Riemann-Hilbert method and N-soliton solutions for the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation
aa r X i v : . [ n li n . S I] A p r Riemann-Hilbert method and N-soliton solutions for the mixed Chen-Lee-Liuderivative nonlinear Schr ¨odinger equation
Fang Fang, Beibei Hu ∗ , Ling Zhang School of Mathematics and Finance, Chuzhou University, Anhui 239000 China
Abstract
In this paper, we aim to investigate the mixed Chen-Lee-Liu derivative nonlinear Schr¨odinger(CLL-NLS) equation viathe Riemann-Hilbert(RH) method. we construct a RH problem base on the Jost solution of the Lax pair. By solvingthis RH problem corresponding to the non reflection case, the N-soliton solution of CLL-NLS equation is obtained,which expression is the ratio of (2 N + × (2 N +
1) determinant and 2 N × N determinant. Keywords:
Riemann-Hilbert method; Chen-Lee-Liu derivative nonlinear Schr¨odinger equation; soliton solutions; boundary con-ditions
1. Introduction
Soliton theory is an important branch of nonlinear science. In physics, solitons are used to describe solitarywaves with elastic scattering characteristics. In mathematics, soliton theory provides a series of methods for solvingnonlinear partial di ff erential equations, which attracts the attention of mathematical physicists. With the developmentof soliton theory, many methods to solve soliton equations with important physical background have been proposed,for example, inverse scattering transform(IST) [1, 2, 3], Darboux transformation(DT) [4], B¨acklund transformation[5], Hirota bilinear method [6, 7, 8], Wroskian technique [9, 10, 11] and so on.The derivative nonlinear schr¨odinger(DNLS) equation describing Alfv´en waves in magnetic field is as follows[12]: iu t = u xx + i ( | u | u ) x = , (1.1)which is one of the most significant equations in physics. Kaup and Newell obtained the solution of Eq.(1.1) by usingIST method[13]. The IST method has obvious advantages in solving the initial value of the soliton equation, butits calculation is large. Fortunately, on the basis of IST method, Riemann-Hilbert(RH) method, a relatively simpleand direct method for solving soliton equation, was proposed by Novikovet et al.[14]. This method is similar to theIST method, the RH method first considers the direct scattering problem, that is, the RH problem is constructed,from the initial data of the soliton equation, the scattering data at the initial time is obtained, and the scattering dataat any time is obtained by using its time evolution law. Finally, the exact solution is established by using the ISTmethod[15]. Since the RH method was proposed, many solutions of soliton equations have been discussed, such as, ∗ Corresponding author. Email: hu [email protected]; hu [email protected]
Preprint submitted to Elsevier April 21, 2020 he coupled DNLS equation [16], the coupled higher-order NLS equation [17], the short pulse(SP) equation [18], thecoupled modified Korteweg-de Vries (mKdV) equation [19], the generalized Sasa-Satsuma equation [20], the two-component Gerdjikov-Ivanov(GI) equation [21], the modified SP equation [22], and so on [23, 24, 25]. In particular,RH method is an e ff ective way to working the initial-boundary value problems(IBVPs) of the integrable systems[26, 27, 28, 29, 30, 31].Recently, Chan et al.[32] reported the the mixed Chen-Lee-Liu derivative nonlinear Sch¨odinger(CLL-NLS) equa-tion as follows ir t + r xx + | r | r − i | r | r x = , (1.2)which is a completely integrable model, and a large number of solutions of Eq.(1.2) are discussed, such as, thesoliton solution by Hirota bilinear method [33], the higher-order soliton, breathers, and rogue wave solutions by DTmethod[34]. In particular, the IBVPs of Eq.(1.2) to be investigated by Fokas method[35].The design structure of this paper is as follows. Section 2, we will construct a basic RH problem based on the Jostsolution of Lax pair. Section 3, we will give the reconstruction of potential function and the law of scattering dataevolution with time. Section 4, the formula of N-soliton solution expressed by determinant ratio is proposed. Section5 is the conclusions.
