symmetry of the NLS equation. In addition, it is clear that the NLS equation has the scalingsymmetry ˜ q ( x , t ) = e (cid:101) q ( x e (cid:101) , t e (cid:101) ) , but the Hirota equation has no scaling invariance. So the properties ofHirota equation will be different from the NLS equation in this aspect.As an integrable physical model, the soliton solution has always been a research hot. In self modulationproblem, the soliton [5] is a single wave packet propagating without distortion of its envelope, which isregarded as a balance between the dispersion and the nonlinear term. If two or more solitons admit thedistinct velocities, then these solitions will interact each other without changing the shape but with theshifting of phase. Otherwise, the solitons will come into being the breathing or periodic effect[6]. If wetake the periodic parameter to infinity, the breather will turn into the multi-pole (i.e. high order) soliton. Date : August 31, 2020. a r X i v : . [ n li n . S I] A ug ctually, the systematic analysis for the high order soliton of NLS equation is a long-standing problem inthe integrable theory, which was solved very recently [7, 8].In the terminology of inverse scattering transform [9], N -solitons are given in the reflectionless caseunder the transmission scattering data a ( λ ) has N distinct simple poles. The multi-soliton with differentvelocities will break up into individual solitons as t → ± ∞ , so it can be used to describe the interactionbetween N solitons. Especially, when N → ∞ , there are infinity solitons, which was first provided by Zhou[10]. The explicit infinity solitons to NLS equation and KdV equation were obtained in the reference [11]and [12] respectively. Especially, in[13], the authors gave a detailed analysis to the NLS equation when N is large. On the other hand, the multi-pole soliton are related to the scattering data a ( λ ) has multiple poles,which can describe the interaction between N solitons of equal amplitude but having a particular chirp dueto they depend on a single spectrum. When t → ± ∞ , they also will break up into N individual solitons.There will not exist multi-pole nonsingular soliton to the Korteweg-de Vries equation, because the spec-tral point is simple and its Lax operator is self-adjoint [14]. However, the focusing NLS equation, it allowsmultiple poles, which was first observed by Zakharov and Shabat [15]. Since then, there are many relativemultiple pole solutions about various equations, such as the sine-Gordon equation [16, 17], the modifiedKorteweg-de Vries equation [18]. Especially, Olmedilla[19] studied the multiple pole solutions to NLS equa-tion by solving Gelfand-Levitan-Marchenko equation and gave the asymptotic analysis to double pole andtriple pole soliton when t → ∞ , but he left over a conjecture about the asymptotics with general multi-polesoliton. Fortunately, Schiebold confirmed this conjecture and presented a detailed description about theasymptotic analysis [8]. Additionally, there are some relative research on the asymptotic behavior on boththe (1+1)-dimensional integrable system and (2+1)-dimensional system [20, 21, 22]. Recently, the multiplepole soliton concept is extended to some nonlocal system and high order (2+1)-dimensional system[23, 24].Thus it is natural to consider the long-time asymptotics of high order soliton to the Hirota equation. In[25, 26], the authors gave the asymptotic expression of the second-order and third-order soliton for Hirotaequation, but they did not give the asymptotic analysis to general high order soliton. Compared withthe soliton in NLS equation, the soliton in Hirota equation has the different velocity due to the high orderdispersion and time-delay corrections. So the multi-soliton for NLS equation and Hirota equation is similar.But for the high order soliton, they are different because the expression is the mixture of exponential andrational functions. And the dynamics of high order soliton between NLS equation and Hirota equationhas the great discrepancy. Thus the long-time asymptotic analysis for the Hirota equation is necessityto further study for the theory of integrable models. We plan to analyze the soliton solution directly toobtain the long-time asymptotics, which is not depending on the Riemann-Hilbert problem(RHP). Beforethis analysis, we need to construct the exact soliton with the aid of Darboux transformation. Now we givea brief introduction about this method.Darboux transformation is an algebraic method, which has several different versions [27, 28]. One of therigorous method is using the loop group [29], which solves the Lax pair by assumption of the holomor-phic property of wave functions with Φ ( λ ; 0, 0 ) = I . For the AKNS system with the su ( ) symmetry, theelementary Darboux matrix can be rewritten as:(3) T ( λ ; x , t ) = I − λ − λ ∗ λ − λ ∗ P ( x , t ) , P ( x , t ) = φ φ †1 φ †1 φ , φ = Φ ( λ ; x , t ) v ,where v is a constant vector. And the new wave function Φ [ ] ( λ ; x , t ) = T ( λ ; x , t ) Φ ( λ ; x , t ) T − ( λ ; 0, 0 ) willsatisfy a new Lax pair with the same shape. The corresponding B¨acklund transformation between old andnew potential functions is(4) q [ ] ( x , t ) = q ( x , t ) − ( λ − λ ∗ ) φ φ ∗ φ †1 φ , φ = ( φ , φ ) T .If | q (
0, 0 ) | = max ( | q ( x , t ) | ) , the module of new solution q [ ] ( x , t ) will attain the maximal value at ( x , t ) =(
0, 0 ) with | q [ ] (
0, 0 ) | = | q (
0, 0 ) | + ( λ ) by taking v = [
1, i ] T . This simple proposition is beneficial toconstruct the solitons with maximal peak. If we want to iterate the Darboux transformation at λ = λ , thenew wave function Φ [ ] ( λ ; x , t ) will appear the removable singularity. So we should redefine the Darbouxtransformation with limit technique so as to get the high order soliton at the same spectral point. Based on his method, the high order soliton to derivative NLS equation [30], the Landau-Lifshitz equation [31], therogue wave to NLS equation[32], the spatial discrete Hirota equation[33] are given systematically. Apartfrom the generalized Darboux transformation method, some other algebraic method, such the KP reductionmethod can also be used to construct the high-order soliton or rogue wave[34, 35].Furthermore, after giving the Darboux transformation and the corresponding B¨acklund transformation,we begin to analyze the long-time asymptotics of high order soliton. There are three steps in the wholeanalysis procedure. Firstly, we extract the leading order term from the exact high-order soliton solutiondeterminant with single spectral parameter directly, which is a crucial factor to the long-time asymptotics.Then we find an interesting fact that if the high order soliton moves along a special characteristic curve,then the limit of the high order can be reduced to a single soliton. Otherwise, it has a vanishing limitation.Therefore, we give the long-time asymptotics on the basis of moving along the characteristic curves. Moreimportantly, with the aid of the asymptotics of Darboux matrices and the result of the asymptotics withsingle spectral parameter, we give a general long-time asymptotics of high order soliton with k spectralparameters, which has never been reported before.The outline of this paper is organized as follows. In Section 2, the Hirota equation is derived fromthe AKNS hierarchy. In section 3, the Darboux matrix involving k spectral parameters λ , λ , · · · , λ k of theHirota equation is given. The RHP for the Darboux matrices is derived to establish the relationship betweentwo approaches. With the aid of B¨acklund transformation, the high order soliton could be constructed bytaking the zero seed solution. In section 4, the asymptotic analysis of high order soliton with single spectralparameter λ is shown by the Theorem 2. Then the long-time asymptotic analysis can be extended to thegeneral case with k spectral parameters, which is given by the Theorem 4. The conclusions and discussionsare involved in the finial section. Consider the AKNS spectral problem(5) ∂∂ x ΦΦΦ ( λ ; x , t ) = U ( λ ; x , t ) ΦΦΦ ( λ ; x , t ) , U ( λ ; x , t ) = i ( λσ + Q ( x , t )) , Q ( x , t ) = (cid:20) r ( x , t ) q ( x , t ) (cid:21) ,where x ∈ R , t = ( t , t , · · · ) ∈ R ∞ , σ = diag ( − ) is the third Pauli matrix and Φ ( λ ; x , t ) is thewave function. Now we introduce the real infinite-dimensional iso-spectral manifold [36] U for the AKNSspectral problem (5). The flow was defined by the following iso-spectral equations(6) ∂∂ t i ΦΦΦ ( λ ; x , t ) = i (cid:16) λ i m ( λ ; x , t ) σ m − ( λ ; x , t ) (cid:17) + ΦΦΦ ( λ ; x , t ) ,where the subscript + denotes the positive power of polynomial with respect to λ , and(7) ΦΦΦ ( λ ; x , t ) = m ( λ ; x , t ) exp (cid:34) i σ (cid:32) λ ( x + t ) + + ∞ ∑ i = t i λ i (cid:33)(cid:35) , m ( λ ; x , t ) = I + m ( x , t ) λ − + · · · ,for λ in the neighborhood of ∞ . Plugging the ansatz (7) into the spectral problem (5), we have(8) i m ( x , t ; λ ) σ m − ( x , t ; λ ) + ∂∂ x m ( x , t ; λ ) m − ( x , t ; λ ) = i ( λσ + Q ( x , t )) .Comparing the coefficients of (8) with respect to λ , we arrive at(9) Q ( x ; t ) = [ σ , m ( x , t )] ,which is useful to reconstruct the solution of integrable equations, where the commutator [ A , B ] = AB − BA . Furthermore, we show that the holomorphic function of λ : A ( λ ; x , t ) = m ( λ ; x , t ) σ m − ( λ ; x , t ) can beexpanded in the neighborhood of ∞ with the form(10) A ( λ ; x , t ) = ∞ ∑ i = A i ( x , t ) λ − i , A = σ , here the matrices A i ( x , t ) can be determined by the recursion relationship(11) ∂∂ x A i ( x , t ) = i [ σ , A i + ( x , t )] + i [ Q ( x , t ) , A i ( x , t )] , i ≥ A off1 = Q ,which was implied by the stationary zero-curvature equation A x = [ U , A ] . Taking the off-diagonal part ofequation (11), it will reduce into(12) A off i + = − σ (cid:18) i ∂∂ x A off i + (cid:104) Q , A diag i (cid:105)(cid:19) .On the other hand, due to a simple identity m ( λ ; x , t )[ σ − I ] m − ( λ ; x , t ) = A ( λ ; x , t ) − I =
0, comparingthe coefficient of λ , we give the diagonal elements of A i as(13) A diag i + = − σ i ∑ k = ( A k A i + − k ) diag , i ≥ A diag1 = Proposition 1.
The matrices A i in the equation (10) can be represented as the differential polynomials of Q with thederivative on the variable x. The first three elements can be represented as A = Q , A = − (cid:16) σ Q + i σ Q x (cid:17) , A = − (cid:16) Q xx + Q − i Q x Q + i QQ x (cid:17) .(14)Furthermore, if we suppose the matrix function m ( λ ; x , t ) is holomorphic in C / Γ , where Γ is the contourin the complex plane, then we have the following propositions: Proposition 2.
The wave functions
ΦΦΦ ( λ ; x , t ) define the following evolution equations: (15) ∂∂ t k ΦΦΦ ( λ ; x , t ) = i (cid:32) k ∑ i = A i λ k − i (cid:33) ΦΦΦ ( λ ; x , t ) . Proof.
