Darboux Transformation for the Hirota equation
DDarboux Transformation for the Hirota equation
Halis Yilmaz † School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, UK Department of Mathematics, Mimar Sinan Fine Arts University, Istanbul, Turkey Department of Mathematics, University of Dicle, 21280 Diyarbakir, Turkey
Abstract
The Hirota equation is an integrable higher order nonlinear Schr¨odinger type equation whichdescribes the propagation of ultrashort light pulses in optical fibers. We present a standard Dar-boux transformation for the Hirota equation and then construct its quasideterminant solutions.The multisoliton and breather solutions of the Hirota equation are given explicitly.
Keywords:
Hirota equation; Darboux transformation; Quasideterminants.2020 Mathematics Subject Classification: 35C08, 35Q55, 37K10
There exists a large class of nonlinear evolution equations which can be solved analytically. Suchequations are called integrable. Integrable equations constitute an important part of the nonlinearwave theory. The simplest integrable equation which describes the dynamics of deep-water gravitywaves is the nonlinear Schr¨odinger (NLS) equation iq t + q xx + 2 | q | q = 0 . (1.1)In 1967, it was first discussed in the general context of nonlinear dispersive waves by Benney andNewell [3]. In 1968, this equation was also derived by Zakharov in his study of modulationalstability of deep water waves [35]. In 1972, Zakharov and Shabat found that the NLS equationhad a Lax pair and could be solved by the inverse scattering transform (IST) method [37]. Thisequation plays an important role in different physical systems as wide as plasma physics [36], waterwaves [3, 4, 35], and nonlinear optics [12, 13]. One of the most interesting applications of the NLSequation is that it can be employed to model for short soliton pulses in optical fibres [16]. However,as the pulses get shorter, various additional effects become important and the NLS model is nolonger appropriate. In order to understand these additional effects, Kodama and Hasegawa [17, 18]suggested a higher-order NLS equation iq t + α q xx + α | q | q + iβ (cid:2) γ q xxx + γ | q | q x + γ q (cid:0) | q | (cid:1) x (cid:3) = 0 , (1.2) † E-mail: [email protected], [email protected] a r X i v : . [ n li n . S I] O c t here the α i , γ i are real constants, β is a real spectral parameter and q is a complex-valued functionof x and t . By choosing β = 0 and α = 2 α = 2 in this equation, we can easily see that the firstthree terms form the standard NLS equation (1.1). Generally, the Kodama-Hasegawa higher-orderNLS equation (1.2) may not be completely integrable if some restrictions are not imposed on thereal constants γ i ( i = 1 , , γ : γ : γ = 0 : 1 : 0), the Kaup-Newell [15] derivative NLSequation ( γ : γ : γ = 0 : 1 : 1), the Hirota [14] NLS equation ( γ : γ : γ = 1 : 6 : 0) and theSasa-Satsuma [30] NLS equation ( γ : γ : γ = 1 : 6 : 3).In this paper, we consider the Hirota [14] NLS equation iq t + α (cid:0) q xx + 2 | q | q (cid:1) + iβ (cid:0) q xxx + 6 | q | q x (cid:1) = 0 , α, β ∈ R , (1.3)in which α = 2 α = 2 α . This equation is commonly known as the Hirota equation (HE), and wewill denote it as such from now on. The HE (1.3) can be used to describe the wave propagationof ultrashort light pulses in optical fibers [1, 17, 18, 21, 24, 33]. It is very interesting to see that theHirota equation (1.3) is the sum of the NLS (1.1) equation ( α = 1 , β = 0) and the complex versionof the modified Korteweg-de Vries (mKdV) equation ( α = 0 , β = 1) q t + q xxx + 6 | q | q x = 0 (1.4)which is completely integrable [14,31]. In the resent years, there has been some interest in solutionsof the HE (1.3) obtained by Darboux − type transformations [2, 20, 29]. These solutions are oftenwritten in terms of determinants. These methods have been used to prove various solutions of theHE (1.3) such as multisolitons, breathers and rogue waves.In 1882, the French mathematician Jean Gaston Darboux [6] introduced a method to solvethe Sturm-Louville equation, which is called Darboux transformation (DT) afterwards. Almost acentury later, in 1979, Matveev [22] realised that the method given by Darboux for the spectralproblem of second order ordinary differential equations can be extended to some important solitonequations. Darboux transformations are one of important tools in studying integrable systems.They provide a universal algorithmic procedure to derive exact solutions of integrable systems.In the present article, we construct for the first time a standard Darboux transformation for theHirota equation (1.3). We underline that the method we use here is based on Darboux’s [6] andMatveev’s original ideas [22, 23]. Therefore, our approach should be considered on its own merits.Furthermore, our solutions for the HE are written in terms of quasideterminants [7, 8] rather thandeterminants. It has been proved that quasideterminants are very useful for constructing exactsolutions of integrable equations [9, 10, 19, 27, 28, 32, 38], enabling these solutions to be expressed ina simple and compact form.This paper is structured as follows. In Section 1.1 below, we give a brief review of quasideter-minants. In Section 2, we establish a 2 × × .1 Quasideterminants In this short section we will list some of the key elementary properties of quasideterminants usedin the paper. The reader is referred to the original papers [7, 8] for a more detailed and generaltreatment.Let M = ( m ij ) be an n × n matrix with entries over a ring (noncommutative, in general) has n quasideterminants written as | M | ij for i, j = 1 , . . . , n . They are defined recursively by | M | ij = m ij − r ji (cid:0) M ij (cid:1) − c ij , (1.5)where r ji represents the row vector obtained from i th row of M with the j th element removed, c ij is the column vector obtained from j th column of M with the i th element removed and M ij is the ( n − × ( n −
1) submatrix obtained by deleting the i th row and the j th column from M . Quasideterminants can be also denoted as shown below by boxing the entry about which theexpansion is made | M | ij = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M ij c ij r ji m ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (1.6)If the entries in M commute, then the quasideterminant | M | ij can be expressed as a ratio ofdeterminants | M | ij = ( − i + j det M det M ij . (1.7) Let us consider the couple Hirota equations q t − iα (cid:0) q