Since we assume that the matrix function m ( λ ; x , t ) is holomorphic in C / Γ , then the matrix function ΦΦΦ ( λ ; x , t ) = m ( λ ; x , t ) exp (cid:2) i σ (cid:0) λ ( x + t ) + ∑ + ∞ i = t i λ i (cid:1)(cid:3) is also analytic in the complex region C / Γ . On theother hand, the matrix function ΦΦΦ ( λ ; x , t ) satisfies the second order differential equation with respect to thevariable x , then the non-tangential limit of ΦΦΦ ( λ ; x , t ) on the contour Γ to the different sides will satisfy therelation ΦΦΦ + ( λ ; x , t ) = ΦΦΦ − ( λ ; x , t ) V ( λ ) , which implies that the derivative of ΦΦΦ ± with respect to the variable t k also satisfy the following same jump relationship ∂∂ t k ΦΦΦ + ( λ ; x , t ) = ∂∂ t k ΦΦΦ − ( λ ; x , t ) V ( λ ) . Then we knowthat (cid:18) ∂∂ t k ΦΦΦ + ( λ ; x , t ) (cid:19) ΦΦΦ − + ( λ ; x , t ) = (cid:18) ∂∂ t k ΦΦΦ − ( λ ; x , t ) (cid:19) ΦΦΦ − − ( λ ; x , t ) on the whole complex plane C , which implies the matrix function (cid:16) ∂∂ t k ΦΦΦ ( λ ; x , t ) (cid:17) ΦΦΦ − ( λ ; x , t ) is holomor-phic in C . By the calculations, we give the asymptotics in the neighborhood of ∞ : (cid:18) ∂∂ t k ΦΦΦ ( λ ; x , t ) (cid:19) ΦΦΦ − ( λ ; x , t ) = (cid:18) ∂∂ t k m ( λ ; x , t ) (cid:19) m − ( λ ; x , t ) + i λ k m ( λ ; x , t ) σ m − ( λ ; x , t )= (cid:16) i λ k m ( λ ; x , t ) σ m − ( λ ; x , t ) (cid:17) + = i (cid:32) k ∑ i = A i λ k − i (cid:33) .(16)By virtue of the Liouville theorem, we get the equations (15). (cid:3) he symmetric reduction for the AKNS spectral problem (5) with q ( x , t ) = r ∗ ( x , t ) can be converted intothe coefficient matrix through the following su ( ) reality condition:(17) U † ( λ ∗ ; x , t ) = − U ( λ ; x , t ) .Moreover, it can be converted into the symmetric relationship for the wave function ΦΦΦ ( λ ; x , t ) :(18) ΦΦΦ † ( λ ∗ ; x , t ) = [ ΦΦΦ ( λ ; x , t )] − .Actually the Hirota equation(1) is a mixture of second order and third order flow in the above integrablehierarchy under the reduction q ∗ ( x , t ) = r ( x , t ) , t = γ t , t = δ t , where t and t i , i ≥ ∂∂ t ΦΦΦ ( λ ; x , t ) = γ ∂∂ t ΦΦΦ ( λ ; x , t ) + δ ∂∂ t ΦΦΦ ( λ ; x , t ) = (cid:34) γ (cid:32) ∑ i = A i λ − i (cid:33) + δ (cid:32) ∑ i = A i λ − i (cid:33)(cid:35) ΦΦΦ ( λ ; x , t ) . In this section, we prepare to construct the exact high order soliton by Darboux transformation method.Last section we have derived the Hirota equation from AKNS hierarchy, the Darboux transformation forthe AKNS system with su ( ) symmetry is well known in the literature [27, 28]. The multi-fold Darbouxmatrix in the frame of loop group, the spectral parameters are different, was given in reference [29]. For thegeneral high order case, the Darboux matrix was given in the literature [32, 37] with the following theorem: Theorem 1.
Suppose we have a smooth solution q ∈ L ∞ ( R ) ∪ C ∞ ( R ) , and the matrix solution Φ ( λ ; x , t ) isholomorphic in the whole complex plane C , the Darboux transformation (20) T N ( λ ; x , t ) = I + Y N M − DY † N , M = X † SX , where Y N = (cid:104) Φ [ ] , Φ [ ] , · · · , Φ [ n − ] , Φ [ ] , Φ [ ] , · · · , Φ [ n − ] , · · · , Φ [ ] k , Φ [ ] k , · · · , Φ [ n k − ] k (cid:105) , D = D · · · D · · · ... ... ... ... · · · D k , D i = λ − λ ∗ i · · · ( λ − λ ∗ i ) λ − λ ∗ i · · · ... ... ... ( λ − λ ∗ i ) ni − ( λ − λ ∗ i ) ni − · · · λ − λ ∗ i , X = X · · · X · · · ... ... ... ... · · · X k , X i = Φ [ ] i Φ [ ] i · · · Φ [ n i − ] i Φ [ ] i · · · Φ [ n i − ] i ... ... . . . ... · · · Φ [ ] i , S = S S · · · S k S S · · · S k ... ... . . . ... S k S k · · · S kk ,(21) and Φ [ k ] i = k ! (cid:16) dd λ (cid:17) k Φ i ( λ ) | λ = λ i , Φ i ( λ i ) belongs to the one dimension linear space: span { vector solution for Lax pair at λ = λ i } , and S i , j = ( ) I λ ∗ i − λ j ( ) I ( λ ∗ i − λ j ) · · · ( n j ) I ( λ ∗ i − λ j ) nj ( ) ( − ) I ( λ ∗ i − λ j ) ( ) ( − ) I ( λ ∗ i − λ j ) · · · ( n j ) ( − ) I ( λ ∗ i − λ j ) nj + ... ... . . . ... ( n i − n i − ) ( − ) ni − I ( λ ∗ i − λ j ) ni ( n i n i − ) ( − ) ni − I λ ∗ i − λ j ) ni + · · · ( n i + n j − n i − ) ( − ) ni − I ( λ ∗ i − λ j ) ni + nj − , onverts the Lax pair (5) and (19) into a new one by replacing the potential functions (22) q [ N ] = q + Y N ,2 M − Y † N ,1 , which is the B¨acklund transformation, where the subscript Y N ,2 denotes the second row vector of Y N , and Y N ,1 denotes the first row vector of Y N . Now we proceed to construct the soliton solutions by the above B¨acklund transformation (22). We startwith a seed solution: q ( x , t ) =
0, solving the Lax pair Eq.(5) and Eq.(19) gives a vector from the span spaceof the fundamental solution:(23) Φ [ k ] i = k ! (cid:16) dd λ (cid:17) k e θ [ i ] (cid:12)(cid:12)(cid:12) λ = λ i δ k ,0 , θ [ i ] : = i λ ( x + ( γλ + δλ ) t ) + n i − ∑ j = a [ j ] i ( λ − λ i ) j − i4 π ,where δ k ,0 is the standard Chrestoffel symbol.By the B¨acklund transformation (22), the single soliton reads(24) q [ ] = λ I sech (cid:16) λ I ( x − v t ) − a [ ] (cid:17) e − ( θ ) − i2 π ,where v = − (cid:16) λ R γ − λ I δ + λ R δ (cid:17) , θ : = θ [ ] (cid:12)(cid:12) λ = λ = i λ ( x + ( γλ + δλ ) t ) + a [ ] − i4 π , λ = λ R + i λ I , Im ( θ ) = λ R x + (cid:16) λ R − λ I (cid:17) γ t + (cid:16) λ R − λ I (cid:17) λ R δ t − π in v , θ stands for v , θ [ i ] depending on λ . Clearly, the velocity of single soliton v is aquadratic function with respect to λ R and λ I , so it has a minimum v min = δ (cid:16) λ I + γ δ (cid:17) .Besides, with the aid of the fourth symmetry in Eq.(2), if q [ ] is a solution to Hirota equation (1), then˜ q [ ] ( x , t ) = λ I e (cid:101) sech (cid:16) λ I e (cid:101) ( x − ˜ v t ) − a [ ] (cid:17) e − ( ˜ θ ) − i2 π (26)is also the solution, where ˜ v = v ( λ I → λ I e (cid:101) , λ R → λ R e (cid:101) + ( e (cid:101) − ) δ γ ) ,Im ( ˜ θ ) = Im ( θ ( λ I → λ I e (cid:101) , λ R → λ R e (cid:101) + ( e (cid:101) − ) δ γ )) .Furthermore, we could derive the high order soliton with the B¨acklund transformation (22). By choosingsome special parameters, we give some examples to describe the dynamical behavior for high order soliton.Suppose N =
2, then there will appear two kinds of soliton, one is n =
2, the other is n = n =
1. Thefirst one depends on single spectral parameter λ , and these two solitons have the same amplitude andvelocity. The second one depends on two spectral parameters λ and λ , so their dynamical properties willbe different from the first one, their velocities and the amplitudes are decided by λ and λ . Moreover, wealso give some higher-order soliton under N = k spectral parameters λ , λ , · · · , λ k with order n , n , · · · , n k respectively, and the maximum amplitude is | q | [ N ] max = k ∑ i = | λ iI | n i .From the figures, if the soliton depends on the same spectral parameter λ i , they will move along withthe same direction. But if the soliton depends on different spectral parameters, their moving directions canbe changed by choosing different spectral values. Especially, if the velocity v = v , then there will appeara more interesting phenomena, two solitons will come into being a breather, four solitons will become twobreathers, which can be seen from Fig.2. It is clear that the dynamical behavior between the multi-soltionand the high order soliton is different. However, they have some common features in some aspects, suchas when t → ∞ , they both break up into individual soliton. Thus what does the long-time asymptotics IGURE Four types of solitons, (a) is second-order solion with λ = − + i, n = a [ ] = a [ ] =
0, (b) is a multi-soliton, with λ = − + i, λ = − − √ + i, n = n = a [ ] = a [ ] = λ = − + i, λ = − − √ + i, n = n = a [ ] = a [ ] = a [ ] = a [ ] =
0, (d) is also high order soliton with λ = − + i, λ = − − √ + i, λ = + i, n = n = n = a [ ] = a [ ] = a [ ] = a [ ] = F IGURE Two types of soliton with the same velocity, one is n = n =
1, the other is n = n =
2. The parameters is λ = − + i, λ = + i, δ = γ = a [ ] = a [ ] = a [ ] = a [ ] = of high-order soliton will be? How to obtain the asymptotic expression when t → ∞ ? We would like todisclose the answers in the following.Before the analysis on the high order soliton, we would like to establish the relationship between theDarboux transformation and Riemann-Hilbert method. In the literature [37, 38, 39], the authors discussthe asymptotic behavior of infinity order of rogue wave and soliton with the robust inverse scatteringmethod. The jump matrix is only related to the Darboux matrix and the dispersion terms, that is to say, theDarboux matrix can be regarded as a special M ( λ ; x , t ) matrix appeared in the RHP. Inspired by this idea, e construct two sectional analytic matrix(27) M [ N ] ( λ ; x , t ) = M [ N ]+ ( λ ; x , t ) = (cid:18) λ − λ λ − λ ∗ (cid:19) − N /2 T N ( λ ; x , t ) , M [ N ] − ( λ ; x , t ) = T N ( λ ; x , t ) Φ ( λ ; x , t ) T − N ( λ ; 0, 0 ) Φ − ( λ ; x , t ) ,which can solve the following RHP: Riemann-Hilbert Problem 1.
Let ( x , t ) ∈ R be arbitrary parameters, and let N ∈ Z ≥ . Find a × matrixfunction M [ N ] ( λ ; x , t ) satisfying the following properties • Analyticity : M [ N ] ( λ ; x , t ) is analytic for λ ∈ C \ ∂ D , and it takes continuous boundary values from theinterior and exterior of ∂ D . • Jump condition : The boundary values on the jump contour ∂ D are related as M [ N ]+ ( λ ; x , t ) = M [ N ] − ( λ ; x , t ) e θσ Q (cid:18) λ − λ λ − λ ∗ (cid:19) N σ Q − e − θσ , • Normalization : M [ N ] ( λ ; x , t ) = I + O ( λ − ) .where Q = √ (cid:20) (cid:21) , θ = i λ ( x + ( γλ + δλ ) t ) , D is a big disk centered at zero, and all spectral parameters λ i , ( i =
1, 2, · · · , k ) are involved in the disk. It needs to be emphasized that the RHP constructed here and in the following are all reflectionless, andthe Darboux matrices depend on single spectral parameter λ .Based on the method in [38, 39], we can unite the dispersion term by defining the following sectionalholomorphic matrix function(28) (cid:102) M [ N ] ( λ ; x , t ) : = M [ N ] ( λ ; x , t ) e θσ Q e − θσ , λ ∈ D M [ N ] ( λ ; x , t ) (cid:18) λ − λ ∗ λ − λ (cid:19) N σ , λ / ∈ D which solves the following RHP: Riemann-Hilbert Problem 2.
Define D ∈ C be a disk centered at the origin containing λ in its interior. Find aunique × matrix function satisfying the following properties: • Analyticity : (cid:102) M [ N ] ( λ ; x , t ) is analytic for λ ∈ C \ ∂ D , and it takes continuous boundary values from theinterior and exterior of ∂ D . • Jump condition : The boundary values on the jump contour ∂ D are related as (cid:102) M [ N ]+ ( λ ; x , t ) = (cid:102) M [ N ] − ( λ ; x , t ) e (cid:18) θ + N log (cid:18) λ − λ λ − λ ∗ (cid:19)(cid:19) σ Q − e − (cid:18) θ + N log (cid:18) λ − λ λ − λ ∗ (cid:19)(cid:19) σ , • Normalization : (cid:102) M [ N ] ( λ ; x , t ) = I + O ( λ − ) . In RHP 1, we construct two sectional analytic matrices M [ N ] ( λ ; x , t ) , and the jump matrix exist in a bigcircle and all spectral parameters are involved in the circle. The classical RHP without reflection coefficientscan be constructed by two sectional analytic matrices in and out of two small circle centered at λ and λ ∗ ,as shown in Fig.3. So we can construct another RHP with the jump in Fig.3. Lemma 1.
For the Darboux matrix T N ( λ ; x , t ) with n = N, two analytic matrices can be constructed as follows: (29) T N ( λ ; x , t ) (cid:34) (cid:16) λ − λ ∗ λ − λ (cid:17) N (cid:35) − N ∑ i = α i ( λ − λ ) i , and (30) T N ( λ ; x , t ) (cid:34) (cid:16) λ − λ ∗ λ − λ (cid:17) N (cid:35) N ∑ i = α ∗ i ( λ − λ ∗ ) i , D + − D + ∂ D + − λ ∗ D − + D − Re( λ )Im( λ ) ∂ D − + F IGURE Definition of D ± , D − + , D + − , and the contour ∂ D − + , ∂ D + − . which are analytic at the neighborhood of λ = λ , λ = λ ∗ respectively, where α i = i ( N − i ) ! ∂ N − i ( λ − λ ∗ ) N e θ ∂λ N − i (cid:12)(cid:12) λ = λ .Proof. Through the kernel condition of the Darboux matrix in theorem 1, we know that the meromorphicvector T N ( λ ; x , t ) (cid:2) e θ , i (cid:3) T can be expanded with the form O (( λ − λ ) N ) at the neighborhood of λ = λ .Furthermore, we know that(31) T N ( λ ; x , t )( λ − λ ) N (cid:34) − i (cid:0) λ − λ ∗ (cid:1) N e θ (cid:0) λ − λ ∗ (cid:1) N (cid:35) is holomorphic at the neighborhood of λ = λ . The principal part at λ = λ for the matrix T N ( λ ; x , t ) (cid:34) (cid:16) λ − λ ∗ λ − λ (cid:17) N (cid:35) can be eliminated by the analytic property of (31): i.e. T N ( λ ; x , t ) − N ∑ i = α i ( λ − λ ) N − i ( λ − λ ) N (cid:16) λ − λ ∗ λ − λ (cid:17) N ,is analytic at the neighborhood of λ = λ , where − i (cid:0) λ − λ ∗ (cid:1) N e θ = N ∑ i = − α i ( λ − λ ) N − i + O (( λ − λ ) N ) .In a similar manner, the equation (30) is analytic at the neighborhood of λ = λ ∗ can be proved. (cid:3) With the aid of lemma 1, we can construct two kinds of sectional analytic matrix as(32) (cid:99) M [ N ] ( λ ; x , t ) = (cid:99) M [ N ]+ ( λ ; x , t ) = T N ( λ ; x , t ) (cid:34) (cid:16) λ − λ ∗ λ − λ (cid:17) N (cid:35) , λ ∈ D + ∪ D − (cid:99) M [ N ] − ( λ ; x , t ) = T N ( λ ; x , t ) (cid:34) (cid:16) λ − λ ∗ λ − λ (cid:17) N (cid:35) − N ∑ i = α i ( λ − λ ) i , λ ∈ D + − (cid:99) M [ N ] − ( λ ; x , t ) = T N ( λ ; x , t ) (cid:34) (cid:16) λ − λ ∗ λ − λ (cid:17) N (cid:35) N ∑ i = α ∗ i ( λ − λ ∗ ) i , λ ∈ D − + which solves the following RHP: iemann-Hilbert Problem 3. Define two small circles D + − , D − + centered at λ = λ and λ = λ ∗ respectively. Finda unique × Matrix (cid:99) M [ N ] ( λ ; x , t ) , satisfying the following properties: • Analyticity : (cid:99) M [ N ] ( λ ; x , t ) is analytic for λ ∈ C \ (cid:0) ∂ D + − (cid:83) ∂ D − + (cid:1) , and it takes continuous boundary valuesfrom the interior and exterior of D − + and D + − . • Jump condition : The boundary values on the jump contour ∂ D + − , ∂ D − + are related as (cid:99) M [ N ]+ ( λ ; x , t ) = (cid:99) M [ N ] − ( λ ; x , t ) v + ( λ ; x , t ) , λ ∈ ∂ D + − (cid:99) M [ N ]+ ( λ ; x , t ) = (cid:99) M [ N ] − ( λ ; x , t ) v − ( λ ; x , t ) , λ ∈ ∂ D − + where v + ( λ ; x , t ) = N ∑ i = α i ( λ − λ ) i , v − ( λ ; x , t ) = N ∑ i = α ∗ i ( λ − λ ∗ ) i ,(33) • Normalization : (cid:99) M [ N ] ( λ ; x , t ) = I + O ( λ − ) . Compared with RHP 2, the jump matrix in the RHP 3 is different in essence, which is a polynomialfunction with respect to x and t . In the literature [37], Bilman, the second author of this work and Millerderived the Lax pair from the RHP 2. Here we consider how to derive the Lax pair by the RHP 3. Proposition 3.
The RHP 3 can derive the Lax pair of Hirota equation.Proof.
According to the the proposition 2, we only need to prove (cid:16)(cid:99) M + ( λ ; x , t ) e θσ (cid:17) x (cid:16)(cid:99) M + ( λ ; x , t ) e θσ (cid:17) − and (cid:16)(cid:99) M + ( λ ; x , t ) e θσ (cid:17) t (cid:16)(cid:99) M + ( λ ; x , t ) e θσ (cid:17) − are holomorphic function in the whole complex plane C . With the jump condition in RHP 3, in the region D + − , we have (cid:16)(cid:99) M [ N ]+ ( λ ; x , t ) e θσ (cid:17) x (cid:16)(cid:99) M [ N ]+ ( λ ; x , t ) e θσ (cid:17) − = (cid:16)(cid:99) M [ N ] − ( λ ; x , t ) v + ( λ ; x , t ) e θσ (cid:17) x (cid:16)(cid:99) M [ N ] − ( λ ; x , t ) v + ( λ ; x , t ) e θσ (cid:17) − = (cid:99) M [ N ] − ( λ ; x , t ) (cid:110) [ v + , x ( λ ; x , t )+ v + ( λ ; x , t )( i λσ )] v − + ( λ ; x , t ) (cid:111) (cid:16)(cid:99) M [ N ] − (cid:17) − ( λ ; x , t )+ (cid:99) M [ N ] − , x ( λ ; x , t )( (cid:99) M [ N ] − ) − ( λ ; x , t ) .(34)Furthermore, with the expression of v + ( λ ; x , t ) , we know [ v + , x ( λ ; x , t )+ v + ( λ ; x , t )( i λσ )] v − + ( λ ; x , t ) = i λ (cid:18) N ∑ i = α i ( λ − λ ) i (cid:19) x − λ (cid:18) N ∑ i = α i ( λ − λ ) i (cid:19) − i λ .(35)Based on the definition of α j in Lemma 1, we know that the (
2, 1 ) element of the matrix at the righthand side in the above equation is analytic in the neighborhood of λ = λ , which deduces that the ma-trix (cid:16)(cid:99) M [ N ]+ ( λ ; x , t ) e θσ (cid:17) x (cid:16)(cid:99) M [ N ]+ ( λ ; x , t ) e θσ (cid:17) − is analytic in the region D + − . Similarly, we can prove that (cid:16)(cid:99) M [ N ]+ ( λ ; x , t ) e θσ (cid:17) x (cid:16)(cid:99) M [ N ]+ ( λ ; x , t ) e θσ (cid:17) − is analytic in the region D − + . Therefore, we conclude that thematrix (cid:16)(cid:99) M [ N ]+ ( λ ; x , t ) e θσ (cid:17) x (cid:16)(cid:99) M [ N ]+ ( λ ; x , t ) e θσ (cid:17) − is holomorphic on the whole complex plane C .In a similar manner, we consider the t -part. By the similar property of x -part, we have (cid:104) v + , t ( λ ; x , t )+ v + ( λ ; x , t ) (cid:16) λ ( γ + δλ ) σ (cid:17)(cid:105) v − + ( λ ; x , t )= λ ( γ + δλ ) (cid:18) N ∑ i = α i ( λ − λ ) i (cid:19) t − λ ( γ + δλ ) (cid:18) N ∑ i = α i ( λ − λ ) i (cid:19) − λ ( γ + δλ ) ,(36) hich implies that (cid:16)(cid:99) M + ( λ ; x , t ) e θσ (cid:17) t (cid:16)(cid:99) M + ( λ ; x , t ) e θσ (cid:17) − is analytic in the region D + − . With the samemethod, we can also prove that it is analytic in the region D − + . Then the Lax pair of Hirota equation can bederived by the proposition 2. (cid:3) The Riemann-Hilbert representation for the Darboux matrix will be useful in the analysis to the largeorder soliton [38, 39]. The further analysis on the large order soliton in different region will be anothertopic, whose difficulty is that the phase term θ is a cubic polynomial with respect to λ .Although we give two kinds of RHP of high order soliton, we intend to analyze the long-time asymp-totics from the solutions determinant directly. Observing the Fig.1, when t → ∞ , Fig.1 (a) and (b) has twoindividual solitons, and (c) and (d) has four individual solitons. We hope to get the asymptotic individualsolitons with the form of single soliton. That is, when t → ∞ , if there are N individual solitons, we shouldconstruct N single solitons as the asymptotic expression for N -th order soliton. Last section, we exhibit some graphics on the high order soltion. Now we are going to proceed to analyzethe long-time asymptotic behavior about the high order soliton. From the figures, we know the N -th ordersoliton can break up into N individual solitons when t → ∞ . So how to find the N individual solitons tomatch the N -th order soliton? In order to tackle with this problem, we first extract the leading order termfrom the high order soliton determinant (22), which contains polynomial function and exponent function.If both two kinds of function have the same degree t , then the long-time asymptotics can be expressed by asingle soliton. Otherwise, it is a vanishing solution. Before the detailed analysis, we give some Lemmas: Lemma 2.