xx + 2 q r (cid:1) + β ( q xxx + 6 qrq x ) = 0 , (2.1) r t + iα (cid:0) r xx + 2 qr (cid:1) + β ( r xxx + 6 qrr x ) = 0 , (2.2)where q = q ( x, t ) and r = r ( x, t ) are complex valued functions. Equations (2.1) and (2.2) reduceto the Hirota equation (1.3) when r = q ∗ . Here the asterisk superscript on q denotes the complexconjugate.The Lax pair [29] for the couple Hirota equations (2.1)-(2.2) is given by L = ∂ x + J λ − R, (2.3) M = ∂ t + 4 βJ λ + 2 U λ − V λ − W, (2.4)where J , R , U , V and W are 2 × J = (cid:18) i − i (cid:19) , R = (cid:18) q − r (cid:19) , U = (cid:18) iα − βq βr − iα (cid:19) , (2.5) V = (cid:18) iβqr αq + iβq x − αr + iβr x − iβqr (cid:19) , W = (cid:18) iαqr + β ( qr x − rq x ) iαq x − β (cid:0) q xx + 2 q r (cid:1) iαr x + β (cid:0) r xx + 2 qr (cid:1) − iαqr − β ( qr x − rq x ) (cid:19) . (2.6)3ere λ is a spectral parameter. It can be seen that the potential matrix R in (2.5) has two symmetryproperties. One is that it is skew-Hermitian: R † = − R . The other one is that SRS − = R ∗ , where S = (cid:18) − (cid:19) . (2.7)Let φ = ( ϕ, ψ ) T be a vector eigenfunction for (2.3)-(2.4) for eigenvalue λ so that L λ ( φ ) = M λ ( φ ) = 0.Using the second symmetry, it may be seen that ˜ φ = Sφ = ( ψ ∗ , − ϕ ∗ ) T is another eigenfunction foreigenvalue λ ∗ such that L λ ∗ ( ˜ φ ) = M λ ∗ ( ˜ φ ) = 0. Using these vector eigenfunctions we can define asquare 2 × θ with 2 × θ = (cid:18) ϕ ψ ∗ ψ − ϕ ∗ (cid:19) , Λ = (cid:18) λ λ ∗ (cid:19) , (2.8)satisfying θ x + J θ Λ − Rθ = 0 , (2.9) θ t + 4 βJ θ Λ + 2 U θ Λ − V θ Λ − W θ = 0 . (2.10) Let us consider the linear operators L = ∂ x + n (cid:88) i =0 u i ∂ iy , M = ∂ t + n (cid:88) i =0 v i ∂ iy , (3.1)where u i , v i are m × m matrices. The standard approach to Darboux transformations [6, 22, 23]involves a gauge operator G θ = θ∂ y θ − , where θ = θ ( x, y, t ) is an invertible m × m matrix solutionto a linear system L ( φ ) = M ( φ ) = 0 . (3.2)If φ is any eigenfunction of L and M then ˜ φ = G θ ( φ ) satisfies the transformed system˜ L ( ˜ φ ) = ˜ M ( ˜ φ ) = 0 , (3.3)where the linear operators ˜ L = G θ LG − θ and ˜ M = G θ M G − θ have the same forms as L and M :˜ L = ∂ x + n (cid:88) i =0 ˜ u i ∂ iy , ˜ M = ∂ t + n (cid:88) i =0 ˜ v i ∂ iy . (3.4) Here, we describe a reduction of the Darboux transformation from (2 + 1) to (1 + 1) dimensions.We choose to eliminate the y -dependence by employing a ‘separation of variables’ technique. Thereader is referred to the paper [25] for a more detailed treatment. We make the ansatz φ = φ r ( x, t ) e λy , (3.5)4 = θ r ( x, t ) e Λ y , (3.6)where λ is a constant scalar and Λ an N × N constant matrix and the superscript r denotesreduced functions, independent of y . Hence in the dimensional reduction we obtain ∂ iy ( φ ) = λ i φ and ∂ iy ( θ ) = θ Λ i and so the operator L and Darboux transformation G become L r = ∂ x + n (cid:88) i =0 u i λ i , (3.7) G r = λ − θ r Λ( θ r ) − , (3.8)where θ r is a matrix eigenfunction of L r such that L r ( θ r ) = 0, with λ replaced by the matrix Λ,that is, θ rx + n (cid:88) i =0 u i θ r Λ i = 0 . (3.9)Below we omit the superscript r for ease of notation. In this section we shall consider iteration of the Darboux transformation and find closed