Denote ϑ = − [ x + λ ( γ + δλ ) t ] + a [ ] , ϑ = − ( γ + δλ ) t − a [ ] , ϑ = δ t − a [ ] ,then we have the following expansion for the function (37) exp ( (cid:101) ( ϑ + (cid:101)ϑ + (cid:101) ϑ )) = + ∞ ∑ i = f l (cid:101) l , where (38) f l = [ l /3 ] ∑ m = [ l − m /2 ] ∑ k = max ( l − m ,0 ) ϑ k k ! ϑ l − k − m ( l − k − m ) ! ϑ k − l + m ( k − l + m ) ! . Proof.
By the formula of combination, we have ∞ ∑ i = (cid:101) i i ! (cid:16) ϑ + (cid:101)ϑ + (cid:101) ϑ (cid:17) i = ∞ ∑ i = i ∑ m = k + k = m , k s ≥ s = ∑ k + k + k = i ϑ k k ! ϑ k k ! ϑ k k ! (cid:101) m + i = ∞ ∑ i = i ∑ m = [ i − m /2 ] ∑ k = i − m ϑ k k ! ϑ i − m − k ( i − m − k ) ! ϑ k + m − i ( k − i + m ) ! (cid:101) m + i = ∞ ∑ l = [ l /3 ] ∑ m = [ l − m /2 ] ∑ k = max ( l − m ,0 ) ϑ k k ! ϑ l − k − m ( l − k − m ) ! ϑ k − l + m ( k − l + m ) ! (cid:101) l (39)which deduces the equation (37). (cid:3) We apply the zero seed solutions to the B¨acklund transformation (22). With the aid of Lemma 2, weobtain the following determinant representation of high order soliton under single spectral parameter λ :(40) q [ N ] = G det M here M : = F †1 B F + B , G : = (cid:20) M F †1,1 e (cid:21) , e : = [
1, 0, 0, · · · , 0 ] , B : = (cid:32) λ − λ ∗ (cid:18) i + j − i − (cid:19) (cid:18) i λ ∗ − λ (cid:19) i + j − (cid:33) ≤ i , j ≤ N , F : = e θ f · · · f N − · · · f N − ... ... ... ...0 0 · · · = e θ (cid:32) I + N − ∑ i = f i E iN (cid:33) , E N : = (cid:0) δ i , j − (cid:1) ≤ i , j ≤ N ,(41)and the subscripts in F and B stand for the matrices depending on the spectral parameter λ , F represents the first row of matrix F , f l is given in equation (38). To analyze the asymptotic behavior ofthe high order single-pole soliton, we rewrite the relative matrices B of the determinant solutions:(42) B = D P N D λ − λ ∗ = D S † N S N D λ − λ ∗ ,where P N = (cid:16) ( i + j − i − ) (cid:17) ≤ i , j ≤ N stands for the N -th order Pascal matrix which can be performed the LU decomposition P N = S † N S N , and D = diag (cid:32) (cid:18) − λ I (cid:19) , · · · , (cid:18) − λ I (cid:19) N − (cid:33) , S N = (cid:18)(cid:18) j − i − (cid:19)(cid:19) ≤ i ≤ j ≤ N .Due to the Lemma 2, the leading polynomial of f l ≥ is f j = ϑ j j ! + lower order terms (l.o.t) of x and t . Lemma 3. (43) Γ j : = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ! 1 ( j + ) ! · · · ( N − ) !1 ( j − ) ! 1 j ! · · · ( N − ) ! ... ... ... ... ( j − N + ) ! · · · · · · j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) · · · ( N − − j ) ! j ! ( j + ) ! · · · ( N − ) ! (cid:19) roof. This determinant can be calculated by subtracting the i − i -th row subsequently,that is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ! 1 ( j + ) ! · · · ( N − ) !1 ( j − ) ! 1 j ! · · · ( N − ) ! ... ... ... ... ( j − N + ) ! · · · · · · j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ! N − − j ( N − ) ( j + ) ! N − − ( j + )( N − ) · · · ( N − ) ! ( j − ) ! N − − j ( N − ) j ! N − − j ( N − ) · · · ( N − ) ! ( j − N + ) ! N − − jj + ( j − N + ) ! N − − jj + · · · ( j + ) ! ∗ ∗ ∗ ∗ j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( N − − j ) ! ( N − ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ! 1 ( j + ) ! · · · ( N − ) !1 ( j − ) ! 1 j ! · · · ( N − ) ! ... ... ... ... ( j − N + ) ! 1 ( j − N + ) ! · · · j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( N − − j ) ! ( N − − j ) ! ( N − ) ! ( N − ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ! 1 ( j + ) ! · · · ( N − ) !1 ( j − ) ! 1 j ! · · · ( N − ) ! ... ... ... ... ( j − N + ) ! 1 ( j − N + ) ! · · · j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) · · · ( N − − j ) ! j ! ( j + ) ! · · · ( N − ) ! (cid:19) .(44) (cid:3) Through the expression (40), we find the leading order term of det ( M ) can be expanded as the linearcombinations of exp ( j ( θ + θ ∗ )) and det ( G ) can be expanded as the linear combinations of exp ( j ( θ + θ ∗ ) + θ ∗ ) . The coefficients of exp ( j ( θ + θ ∗ )) and exp ( j ( θ + θ ∗ ) + θ ∗ ) are polynomial function withrespect to x and t . In order to study the asymptotic behavior, we need to give the leading order terms ofdet ( M ) and det ( G ) . Lemma 4.
The denominator of q [ N ] : det ( F †1 B F + B ) can be represented as det ( F †1 B F + B ) = N − ∑ j = e ( N − j )( θ + θ ∗ ) A j ( x , t ) + A ( x , t ) + l.o.t(45) where (46) A j ( x , t ) = ( − i ) N ( λ I ) − ( N − j ) − j Γ j | ϑ | j ( N − j ) . Proof.
Through the formula (42) and the determinant formulas det ( KL ) = det (cid:18) K − L I (cid:19) , the determinantof matrix F †1 B F + B can be rewritten as | F †1 B F + B | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − i2 λ I (cid:19) (cid:16) F †1 D S † N , D S † N (cid:17) (cid:20) S N D F S N D (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) − i2 λ I (cid:19) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F †1 D S † N D S † N − S N D F I N − S N D I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) − i2 λ I (cid:19) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F †1 D + l.o.t D S † N − D F + l.o.t I N − S N D I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,(47)since the matrix S N D F = D F + l.o.t and the leading order term of every entry f j in F is ϑ j j ! . Wecalculate this determinant by the general Laplace expansion method. In this case, we choose the first N olumn, the determinant has ( NN ) terms. The leading order term is a special choice, which starts from N + N + j row and ends up with 2 N + N − j row, where ( j = · · · , N ) . It needs tobe emphasized that j = N + N row, and j = N indicates N + N row. Asto the complementary cofactor matrix, we continue to calculate it with the Laplace expansion method. Forsimplicity, we choose the first N row, and the determinant can be given easily. Next, we give a detailedcalculation about the leading order coefficient for e ( N − j )( θ ∗ + θ ) . Following the given rule, the coefficient is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − ( ϑ ) · · · − ϑ j − ( j − ) ! − ϑ j j ! · · · − ϑ N − ( N − ) ! − (cid:16) − λ I (cid:17) · · · − (cid:16) − λ I (cid:17) ϑ j − ( j − ) ! − (cid:16) − λ I (cid:17) ϑ j − ( j − ) ! · · · − (cid:16) − λ I (cid:17) ϑ N − ( N − ) ! ... ... ... ... ... ... ...0 0 · · · − (cid:16) − λ I (cid:17) N − j − ϑ j − N ( j − N ) ! − (cid:16) − λ I (cid:17) N − j − ϑ j − N + ( j − N + ) ! · · · − (cid:16) − λ I (cid:17) N − j − ϑ j j ! − ( ) (cid:16) − λ I (cid:17) − ( ) (cid:16) − λ I (cid:17) · · · − ( j − ) (cid:16) − λ I (cid:17) j − − ( j ) (cid:16) − λ I (cid:17) j · · · − ( N − ) (cid:16) − λ I (cid:17) N − − ( ) (cid:16) − λ I (cid:17) · · · − ( j − ) (cid:16) − λ I (cid:17) j − − ( j ) (cid:16) − λ I (cid:17) j · · · − ( N − ) (cid:16) − λ I (cid:17) N − ... ... ... ... ... ... ...0 0 · · · − ( j − j − ) (cid:16) − λ I (cid:17) j − − ( jj − ) (cid:16) − λ I (cid:17) j · · · − ( N − j − ) (cid:16) − λ I (cid:17) N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · ( ) (cid:16) − λ I (cid:17) · · · ϑ ∗ − λ I · · · ( ) (cid:16) − λ I (cid:17) ( ) (cid:16) − λ I (cid:17) · · · ϑ ∗ , j − ( j − ) ! − λ I ϑ ∗ , j − ( j − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ ,2 j − N ( j − N ) ! ( j − ) (cid:16) − λ I (cid:17) j − ( j − ) (cid:16) − λ I (cid:17) j − · · · ( j − j − ) (cid:16) − λ I (cid:17) j − ϑ ∗ , j j ! − λ I ϑ ∗ , j − ( j − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ ,2 j − n + ( j − N + ) ! ( j ) (cid:16) − λ I (cid:17) j ( j ) (cid:16) − λ I (cid:17) j − · · · ( jj − ) (cid:16) − λ I (cid:17) j ... ... ... ... ... ... ... ... ϑ ∗ , N − ( N − ) ! − λ I ϑ ∗ , N − ( N − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ , j j ! ( n − ) (cid:16) − λ I (cid:17) N − ( n − ) (cid:16) − λ I (cid:17) N − · · · ( N − j − ) (cid:16) − λ I (cid:17) N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · ( − ) j + N + N (cid:18) − i2 λ I (cid:19) N + l.o.t(48) Furthermore, we still use the Laplace expansion method to calculate both two determinants in Eq. (48). Theleading order term in the first determinant is choosing 1 to j column, N − j + N row, as the dottedline in the determinant. The second determinant can be given similarly, so Eq. (48) becomes (cid:18) − i2 λ I (cid:19) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ) (cid:16) − λ I (cid:17) ( ) (cid:16) − λ I (cid:17) · · · ( j − ) (cid:16) − λ I (cid:17) j − ( ) (cid:16) − λ I (cid:17) · · · ( j − ) (cid:16) − λ I (cid:17) j − ... ... ... ...0 0 · · · ( j − j − ) (cid:16) − λ I (cid:17) j − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ j j ! · · · ϑ N − ( N − ) ! − λ I ϑ j − ( j − ) ! · · · − λ I ϑ N − ( N − ) ! ... ... ... (cid:16) − λ I (cid:17) N − j − ϑ j − N + ( j − N + ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ j j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ) (cid:16) − λ I (cid:17) · · · ( ) (cid:16) − λ I (cid:17) ( ) (cid:16) − λ I (cid:17) · · · ( j − ) (cid:16) − λ I (cid:17) j − ( j − ) (cid:16) − λ I (cid:17) j − · · · ( j − j − ) (cid:16) − λ I (cid:17) j − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ ∗ , j j ! − λ I ϑ ∗ , j − ( j − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ ,2 j − N + ( j − N + ) ! ... ... ... ... ϑ ∗ , N − ( N − ) ! − λ I ϑ ∗ , N − ( N − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ , j j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + l.o.t = (cid:32)(cid:18) − λ I (cid:19) j ( j − )+( N − j )( N − j − ) (cid:18) − i2 λ I (cid:19) N | ϑ | j ( N − j ) (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ! 1 ( j + ) ! · · · ( N − ) !1 ( j − ) ! 1 j ! · · · ( N − ) ! ... ... ... ... ( j − N + ) ! · · · · · · j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + l.o.t.(49) With Lemma 3, we get Eq.(45). It completes the proof. (cid:3)
Lemma 5.
The numerator of q [ N ] can be written as det G = N − ∑ j = e θ ∗ e ( N − j − )( θ + θ ∗ ) B j ( x , t ) + l.o.t(50) here (51) B j ( x , t ) = ( − ) N − j ( − i ) N − ( ϑ ) ( N − j − )( j + ) ( ϑ ∗ ) j ( N − j ) ( λ I ) − ( N − j − ) − j − N + Γ j Γ j + Proof.
The determinant of G can be written asdet G = − ( λ − λ ∗ ) e − θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( θ + θ ∗ ) λ − λ ∗ D D · · · D N e ( θ + θ ∗ ) λ − λ ∗ f ∗ D D · · · D N ... ... ... ... ... e ( θ + θ ∗ ) λ − λ ∗ f ∗ N − D N D N · · · D NN (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − ( λ − λ ∗ ) e − θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ − λ ∗ (cid:16) F †1 D S †1 , D S †1 (cid:17) (cid:34) ( S D F ) [ ] + e θ ( S − †1 ) ( S D ) [ ] (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − ( λ − λ ∗ ) e − θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ − λ ∗ (cid:16) F †1 D + l.o.t, D S †1 (cid:17) (cid:34) ( D F ) [ ] + e θ ( S − †1 ) + l.o.t ( S D ) [ ] (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − (cid:18) − i2 λ I (cid:19) N − e − θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F †1 D + l.o.t D S † N ( D F ) [ ] + e θ ( S − †1 ) + l.o.t I N − ( S D ) [ ] I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (52)where ( · ) [ ] represents the matrix · by replacing elements of the first column with zeros and the symbol ( · ) denotes taking the first column of the matrix ( · ) and setting the other elements as zeros. Similar tothe analysis in the denominator, we continue to calculate this determinant with the Laplace expansionmethod. According to the method in the denominator, we give a detailed calculation about the leadingorder coefficient of e θ ∗ e ( N − j − )( θ + θ ∗ ) , which is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − ϑ · · · − ϑ j j ! − ϑ j + ( j + ) ! · · · − ϑ N − ( N − ) ! λ I · · · λ I ϑ j − ( j − ) ! 12 λ I ϑ j ( j ) ! · · · λ I ϑ N − ( N − ) ! ... ... ... ... ... ... ... ( − ) N − j · · · − (cid:16) − λ I (cid:17) N − j − ϑ j + − N ( j + − N ) ! − (cid:16) − λ I (cid:17) N − j − ϑ j − N + ( j − N + ) ! · · · − (cid:16) − λ I (cid:17) N − j − ϑ j j ! − ( ) (cid:16) − λ I (cid:17) · · · − ( j ) (cid:16) − λ I (cid:17) j − ( j + ) (cid:16) − λ I (cid:17) j + · · · − ( N − ) (cid:16) − λ I (cid:17) N − − ( ) (cid:16) − λ I (cid:17) · · · − ( j ) (cid:16) − λ I (cid:17) j − ( j + ) (cid:16) − λ I (cid:17) j + · · · − ( N − ) (cid:16) − λ I (cid:17) N − ... ... ... ... ... ... ...0 0 · · · − ( jj − ) (cid:16) − λ I (cid:17) j − ( j + j − ) (cid:16) − λ I (cid:17) j + · · · − ( n − j − ) (cid:16) − λ I (cid:17) N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · ( ) (cid:16) − λ I (cid:17) · · · ϑ ∗ − λ I · · · ( ) (cid:16) − λ I (cid:17) ( ) (cid:16) − λ I (cid:17) · · · ϑ ∗ , j − ( j − ) ! − λ I ϑ ∗ , j − ( j − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ ,2 j − n ( j − N ) ! ( j − ) (cid:16) − λ I (cid:17) j − ( j − ) (cid:16) − λ I (cid:17) j − · · · ( j − j − ) (cid:16) − λ I (cid:17) j − ϑ ∗ , j j ! − λ I ϑ ∗ , j − ( j − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ ,2 j − N + ( j − N + ) ! ( j ) (cid:16) − λ I (cid:17) j ( j ) (cid:16) − λ I (cid:17) j − · · · ( jj − ) (cid:16) − λ I (cid:17) j ... ... ... ... ... ... ... ... ϑ ∗ , N − ( N − ) ! − λ I ϑ ∗ , N − ( N − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ , j j ! ( N − ) (cid:16) − λ I (cid:17) N − ( n − ) (cid:16) − λ I (cid:17) N − · · · ( N − j − ) (cid:16) − λ I (cid:17) N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · ( − ) j + N + N + (cid:18) − i2 λ I (cid:19) N − + l.o.t(53) The leading order term in the second determinant in Eq.(53) can be calculated with the same method inEq.(48), while the first determinant has a little difference, whose leading order term can be given with a ifferent choice, as shown with the dotted line. With the Lemma 3, Eq.(53) becomes ( − ) N (cid:18) − i2 λ I (cid:19) N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( − ) N − j + · · · (cid:16) − λ I (cid:17) N − j − ϑ j + − N ( j + − N ) ! − ( ) λ I · · · ( j ) (cid:16) − λ I (cid:17) j − ( ) λ I · · · ( j ) (cid:16) − λ I (cid:17) j ... ... ... ...0 0 · · · ( jj − ) (cid:16) − λ I (cid:17) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ j + ( j + ) ! · · · ϑ N − ( N − ) ! − λ I ϑ j j ! · · · − λ I ϑ N − ( N − ) ! ... ... ... (cid:16) − λ I (cid:17) N − j − ϑ j − N + ( j − N + ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ j + ( j + ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ) (cid:16) − λ I (cid:17) · · · ( ) (cid:16) − λ I (cid:17) ( ) (cid:16) − λ I (cid:17) · · · ( j − ) (cid:16) − λ I (cid:17) j − ( j − ) (cid:16) − λ I (cid:17) j − · · · ( j − j − ) (cid:16) − λ I (cid:17) j − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ ∗ , j j ! − λ I ϑ ∗ , j − ( j − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ ,2 j − n + ( j − n + ) ! ... ... ... ... ϑ ∗ , N − ( N − ) ! − λ I ϑ ∗ , N − ( N − ) ! · · · (cid:16) − λ I (cid:17) N − j − ϑ ∗ , j j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + l.o.t = ( − ) j + ( ϑ ) ( N − j − )( j + ) ( ϑ ∗ ) j ( N − j ) (cid:18) − λ I (cid:19) j +( N − j − ) (cid:18) − i2 λ I (cid:19) N − Γ j Γ j + + l.o.t(54) which is the leading order coefficient of e θ ∗ e ( N − j − )( θ + θ ∗ ) . It completes this proof. (cid:3) Based on the leading order term of the numerator and the denominator, we begin to analyze the long-time asymptotics of the soliton.
Lemma 6.
The high order soliton q [ N ] has non-zero limit along the following characteristic curves:x = s + v t + N + − κ λ I log ( t ) , ( κ =
1, 2, · · · , N ) , as t → ± ∞ . Otherwise, it has the vanishing limitation.Proof. Obviously, the leading order term of the denominator and the numerator contain polynomial func-tion and exponential function. In order to find the asymptotic behavior when t → ∞ , we should match the t degree in both two functions. Now we choose any two terms in the denominator to show how to decidethis characteristic curve: e ( N − j i )( θ + θ ∗ ) A j i , ( i =
1, 2 ) .(55)Suppose the characteristic curve is(56) x = s + v t + µ log ( t ) .If the high order soliton moves along this characteristic curve, Eq.(55) becomes | C | j i ( N − j i ) (cid:16) ( − i ) N ( λ I ) − ( N − j i ) − j i (cid:17) t ( N − j i )( j i − λ I µ ) e − λ I ( N − j i ) s + O (cid:16) t ( N − j i )( j i − λ I µ ) − log ( t ) (cid:17) ,(57)where C = δλ I + γλ I + δλ R λ I .If both two terms in Eq. (57) can be matched well, they should have the same power of t , that is2 ( N − j )( j − λ I µ ) = ( N − j )( j − λ I µ ) ,which deduces(58) µ = j + j − N λ I .Moreover, we need to determine the constraint that j and j should satisfy. To achieve this aim, wediscuss the leading order term of the denominator det ( M ) and numerator det ( G ) along this characteristic urves with given µ in Eq.(58). The power of t becomes t − (cid:16) j − j + j (cid:17) + ( j + j − N ) , j =
0, 1, · · · , N − t − (cid:16) j − j + j − (cid:17) − ( j + j − N + )( j + j − N − ) ,(59)where the first formula in Eq.(59) is the denominator term and the second formula is the numerator term.It can be seen the power of t is a quadratic function with respect to j . Obviously, the power of t in thedenominator can reach its maximum ( j + j − N ) at j = j + j , and the numerator can reach its maximum ( j + j − N ) − at j = j + j − . If j + j is an integer, then the asymptotic behavior of q [ N ] along this curveEq.(56) is vanishing as t → ∞ because the power of t in the denominator is higher than the one in thenumerator. Otherwise, j + j − is an integer. In this case, if both terms with j = j and j = j become theleading order simultaneously, j and j must satisfy | j − j | =
1. Under this condition, the asymptoticbehavior of q [ N ] will tend to the single soliton, so we have µ = j − − N λ I , ( j =
1, 2, · · · , N ) , we rewrite j = κ . It completes this proof. (cid:3) From Lemma 6, we get a conclusion that if the high order soliton q [ N ] has non-zero limit, then the highorder soliton must move along a specific characteristic curves, and the matching terms must be two adjacentterms. Now we give an example to verify this fact. Example 1.
When N = λ = i, a [ ] = a [ ] = , the second-order soliton is (60) q [ ] = (cid:16) B ( x , t ) + i (cid:17) e − x + δ t + γ t + i2 π + (cid:16) B ( x , t ) + i (cid:17) e − x + δ t + γ t + i2 π A ( x , t ) e − x + δ t + (cid:16) A ( x , t ) − (cid:17) e − x + δ t + A ( x , t ) , where B ( x , t ) = i x − γ t − δ t , B ( x , t ) = − i x − γ t + δ t , A ( x , t ) = −
116 , A ( x , t ) = − (cid:16) x − δ tx + δ t + γ t (cid:17) . As given in the lemma 6, taking N = κ = and N = κ = , then the characteristic curves arex = s + δ t −
12 log ( t ) , x = s + δ t +
12 log ( t ) .(61) Along the first curve, q [ ] becomesq [ ] = (cid:2) ( i γ − δ ) t + O ( t log ( t )) (cid:3) e − s + γ t + (cid:2) ( i γ + δ ) t + O ( t log ( t )) (cid:3) e − s + γ t t e − s + [ ( γ + δ ) t + O ( t log ( t ))] e − s + = γ t + i arg ( − γ δ ) sech (cid:18) (cid:18) x − δ t +
12 log ( t ) (cid:19) + ( ) +
12 log ( δ + γ ) (cid:19) + O ( log ( t ) / t ) .(62) Along the second curve, q [ ] becomesq [ ] = (cid:2) ( i γ − δ ) t − + O ( t − log ( t )) (cid:3) e − s + γ t + (cid:2) ( i γ + δ ) + O ( t − log ( t )) (cid:3) e − s + γ t t − e − s + [ ( γ + δ ) + O ( t − log ( t ))] e − s + = γ t + i arg ( γ δ ) sech (cid:18) (cid:18) x − δ t −
12 log ( t ) (cid:19) − ( ) −
12 log ( δ + γ ) (cid:19) + O ( log ( t ) / t ) .(63) herefore, if the high order soliton moves along these two characteristic curves, the long-time asymptotics of q [ ] isq [ ] = γ t + i arg ( − γ δ ) sech (cid:18) (cid:18) x − δ t +
12 log ( t ) (cid:19) + ( ) +
12 log ( δ + γ ) (cid:19) + γ t + i arg ( γ δ ) sech (cid:18) (cid:18) x − δ t −
12 log ( t ) (cid:19) − ( ) −
12 log ( δ + γ ) (cid:19) + O ( log ( t ) / t ) .(64)With the Lemma 6 and the Example 1, we give a theorem about the asymptotic expression of the N -thorder soliton with single spectral parameter λ . Theorem 2.
As t → ± ∞ , the long-time asymptotics of the high order soliton q [ N ] can be represented asq [ N ] ± = [ N + ] ∑ κ = λ I sech (cid:16) λ I s l , ± ± ∆ [ κ ] l − a [ ] (cid:17) e − ( θ ) ± i N + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i π ( + κ )+ [ N ] ∑ κ = λ I sech (cid:16) λ I s r , ± ∓ ∆ [ κ ] r − a [ ] (cid:17) e − ( θ ) ∓ i N + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i π ( N − + κ ) + O ( log ( t ) / t ) ,(65) where the subscript ± stands for t → ± ∞ ands i , l , ± = x ∓ (cid:18) N + − κ λ iI (cid:19) log ( | t | ) − v i t , s i , r , ± = x ± (cid:18) N + − κ λ iI (cid:19) log ( | t | ) − v i t , ∆ [ κ ] l = log (cid:18) ( N − κ ) ! ( κ − ) ! (cid:19) − ( N + − κ ) ( log ( λ I ) + log ( | C | )) , ∆ [ κ ] r = log (cid:18) ( N − κ ) ! ( κ − ) ! (cid:19) − ( N + − k ) ( log ( λ I ) + log ( | C | )) . and the subscript l , r means the the left characteristic curves or the right one.Proof. With Lemma 6, if the high order soliton moves along a specific curve(66) x = s l , + + (cid:18) N + − κ λ I (cid:19) log ( t ) + v t ,for every κ =
1, 2, · · · , (cid:104) N + (cid:105) , then the high order soliton q [ N ] becomes q [ N ] = ( − ) κ λ I C N + − κ e λ I s l , + − a [ ] − ( θ ) t κ ( κ − ) + O ( t κ ( κ − ) − log ( t )) (cid:104) e λ I s l , + − a [ ] ( λ I ) κ − N − (cid:16) ( N − κ ) ! ( κ − ) ! (cid:17) + | C | N + − κ ( λ I ) N + − κ (cid:16) ( κ − ) ! ( N − κ ) ! (cid:17)(cid:105) t κ ( κ − ) + O ( t κ ( κ − ) − log ( t ))= λ I sech (cid:16) λ I s l , + + ∆ [ κ ] l − a [ ] (cid:17) e − ( θ )+ i N + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i π ( + κ ) + O ( log ( t ) / t ) .(67) Similarly, along the other characteristic curves x = s r , + − (cid:18) N + − κ λ I (cid:19) log ( t ) + v t , (cid:18) κ =
1, 2, · · · , (cid:20) N (cid:21)(cid:19) ,the asymptotic expression is q [ N ] = i ( − ) N − + κ λ I ( C ∗ ) N + − κ e λ I s r , + − a [ ] − ( θ ) t ( κ − N )( κ − N − ) + O ( t ( κ − N )( κ − N − ) − log ( t )) (cid:104) ( κ − ) ! ( N − κ ) ! e λ I s r , + − a [ ] | C | N + − κ ( λ I ) N + − κ + ( N − κ ) ! ( κ − ) ! ( λ I ) κ − N − (cid:105) t ( κ − N )( κ − N − ) + O ( t ( κ − N )( κ − N − ) − log ( t ))= λ I sech (cid:16) λ I s r , + − ∆ [ κ ] r − a [ ] (cid:17) e − ( θ ) − i N + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i π ( N − + κ ) + O ( log ( t ) / t ) .(68) f the high order soliton moves along the other curves x = s + vt + β log ( t ) with v (cid:54) = v or β (cid:54) = N + − κ λ I ,then its asymptotic solution is(69) q [ N ] = O ( e − λ I | v − v | t ) ,and(70) q [ N ] = O ( t − ) ,respectively, both of which are vanishing when t → ∞ . Hence, the global long-time asymptotics of q [ N ] as t → + ∞ is q [ N ] = [ N + ] ∑ κ = λ I sech (cid:16) λ I s l , + + ∆ [ κ ] l − a [ ] (cid:17) e − ( θ )+ i N + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i π ( + κ )+ [ N ] ∑ κ = λ I sech (cid:16) λ I s r , + − ∆ [ κ ] r − a [ ] (cid:17) e − ( θ ) − i N + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i π ( N − + κ ) + O ( log ( t ) / t ) .(71)When t → − ∞ , its asymptotic expression can be given in a similar manner. (cid:3) Next, we will give several examples to verify the above asymptotic expressions by numerical plotting.
Example 2.
Firstly, we give the even-th order figures by choosing λ = i, δ = γ = a [ j ] = ( j =
0, 1, · · · , 7 ) .With the simple calculation, we konw the velocity is v = If we use a simple transformation ξ = x − t, then thevelocity is zero on the ( ξ , t ) -plane. Secondly, we give the odd-th order figures by choosing the same parameters: F IGURE The upper is the even-th order soliton, (a) is second-order soliton, (b) is fourth-ordersoliton, (c) is sixth-order soliton. The below is the comparison between the exact solution and thenumerical result.
In the above figures, we choose a [ j ] = ( j =
0, 1, · · · , 7 ) to attain the maximal value at ( x , t ) = (
0, 0 ) .Actually, the peak of N -order soliton can be calculated with the B¨acklund transformation (22). Proposition 4.
When x = t = a [ j ] = ( j =
0, 1, · · · , N − ) , the high order solution q [ N ] has the maximumamplitude | q [ N ] (
0, 0 ) | = | λ I | N. IGURE The upper is the odd-th order solidon, (a) is third-order soliton, (b) is fifth-order soli-ton, (c) is seventh-order soliton. The below is the comparison between the exact solution and thenumerical result.
Proof.
When x = t = F = I , so the determinant det ( M ) = det ( B ) . With the decomposition Eq.(42),it is obvious that(72) det ( B ) = ( − i ) N N ( − N ) ( λ I ) − N .Moreover, with the Laplace expansion, we havedet G = ( − i ) N ( λ I ) − N N ( − N ) N ,(73)so (cid:12)(cid:12)(cid:12) q [ N ] (cid:12)(cid:12)(cid:12) (
0, 0 ) = | λ I | N . (cid:3) After giving the long-time asymptotics to the high order soliton with single spectral parameter λ , weintend to study the asymptotics to the high order soliton with multiple spectral parameters. For the simplic-ity, we first discuss the asymptotic expression with two spectral parameters. To avoid the tedious analysison the determinants, we would like to utilize the iterated Darboux matrix to analyze the dynamics. Thenwe give the following lemma Lemma 7.
Suppose there are two spectral parameters λ and λ , we can rewrite the Darboux matrices in Eq. (20) tothe following form (74) T N ( λ ; x , t ) = T [ n ] T [ n − ] · · · T [ ] T [ n ] T [ n − ] · · · T [ ] , or (75) T N ( λ ; x , t ) = (cid:101) T [ n ] (cid:101) T [ n − ] · · · (cid:101) T [ ] (cid:101) T [ n ] (cid:101) T [ n − ] · · · (cid:101) T [ ] , here T [ j ] i = (cid:18) I − λ i − λ ∗ i λ − λ ∗ i P [ j ] i (cid:19) , ( i =
1, 2; j = · · · , n i ) , P [ j ] i = Φ [ j − ] i (cid:16) Φ [ j − ] i (cid:17) † (cid:16) Φ [ j − ] i (cid:17) † Φ [ j − ] i , Φ [ j ] = lim (cid:101) → (cid:16) T [ j ] T [ j − ] · · · T [ ] (cid:17) Φ [ ] (cid:101) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ = λ + (cid:101) , T [ ] i = I , Φ [ j ] = lim (cid:101) → (cid:16) T [ j ] T [ j − ] · · · T [ ] T [ n ] T [ n − ] · · · T [ ] (cid:17) Φ [ ] (cid:101) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ = λ + (cid:101) , (cid:101) T [ j ] i = (cid:18) I − λ i − λ ∗ i λ − λ ∗ i (cid:101) P [ j ] i (cid:19) , ( i =
1, 2; j = · · · , n i ) , (cid:101) P [ j ] i = (cid:101) Φ [ j − ] i (cid:16) (cid:101) Φ [ j − ] i (cid:17) † (cid:16) (cid:101) Φ [ j − ] i (cid:17) † (cid:101) Φ [ j − ] i , (cid:101) Φ [ j ] = lim (cid:101) → (cid:16)(cid:101) T [ j ] (cid:101) T [ j − ] · · · (cid:101) T [ ] (cid:17) Φ [ ] (cid:101) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ = λ + (cid:101) , (cid:101) T [ ] i = I , (cid:101) Φ [ j ] = lim (cid:101) → (cid:16)(cid:101) T [ j ] (cid:101) T [ j − ] · · · (cid:101) T [ ] (cid:101) T [ n ] (cid:101) T [ n − ] · · · (cid:101) T [ ] (cid:17) Φ [ ] (cid:101) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ = λ + (cid:101) .(76) If the soliton moves along the trajectory ξ = x − vt with v < v , the asymptotics of the Darboux matrices Eq. (74) can be represented as (77) T [ n ] T [ n − ] · · · T [ ] = (cid:34)(cid:16) λ − λ λ − λ ∗ (cid:17) n
00 1 (cid:35) + O (cid:16) t n − e − λ I | v − v | t (cid:17) , t → + ∞ , (cid:34) (cid:16) λ − λ λ − λ ∗ (cid:17) n (cid:35) + O (cid:16) | t | n − e λ I | v − v | t (cid:17) , t → − ∞ . Similarly, if the soliton moves along the trajectory ξ = x − vt with v > v , then the asymptotics of the Darbouxmatrices in Eq. (75) is (78) (cid:101) T [ n ] (cid:101) T [ n − ] · · · (cid:101) T [ ] = (cid:34) (cid:16) λ − λ λ − λ ∗ (cid:17) n (cid:35) + O (cid:16) t n − e − λ I | v − v | t (cid:17) , t → + ∞ , (cid:34)(cid:16) λ − λ λ − λ ∗ (cid:17) n
00 1 (cid:35) + O (cid:16) | t | n − e λ I | v − v | t (cid:17) , t → − ∞ . Proof.