formexpressions for these in terms of quasideterminants.Let L be an operator, form invariant under the reduced Darboux transformation G θ = λ − θ Λ θ − discussed above.Let φ = φ ( x, t ) be a general eigenfunction of L such that L ( φ ) = 0. Then˜ φ = G θ ( φ )= λφ − θ Λ θ − φ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ φθ Λ λφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is an eigenfunction of ˜ L = G θ LG − θ so that ˜ L ( ˜ φ ) = λ ˜ φ . Let θ i for i = 1 , . . . , n, be a particularset of invertible eigenfunctions of L so that L ( θ i ) = 0 for λ = Λ i , and introduce the notationΘ = ( θ , . . . , θ n ). To apply the Darboux transformation a second time, let θ [1] = θ and φ [1] = φ bea general eigenfunction of L [1] = L . Then φ [2] = G θ [1] (cid:0) φ [1] (cid:1) and θ [2] = φ [2] | φ → θ are eigenfunctionsfor L [2] = G θ [1] L [1] G − θ [1] .In general, for n ≥
1, we define the n th Darboux transform of φ by φ [ n +1] = λφ [ n ] − θ [ n ] Λ n θ − n ] φ [ n ] , (3.10)in which θ [ k ] = φ [ k ] | φ → θ k . For example, φ [2] = λφ − θ Λ θ − φ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ φθ Λ λφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , [3] = λφ [2] − θ [2] Λ θ − φ [2] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ θ φθ Λ θ Λ λφθ Λ θ Λ λ φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . After n iterations, we get φ [ n +1] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ θ . . . θ n φθ Λ θ Λ . . . θ n Λ n λφθ Λ θ Λ . . . θ n Λ n λ φ ... ... . . . ... ... θ Λ n θ Λ n . . . θ n Λ nn λ n φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.11) In this section we determine the specific effect of the Darboux transformation G θ = λ − θ Λ θ − on the operator L given by (2.3). Corresponding results hold for the operator M given by (2.4).Here the eigenfunction θ is the solution of the linear system (2.9)-(2.10) is given explicitly with theeigenvalue Λ in (2.8). From ˜ LG θ = G θ L , the operator L = ∂ x + J λ − R is transformed to a newoperator ˜ L in which J is unchanged and˜ R = R − (cid:2) J, θ Λ θ − (cid:3) . (4.1)For notational convenience, we introduce a 2 × Q such that R = [ J, Q ], and hence Q = 12 i (cid:18) qr (cid:19) , (4.2)where the entries left blank are arbitrary and do not contribute to R . From (4.1) it follows that˜ Q = Q − θ Λ θ − (4.3)which can be written in a quasideterminant structure as˜ Q = Q + (cid:12)(cid:12)(cid:12)(cid:12) θ I θ Λ 0 (cid:12)(cid:12)(cid:12)(cid:12) . (4.4)We rewrite (4.3) as Q [2] = Q [1] − θ [1] Λ θ − (4.5)where Q [1] = Q , Q [2] = ˜ Q , θ [1] = θ = θ and Λ = Λ. Then after n repeated Darboux transforma-tions, we have Q [ n +1] = Q [ n ] − θ [ n ] Λ n θ − n ] (4.6)6n which θ [ k ] = φ [ k ] | φ → θ k . We express P [ n +1] in quasideterminant form as Q [ n +1] = Q + (cid:12)(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 . . . θ n θ Λ θ Λ . . . θ n Λ n ... ... . . . ... ... θ Λ n − θ Λ n − . . . θ n Λ n − n θ Λ n − θ Λ n − . . . θ n Λ n − n I θ Λ n θ Λ n . . . θ n Λ 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)(cid:12) , (4.7)where each θ i , Λ i as a 2 × θ i = (cid:18) ϕ i ψ ∗ i ψ i − ϕ ∗ i (cid:19) , Λ i = (cid:18) λ i λ ∗ i (cid:19) (4.8)in which i = 1 , ..., n . Now let Θ ( n ) be a 2 × n matrix such thatΘ ( n ) = ( θ Λ n , . . . , θ n Λ nn ) = (cid:18) ϕ ( n ) ψ ( n ) (cid:19) , (4.9)where ϕ ( n ) = ( λ n ϕ , λ ∗ n ψ ∗ , . . . , λ nn ϕ n , λ ∗ nn ψ ∗ n ) ,ψ ( n ) = ( λ n ψ , − λ ∗ n ϕ ∗ , . . . , λ nn ψ n , − λ ∗ nn ϕ ∗ n )denote 1 × n row vectors. Thus, (4.7) can be written as Q [ n +1] = Q + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Θ E Θ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.10)where (cid:98) Θ = (cid:16) θ i Λ j − i (cid:17) i,j =1 ,...,n and E = ( e n − , e n ) denote 2 n × n and 2 n × e i represents a column vector with 1 in the i th row and zeros elsewhere. Hence, we obtain Q [ n +1] = Q + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Θ e n − ϕ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Θ e n ϕ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Θ e n − ψ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Θ e n ψ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.11)Here we immediately see that a quasideterminant solution q [ n +1] of the Hirota equation (1.3) alongwith its complex conjugate r [ n +1] can be expressed as q [ n +1] = q + 2 i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Θ e n ϕ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , r [ n +1] = r + 2 i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Θ e n − ψ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.12)where it can be easily shown that the reduction r [ n +1] = q ∗ [ n +1] holds.7 .1 Explicit solutions In order to construct explicit solutions for the Hirota equation (1.3), we consider the quasideter-minant solution given by (4.12) in which we obtain q [ n +1] = q + 2 i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ψ ∗ . . . ϕ n ψ ∗ n ψ − ϕ ∗ . . . ψ n − ϕ ∗ n ϕ λ ψ ∗ λ ∗ . . . ϕ n λ n ψ ∗ n λ ∗ n ψ λ − ϕ ∗ λ ∗ . . . ψ n λ n − ϕ ∗ n λ ∗ n ϕ λ n − ψ ∗ λ ∗ n − . . . ϕ n λ n − n ψ ∗ n λ ∗ n − n ψ λ n − − ϕ ∗ λ ∗ n − . . . ψ n λ n − n − ϕ ∗ n λ ∗ n − n ϕ λ n ψ ∗ λ ∗ n . . . ϕ n λ nn ψ ∗ n λ ∗ 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)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.13)Here ϕ j and ψ j are scalar functions such that the eigenfunction φ j = ( ϕ j , ψ j ) T denotes n distinctsolutions of the spectral problem L ( φ j ) = M ( φ j ) = 0 with the associated eigenvalue λ j , where theoperators L , M are given by (2.3)-(2.4) so that φ j,x + J φ j λ j − Rφ j = 0 ,φ j,t + 4 βJ φ j λ j + 2 U φ j λ j − V φ j λ j − W φ j = 0 , (4.14)in which j = 1 , . . . , n and J , R , U , V , W are 2 × n = 1 , . . . ,
3. For theone-fold ( n = 1), two-fold ( n = 2) and three-fold ( n = 3) Darboux transformations, the solution(4.13) yields q [2] = q + 2 i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ψ ∗ ψ − ϕ ∗ ϕ λ ψ ∗ λ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.15) q [3] = q + 2 i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ψ ∗ ϕ ψ ∗ ψ − ϕ ∗ ψ − ϕ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.16)and q [4] = q + 2 i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ψ ∗ ϕ ψ ∗ ϕ ψ ∗ ψ − ϕ ∗ ψ − ϕ ∗ ψ − ϕ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.17)respectively. The quasideterminant solutions (4.15)-(4.16) can be expanded as q [2] = q − i ( λ − λ ∗ ) ϕ ψ ∗ | ϕ | + | ψ | (4.18)8nd q [3] = q − i Λ (cid:16) Π | ϕ | + Π ∗ | ψ | (cid:17) ϕ ψ ∗ + Λ (cid:16) Λ | ϕ | + Λ ∗ | ψ | (cid:17) ϕ ψ ∗ | λ − λ | | ϕ ϕ ∗ + ψ ψ ∗ | + | λ − λ ∗ | | ϕ ψ − ϕ ψ | , (4.19)whereΛ = λ − λ ∗ , Λ = λ − λ ∗ , Λ = ( λ − λ ) ( λ − λ ∗ ) , Π = ( λ − λ ) ( λ ∗ − λ ) . Moreover, the solution (4.17) can be expressed in terms of determinants such that q [4] = q − i D ∆ , (4.20)in which D = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ψ ∗ ϕ ψ ∗ ϕ ψ ∗ ψ − ϕ ∗ ψ − ϕ ∗ ψ − ϕ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.21)∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ψ ∗ ϕ ψ ∗ ϕ ψ ∗ ψ − ϕ ∗ ψ − ϕ ∗ ψ − ϕ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ϕ λ ψ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ ψ λ − ϕ ∗ λ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.22) For q = r = 0, the spectral problem (4.14) becomes φ j,x + J φ j λ j = 0 ,φ j,t + (cid:16) βλ j + 2 αλ j (cid:17) J φ j = 0 , (5.1)which has solution φ j = ( ϕ j , ψ j ) T such that ϕ j ( x, t, λ j ) = e − i [ λ j x + ( αλ j +4 βλ j ) t ] ,ψ j ( x, t, λ j ) = e i [ λ j x + ( αλ j +4 βλ j ) t ] , (5.2)where j = 1 , . . . , n . 9 ase I ( n = 1 ) By letting λ = ξ + iη and substituting the functions ϕ and ψ given by (5.2) into (4.18), weobtain the one-soliton solution of the Hirota equation (1.3) as q [2] = 2 ηe − i [ ξx +2 ( α [ ξ − η ] +2 β [ ξ − ξη ]) t ] sech (cid:0) ηx + 8 (cid:2) αξη + β (cid:0) ξ η − η (cid:1) t (cid:3)(cid:1) (5.3)which yields (cid:12)(cid:12) q [2] (cid:12)(cid:12) = 4 η sech (cid:0) ηx + 8 (cid:2) αξη + β (cid:0) ξ η − η (cid:1)(cid:3) t (cid:1) . (5.4)This solution is plotted in Fig. 1. ( a ) ( b ) Fig. 1. (Color online) One-soliton solution | q [2] | of the HE (1.3) when α = β = 1, ξ = 0 . , η = 1 . a ) describes its surface and ( b ) gives its profiles at different times t = − . t = 0(blue), t = 1 . Case II ( n = 2 ) Let λ = ξ + η and λ = ξ + η such that η η (cid:54) = 0. By substituting the corresponding eigenfunctions ϕ , ψ and ϕ , ψ , given by (5.2), into (4.19), we obtain the two-soliton solution of the Hirotaequation (1.3) as q [3] = 4 (cid:0) η − η (cid:1) η e − ig cosh f − η e − ig cosh f ( η − η ) cosh F + ( η + η ) cosh F − η η cos F (5.5)which yields (cid:12)(cid:12) q [3] (cid:12)(cid:12) = 16 (cid:0) η − η (cid:1) η cosh f + η cosh f − η η cosh f cosh f cos F (cid:104) ( η − η ) cosh F + ( η + η ) cosh F − η η cos F (cid:105) , (5.6)where f = 2 η (cid:2) x + 4 (cid:0) αξ + β (cid:2) ξ − η (cid:3)(cid:1) t (cid:3) , = 2 η (cid:2) x + 4 (cid:0) αξ + β (cid:2) ξ − η (cid:3)(cid:1) t (cid:3) ,g = 2 ξx + 4 (cid:2) α (cid:0) ξ − η (cid:1) + 2 βξ (cid:0) ξ − η (cid:1)(cid:3) t,g = 2 ξx + 4 (cid:2) α (cid:0) ξ − η (cid:1) + 2 βξ (cid:0) ξ − η (cid:1)(cid:3) t and F = f + f , F = f − f , F = g − g such that F = 2 ( η + η ) (cid:2) x + 4 (cid:0) αξ + β (cid:2) ξ + η η − η − η (cid:3)(cid:1) t (cid:3) ,F = 2 ( η − η ) (cid:2) x + 4 (cid:0) αξ + β (cid:2) ξ − η η − η − η (cid:3)(cid:1) t (cid:3) ,F = 4 (cid:0) η − η (cid:1) [ α + 6 βξ ] t. By choosing appropriate parameters, the two-soliton solution of the Hirota equation (1.3) is plottedin Fig. 2. ( a ) ( b ) Fig. 2. (Color online) Two-soliton solution | q [3] | of the HE (1.3) when α = β = 1, ξ = 0 . η = 0 . η = 1 .