We just need to consider one peculiar case. The other cases can be considered similarly. Based on theidea[40], when t → + ∞ , if the soliton moves along the trajectory ξ = x − vt with v < v , λ I >
0, followingEq.(76), we have Φ [ ] = (cid:20) − θ (cid:21) = (cid:20) (cid:21) + O ( e − λ I | v − v | t ) , t → + ∞ ,(79)where θ : = θ [ ] (cid:12)(cid:12) λ = λ = i λ ( x + ( γλ + δλ ) t ) + a [ ] − i4 π , λ = λ R + i λ I . Correspondingly, the asymp-totics for P [ ] and T [ ] can be represented as P [ ] = Φ [ ] (cid:16) Φ [ ] (cid:17) † (cid:16) Φ [ ] (cid:17) † Φ [ ] = (cid:20) (cid:21) + O ( e − λ I | v − v | t ) , t → + ∞ , T [ ] = (cid:18) I − λ − λ ∗ λ − λ ∗ P [ ] (cid:19) = (cid:34)(cid:16) λ − λ λ − λ ∗ (cid:17)
00 1 (cid:35) + O ( e − λ I | v − v | t ) , t → + ∞ .(80) urthermore, with Eq.(76), Eq.(79) and Eq.(80), we have Φ [ ] = (cid:16) I − P [ ] (cid:17) (cid:34) dd λ e − θ [ ] (cid:12)(cid:12)(cid:12) λ = λ (cid:35) + P [ ] λ − λ ∗ (cid:20) − θ (cid:21) = λ − λ ∗ (cid:20) (cid:21) + O (cid:16) t e − λ I | v − v | t (cid:17) , t → + ∞ , P [ ] = Φ [ ] (cid:16) Φ [ ] (cid:17) † (cid:16) Φ [ ] (cid:17) † Φ [ ] = (cid:20) (cid:21) + O (cid:104) t e − λ I | v − v | t (cid:105) , t → + ∞ , T [ ] = (cid:18) I − λ − λ ∗ λ − λ ∗ P [ ] (cid:19) = (cid:34)(cid:16) λ − λ λ − λ ∗ (cid:17)
00 1 (cid:35) + O (cid:104) t e − λ I | v − v | t (cid:105) , t → + ∞ .(81)In succession, after n -fold iteration, we have T [ n ] T [ n − ] · · · T [ ] = (cid:34)(cid:16) λ − λ λ − λ ∗ (cid:17) n
00 1 (cid:35) + O (cid:16) t n − e − λ I | v − v | t (cid:17) , t → + ∞ ,(82)which deduces the equation in (77).If the soliton moves along the trajectory ξ = x − vt with v > v , in this case, we use the Darboux matricesin Eq.(75), then its asymptotics can be given in a similar manner. Similarly, the asymptotics of the Darbouxmatrices when t → − ∞ can also be presented. It completes the proof. (cid:3) Based on the theorem 2 and the asymptotics of Darboux matrices in Eq.(77) and Eq.(78), the high ordersoliton q [ N ] can be expressed as a sum of n single soltions along the specific characteristic curves with thevector Φ [ ] defined as Φ [ ] = (cid:34)(cid:16) λ − λ λ − λ ∗ (cid:17) n e θ (cid:35) , t → + ∞ , (cid:34)(cid:16) λ − λ ∗ λ − λ (cid:17) n e θ (cid:35) , t → − ∞ .(83)Additionally, q [ N ] can also be expressed as a sum of n single solitons along another specific characteristiccurves with the vector Φ [ ] defined as Φ [ ] = (cid:34)(cid:16) λ − λ λ − λ ∗ (cid:17) − n e θ (cid:35) , t → + ∞ , (cid:34)(cid:16) λ − λ ∗ λ − λ (cid:17) − n e θ (cid:35) , t → − ∞ .(84)Now we can give the asymptotic expression with two spectral parameters λ and λ . heorem 3. Suppose there are two spectral parameters λ , λ and v < v , then the long-time asymptotic behaviorof q [ N ] is q [ N ] ± = (cid:104) n + (cid:105) ∑ κ = λ I sech (cid:16) λ I s l , ± ± ∆ [ κ ] λ , λ , l − a [ ] (cid:17) e − ( θ ) ± i n + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) ± i n (cid:18) arg (cid:18) λ − λ ∗ λ − λ λ ∗ − λ ∗ λ ∗ − λ (cid:19)(cid:19) + i π ( + κ )+ [ n ] ∑ κ = λ I sech (cid:16) λ I s r , ± ∓ ∆ [ κ ] λ , λ , r − a [ ] (cid:17) e − ( θ ) ∓ i n + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) ± i n (cid:18) arg (cid:18) λ − λ ∗ λ − λ λ ∗ − λ ∗ λ ∗ − λ (cid:19)(cid:19) + i π ( n − + κ )+ (cid:104) n + (cid:105) ∑ κ = λ I sech (cid:16) λ I s l , ± ± ∆ [ κ ] λ , λ , l − a [ ] (cid:17) e − ( θ ) ± i n + − k (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) ± i n (cid:18) arg (cid:18) λ − λ λ − λ ∗ λ ∗ − λ λ ∗ − λ ∗ (cid:19)(cid:19) + i π ( + κ )+ [ n ] ∑ κ = λ I sech (cid:16) λ I s r , ± ∓ ∆ [ κ ] λ , λ , r − a [ ] (cid:17) e − ( θ ) ∓ i n + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) ± i n (cid:18) arg (cid:18) λ − λ λ − λ ∗ λ ∗ − λ λ ∗ − λ ∗ (cid:19)(cid:19) + i π ( n − + κ )+ O ( log ( t ) / t ) .(85) where ∆ [ κ ] λ i , λ j , l = log (cid:18) ( n i − κ ) ! ( k − ) ! (cid:19) − ( n i + − κ ) ( log ( λ iI | C i | )) − n j ( sgn ( j − i )) log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ i − λ j λ i − λ ∗ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ [ κ ] λ i , λ j , r = log (cid:18) ( n i − κ ) ! ( κ − ) ! (cid:19) − ( n i + − κ ) ( log ( λ iI | C i | )) + n j ( sgn ( j − i )) log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ i − λ j λ i − λ ∗ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and sgn (x) is a sign function defined by sgn ( x ) : = − i f x < i f x = i f x > Proof.
Based on the analysis in Lemma 4 and the Lemma 7, we give the asymptotics behavior when t → + ∞ . The other case can be given similarly. If the high order soliton moves along the trajectory of soliton ,then the new vector Φ [ ] is defined as Eq.(83). Under this condition, the leading order terms of the denom-inator becomes n − ∑ j = e ( n − j )( θ + θ ∗ ) (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) λ − λ λ − λ ∗ (cid:12)(cid:12)(cid:12)(cid:12) n ( n − j ) A j ( x , t ) (cid:33) + A + l.o.t.(86)Similarly, the leading order of the numerator is n − ∑ j = ( − ) j + e θ ∗ e ( n − j − )( θ + θ ∗ ) (cid:18) λ ∗ − λ ∗ λ ∗ − λ (cid:19) n ( n − j ) (cid:18) λ − λ λ − λ ∗ (cid:19) n ( n − j − ) B j ( x , t ) + l.o.t.(87)If the soliton moves along the following characteristic curve x = s l , + + (cid:18) n + − κ λ I (cid:19) log ( t ) + v t , (cid:18) κ =
1, 2, · · · , (cid:20) n + (cid:21)(cid:19) , x = s r , + − (cid:18) n + − κ λ I (cid:19) log ( t ) + v t , (cid:16) κ =
1, 2, · · · , (cid:104) n (cid:105)(cid:17) ,then the long-time asymptotics of high order soliton q [ N ] is q [ N ] = λ I sech (cid:16) λ I s l , + + ∆ [ κ ] λ , λ , l − a [ ] (cid:17) e − ( θ )+ i n + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i n (cid:18) arg (cid:18) λ − λ ∗ λ − λ λ ∗ − λ ∗ λ ∗ − λ (cid:19)(cid:19) + i π ( + κ )+ O ( log ( t ) / t ) ,or q [ N ] = λ I sech (cid:16) λ I s r , + − ∆ [ κ ] λ , λ , r − a [ ] (cid:17) e − ( θ ) − i n + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i n (cid:18) arg (cid:18) λ − λ ∗ λ − λ λ ∗ − λ ∗ λ ∗ − λ (cid:19)(cid:19) + i π ( n − + κ )+ O ( log ( t ) / t ) .(88) imilarly, if the soliton moves along another characteristic curves x = s l , + + (cid:18) n + − κ λ I (cid:19) log ( t ) + v t , (cid:18) κ =
1, 2, · · · , (cid:20) n + (cid:21)(cid:19) , x = s r , + − (cid:18) n + − κ λ I (cid:19) log ( t ) + v t , (cid:16) κ =
1, 2, · · · , (cid:104) n (cid:105)(cid:17) ,the asymptotic expression is q [ N ] = λ I sech (cid:16) λ I s l , + + ∆ [ κ ] λ , λ , l − a [ ] (cid:17) e − ( θ )+ n + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i n (cid:18) arg (cid:18) λ − λ λ − λ ∗ λ ∗ − λ λ ∗ − λ ∗ (cid:19)(cid:19) + i π ( + κ )+ O ( log ( t ) / t ) ,or q [ N ] = λ I sech (cid:16) λ I s r , + − ∆ [ κ ] λ , λ , r − a [ ] (cid:17) e − ( θ ) − n + − κ (cid:18) arg (cid:18) C C ∗ (cid:19)(cid:19) + i n (cid:18) arg (cid:18) λ − λ λ − λ ∗ λ ∗ − λ λ ∗ − λ ∗ (cid:19)(cid:19) + i π ( n − + κ )+ O ( log ( t ) / t ) .(89) If the high order soliton moves along the other curves x = s + vt + β log ( t ) with v (cid:54) = v , v (cid:54) = v or β (cid:54) = n + − κ λ I , β (cid:54) = n + − κ λ I , then its asymptotic solution is(90) q [ N ] = O ( e − a | v − b | t ) ,and(91) q [ N ] = O ( t − ) ,respectively, where a = min ( λ I , λ I ) , b = min ( | v − v | , | v − v | ) , both of which are vanishing when t → ∞ .The asymptotic behavior of q [ N ] when t → − ∞ can also be given in a similar method. Then the globalasymptotics to q [ N ] can be given in Eq.(85). It completes the proof. (cid:3) In this case, we only present one numerical figure to verify the above asymptotic expressions.F
IGURE The comparison between the exact solution and asymptotic solution by choosing theparameters λ = + i, λ = + i, δ = γ = a [ ] = a [ ] = a [ ] = a [ ] = a [ ] = ξ = x − t /27, their order is n = n = In Fig.6, we choose the parameters λ = + i, λ = + i, δ = γ = a [ ] = a [ ] = a [ ] = a [ ] = a [ ] =
0, then θ = − x + t + i (cid:18) x − t − π (cid:19) , θ = − x + t − t + i (cid:18) x − t − π (cid:19) . t is clear v = , v = , v > v . If the soliton moves along the trajectory of soliton , we have θ → − ∞ , t → + ∞ , θ → + ∞ , t → − ∞ .(92)With the asymptotics in Eq.(84), we know Φ [ ] has asymptotically form of (cid:20) (cid:21) and (cid:20) (cid:21) as t → + ∞ and − ∞ respectively. When t >
0, soliton is located on the left of soliton 1, but when t <
0, soliton is in theleft due to soliton moves faster. Conversely, if the soliton moves along the trajectory of soliton , we know Φ [ ] has an opposite asymptotics, which approaches (cid:20) (cid:21) and (cid:20) (cid:21) when t → + ∞ and − ∞ respectively. It isinteresting that the soliton either decays or grows exponentially, as t → ± ∞ , along any direction except itsown trajectory. In order to observe these five soliton when t is large, we choose the velocity v , v be close.Otherwise, one soliton will go far away to the other ones when t is large, it becomes difficult to see all ofthe solitons in a short range in space.Next, we will give a general asymptotic expression with k spectral parameters λ , λ , · · · , λ k . Theorem 4.
If there are k spectral parameters λ , λ , · · · , λ k with the order n , n , · · · , n k respectively. Supposetheir velocities satisfying v < v < . . . < v k , then the long-time asymptotics of q [ N ] is q [ N ] ± = k ∑ i = (cid:20) ni + (cid:21) ∑ κ = λ iI sech (cid:16) λ iI s i , l , ± ± ∆ [ κ ] λ i , ··· , λ k , λ ··· , λ i − l − a [ ] i (cid:17) e − ( θ i ) ± i ni + − κ (cid:32) arg (cid:32) C i C ∗ i (cid:33)(cid:33) ± i k ∑ j = j (cid:54) = i nj ( j − i ) arg λ i − λ ∗ j λ i − λ j λ ∗ i − λ ∗ j λ ∗ i − λ j + i π (cid:16) + κ (cid:17) + k ∑ i = (cid:104) ni (cid:105) ∑ κ = λ iI sech (cid:16) λ iI s i , r , ± ∓ ∆ [ κ ] λ i , ··· , λ k , λ ··· , λ i − r − a [ ] i (cid:17) e − ( θ i ) ∓ i ni + − κ (cid:32) arg (cid:32) C i C ∗ i (cid:33)(cid:33) ± i k ∑ j = j (cid:54) = i nj ( j − i ) arg λ i − λ ∗ j λ i − λ j λ ∗ i − λ ∗ j λ ∗ i − λ j + i π (cid:16) ni − + κ (cid:17) + O ( log ( t ) / t ) (93) where ∆ [ κ ] λ i , ··· , λ k , λ , ··· , λ i − , l = log (cid:18) ( n i − κ ) ! ( κ − ) ! (cid:19) − ( n i + − κ ) ( log ( λ iI ) + log ( | C i | )) − k ∑ j = j (cid:54) = i n j sgn ( j − i ) log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ i − λ j λ i − λ ∗ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ [ κ ] λ i , ··· , λ k , λ , ··· , λ i − , r = log (cid:18) ( n i − κ ) ! ( k − ) ! (cid:19) − ( n i + − κ ) ( log ( λ iI ) + log ( | C i | )) + k ∑ j = j (cid:54) = i n j sgn ( j − i ) log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ i − λ j λ i − λ ∗ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ i : = θ [ i ] (cid:12)(cid:12) λ = λ i = i λ i ( x + ( γλ i + δλ i ) t ) + a [ ] i − i4 π , λ i = λ iR + i λ iI . Proof.
Similar to the analysis with two spectral parameters, we can give its asymptotics along the trajectoryof every soliton i , ( i =
1, 2, · · · , k ) . If the soliton moves along the trajectory soliton i , that is Re ( θ i ) = O ( ) .When t → + ∞ , we have θ j → (cid:26) − ∞ , j < i , + ∞ , j > i .Then we can define a new Φ [ ] i as Φ [ ] i = k ∏ j = j (cid:54) = i (cid:18) λ i − λ j λ i − λ ∗ j (cid:19) n j sgn ( j − i ) e θ i Based on the analysis in Theorem 3, if the high order soliton moves along the characteristic curve x = s + v i t + n i + − κ λ iI log ( | t | ) , (cid:18) i =
1, 2, · · · , k , κ =
1, 2, · · · , (cid:20) n i + (cid:21)(cid:19) , x = s + v i t − n i + − κ λ iI log ( | t | ) , (cid:16) i =
1, 2, · · · , k , κ =
1, 2, · · · , (cid:104) n i (cid:105)(cid:17) ,(94) hen the asymptotics to q [ N ] is q [ N ] = λ iI sech (cid:16) λ iI s i , l , ± ± ∆ [ κ ] λ i , ··· , λ k , λ , ··· , λ i − , l − a [ ] i (cid:17) · e − ( θ i ) ± i ni + − κ (cid:18) arg (cid:18) C i C ∗ i (cid:19)(cid:19) ± i k ∑ j = j (cid:54) = i nj sgn ( j − i ) (cid:32) arg (cid:32) λ i − λ ∗ j λ i − λ j λ ∗ i − λ ∗ j λ ∗ i − λ j (cid:33)(cid:33) + i π ( + κ )+ O ( log ( t ) / t ) or q [ N ] = λ iI sech (cid:16) λ iI s i , r , ± ∓ ∆ [ κ ] λ i , ··· , λ k , λ , ··· , λ i − , r − a [ ] i (cid:17) · e − ( θ i ) ∓ i ni + − κ (cid:18) arg (cid:18) C i C ∗ i (cid:19)(cid:19) ± i k ∑ j = j (cid:54) = i nj sgn ( j − i ) (cid:32) arg (cid:32) λ i − λ ∗ j λ i − λ j λ ∗ i − λ ∗ j λ ∗ i − λ j (cid:33)(cid:33) + i π ( n i − + κ )+ O ( log ( t ) / t ) (95)Otherwise, if the high order soliton moves along the other curves x = s + vt + β log ( t ) with v (cid:54) = v i , or β (cid:54) = n i + − κ λ iI ( i =
1, 2, · · · , k ) , then its asymptotic solution is(96) q [ N ] = O ( e − a | v − b | t ) ,and(97) q [ N ] = O ( t − ) ,respectively, where a = min ( λ iI , ( i =
1, 2, · · · , k )) , b = min ( | v − v i | )( i =
1, 2, · · · , k ) , which is an exponen-tial decay or algebraic decay. The asymptotic behavior of q [ N ] when t → − ∞ can also be calculated in asimilar method. Then the long-time asymptotics of q [ N ] can be given in Eq.(93). It completes the proof. (cid:3) Now we choose three spectral parameters λ , λ , λ to verify this asymptotic behavior Eq.(93)F IGURE The comparison between the exact solution and asymptotic solution by choosing theparameters λ = + i, λ = + i, λ = + i, δ = γ = a [ ] = a [ ] = a [ ] = a [ ] = a [ ] = a [ ] = ξ = x − t /27, n = n = n = In this case, we still set the velocities v , v , v be closed so as to observe them in a short range in space.Their velocities satisfy v > v > v , so soliton is always located in the middle of soliton and soliton ,whose dynamical behavior is consistent with the theorem 4. In this work, we utilize the Darboux transformation to construct the exact solitons of Hirota equation.The crucial part in our work is to analyze the long-time asymptotics on the high order soliton based on theformulas of determinant and the iterated Darboux matrices. irstly, we give the AKNS hierarchy with su ( ) symmetry, which can deduce the Hirota equation. Thenwe give the Riemann-Hilbert representation of Darboux matrices, which establish the relationships be-tween Darboux transformation with the classical inverse scattering transform and robust inverse scatteringtransform.Secondly, the high order soliton with single spectral parameter λ is analyzed by the determinant. Thekey idea about the asymptotic analysis is looking for the characteristic curves. If the soliton moves alongthis characteristic curve, then the high order soliton can reduce to a single soliton. Otherwise, it has thevanishing limitation. We obtain the exact high order soliton and give its leading order term from the solitonsdeterminant directly. With the aid of the leading order term, we give the asymptotic behavior with singlespectral parameter λ . In this case, we present a detailed calculation about how to obtain the asymptoticexpression for t → ∞ .Furthermore, we give the asymptotic expression to high order soliton with two spectral parameters λ and λ . In this case, by using the asymptotics of Darboux matrices, we convert the problem with twospectral parameters to the problem with single spectral parameter. Thus the asymptotics to high-ordersoliton with two spectral parameters can be given with the result of theorem 2. Based on this analysis, wealso give the general asymptotic expression to high order soliton with k spectral parameters. For each case,we give the numeral plotting to verify the conclusion of the theorems.In this paper, we just consider the finite order soliton and analyze the long-time asymptotics. The anal-ysis method proposed in this work can be readily extended to the other integrable models, such as thesine-Gordan equation, modified KdV equation, derivative NLS equation and so on. Very recently, there ex-ist some progresses on the large order soliton for the NLS equation [37, 38, 39]. For the large order soliton,the complete description on the high order soliton in different regions is also an interesting problem, whichis deserved to explore by the RHP in the third section in the future studying. Acknowledgement
Project supported by the National Natural Science Foundation of China (Grant No. 11771151), theGuangzhou Science and Technology Program of China (Grant No. 201904010362), and the FundamentalResearch Funds for the Central Universities of China (Grant No. 2019MS110).
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CHOOL OF M ATHEMATICS , S
OUTH C HINA U NIVERSITY OF T ECHNOLOGY , G
UANGZHOU , C
HINA , 510641S
CHOOL OF M ATHEMATICS , S
OUTH C HINA U NIVERSITY OF T ECHNOLOGY , G
UANGZHOU , C
HINA , 510641
E-mail address : [email protected]@scut.edu.cn