1. ( a ) Surface diagram. ( b ) Contour diagram. Case III ( n = 3 ) In this case, we have three eigenvalues λ , λ and λ . Let us choose λ = i , λ = 2 i and λ =3 i . By substituting the corresponding eigenfunctions ( ϕ , ψ ) T , ( ϕ , ψ ) T and ( ϕ , ψ ) T , given by(5.2), into (4.20), we obtain the three-soliton solution of the Hirota equation (1.3). By choosingappropriate parameters, this solution is plotted in Fig. 3. In this subsection, for q, r (cid:54) = 0 and r = q ∗ , we take q = ce iµ as a plane wave solution of the Hirotaequation (1.3), where µ = ax + bt in which a, b, c ∈ R under the condition b = α (cid:0) c − a (cid:1) + β (cid:0) a − ac (cid:1) . We use this as a seed solution. Substituting q = ce iµ into the linear system (4.14)11 a ) ( b ) Fig. 3. (Color online) Three-soliton solution | q [4] | of the HE (1.3) when λ = i , λ = 2 i and λ = 3 i . ( a ) Surface diagram. ( b ) Density diagram.and then solving for the eigenfunction φ j = ( ϕ j , ψ j ) T , we obtain ϕ j ( x, t, λ j ) = e iµ (cid:16) c j e iγ j + e j e − iγ j (cid:17) ,ψ j ( x, t, λ j ) = e − iµ (cid:16) (cid:101) c j e iγ j + (cid:101) e j e − iγ j (cid:17) , (5.7)where γ j = s j ( x + k j t ) , (cid:101) c j = i c j c ( a + 2 λ j + s j ) , (cid:101) e j = i e j c ( a + 2 λ j − s j )in which s j = (cid:113) ( a + 2 λ j ) + 4 c , k j = α (2 λ j − a ) + β (cid:16) a − aλ j + 4 λ j − c (cid:17) and c j , e j arearbitrary constants such that j = 1 , . . . , n . Case IV ( n = 1 ) Let the eigenvalue λ = ξ + iη . For simplicity, choose a = − ξ and c = e = c . Substituting theseed solution q = ce iµ and the functions ϕ , ψ given by (5.7), into (4.18), we obtain the followingbreather solution q [2] = ce iµ (cid:18) − η η cosh(Ω t ) − iω sinh(Ω t ) + η cos[2 ω ( x + Γ t )] + ω sin[2 ω ( x + Γ t )] c cosh(Ω t ) + η cos[2 ω ( x + Γ t )] + ηω sin[2 ω ( x + Γ t )] (cid:19) , (5.8)where µ = − ξx + (cid:2) α (cid:0) c − ξ (cid:1) + 4 β (cid:0) c ξ − ξ (cid:1)(cid:3) t, a ) ( b ) Fig. 4. (Color online) Breather solution | q [2] | of the HE (1.3) when α = β = 1, ξ = 0 .
04 and η = 0 .
76. ( a ) Surface diagram. ( b ) Density diagram. ω = (cid:112) c − η , Ω = 4 ηω ( α + 6 βξ ) , Γ = 4 αξ + 2 β (cid:0) ξ − η − c (cid:1) . Thus, we have (cid:12)(cid:12) q [2] (cid:12)(cid:12) = c F + G H , (5.9)where F = (cid:0) η − c (cid:1) cosh(Ω t ) + η cos[2 w ( x + Γ t )] + ηw sin[2 w ( x + Γ t )] ,G = 2 ηw sinh(Ω t ) H = c cosh(Ω t ) + η cos[2 w ( x + Γ t )] + ηw sin[2 w ( x + Γ t )] . Fig. 4 shows the dynamical evolution of the breather solution of the Hirota equation (1.3).
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