2. The Riemann-Hilbert problem
In this section, we shall construct a RH problem for the CLL-NLS equation, by using IST method. First of all, weintroduce the coupled CLL-NLS equation as following: r t − ir xx + ir q + rqr x = , q t + iq xx − iq r + qrq x = . (2.1)which reduces to the CLL-NLS equation while q = − r ∗ and the ∗ denotes complex conjugation. These two equationsin (2.1) are the compatibility condition of the following Lax pair[34, 35, 36, 37] Φ x = U Φ = i ( λ −
12 ) σ + λ Q + iQ σ ! Φ , (2.2a) Φ t = V Φ = [ − i ( λ −
12 ) σ − λ Q − i λ Q σ + λ Q + i σ Q x − Q ! − iQ σ +
14 ( QQ x − Q x Q )] Φ , (2.2b)with σ = − , Q = r − r ∗ . (2.3)where Φ = Φ ( x , t ; λ ) is a matrix function of the complex spectral parameter, λ is the spectral parameter. Q is namedpotential function.For the sake of convenience, we introduce a new matrix function J = J ( x , t ; λ ) defined by Φ = Je i ( λ − ) σ x − i ( λ − ) σ t (2.4)2bviously, we can check that Je i ( λ − ) σ x − i ( λ − ) σ t satisfies Eqs.(2.2a),(2.2b). Inserting (2.4) into (2.2a)-(2.2b), theform of the Lax pair (2.2a)-(2.2b) becomes J x = i ( λ −
12 )[ σ, J ] + U J , (2.5a) J t = − i ( λ −
12 ) [ σ, J ] + V J . (2.5b)where [ σ, J ] = σ J − J σ is the commutator. U = λ Q + iQ σ V = − λ Q − i λ Q σ + λ Q + i σ Q x − Q ! − iQ σ +
14 ( QQ x − Q x Q )As usual, in the following scattering process, we only concentrate on the x -part of the Lax pair (2.2a). Indeed the x -part of the Lax pair (2.2a) allows us to take use of the existing symmetry relations of the potential Q. Consequently,we shall treat the time t as a dummy variable and omit it. Now, we calculate two Jost solutions J ± = J ± ( x , λ ) ofEq.(2.5a) for λ ∈ R . Based on the properties of the Jost solutions J ± = J ± ( x , λ ) J + = ([ J + ] , [ J + ] ) , (2.7) J − = ([ J − ] , [ J − ] ) , (2.8)with the boundary conditions J + → I , x → −∞ , (2.9a) J − → I , x → + ∞ . (2.9b)the subscripts in J ( x , λ ) represent which end of the x -axis the boundary conditions are set. Where [ J ± ] n ( n = , n -th column vector of J ± , I = diag { , } is the 2 × λ ∈ R into the Volterra integral equations. J + ( x , λ ) = I − Z + ∞ x e i ( λ − ) ˆ σ ( x − ξ ) U J + ( ξ, λ ) d ξ, (2.10) J − ( x , λ ) = I + Z x −∞ e i ( λ − ) ˆ σ ( x − ξ ) U J − ( ξ, λ ) d ξ, (2.11)where ˆ σ represents a matrix operator acting on 2 × X by ˆ σ X = [ σ, X ] and by e x ˆ σ X = e x σ Xe − x σ . Moreover,after simple analysis, we find that [ J + ] , [ J − ] are analytic for λ ∈ D + and continuous for λ ∈ D + S R S i R . Similarly[ J − ] , [ J + ] are analytic for λ ∈ D − and continuous for λ ∈ D − S R S i R , here D + = ( λ | arg λ ∈ (cid:18) , π (cid:19) [ π, π !) D − = ( λ | arg λ ∈ (cid:18) π , π (cid:19) [ π , π !) . Secondly, let us investigate the properties of J ± . We deduce the determinants of J ± are constants for all x on thebasis of the Abel’s identity and Tr( U ) =
0. Furthermore, due to the boundary conditions Eq.(2.9a),(2.9b), we havedet J ± = , λ ∈ R [ i R . (2.13)3y introducing a new function E ( x , λ ) = e i ( λ − ) σ x , we find that spectral problem Eq.(2.5a) exists two fundamentalmatrix solutions J + E and J − E , which are linearly related by a 2 × S ( λ ) J − E = J + E · S ( λ ) , λ ∈ R [ i R . (2.14)From Eq.(2.13) and (2.14), we know that det S ( λ ) = . (2.15)Furthermore, let x go to + ∞ , the 2 × S ( λ ) is given as S ( λ ) = lim x → + ∞ E − J − E = I + Z + ∞−∞ E − U J − Ed ξ, λ ∈ R [ i R . From the analytic property of J − , we find that s can be analytically extended to D + , s allows analytic extensionsto D − . Generally speaking, s , s can only be defined in the R S i R .In what follows, we shall construct a RH problem for the CLL-NLS equation. Firstly, using the analytic propertiesof J ± , we define a new Jost solution P = P ( x , λ ) as P = ([ J + ] , [ J − ] ) . (2.16)which is obviously analytic for λ ∈ D + . In addition, from Eqs. (2.10) and (2.11), we have P → I , λ → + ∞ , λ ∈ D + . (2.17)Next, we introduce the limit of P P + = lim λ → Γ P Γ = R [ i R . (2.18)From (2.14) and (2.16), we can get P + = J + e i (2 λ − x s s . (2.19)To obtain the analytic counterpart of P + in D − , denoted by P , we consider the inverse matrices J − ± defined as J − + = [ J − + ] [ J − + ] , J − − = [ J − − ] [ J − − ] , (2.20)here [ J − ± ] n ( n = ,
2) denote the n -th row vector of J − ± . Then we can see that [ J − + ] , [ J − − ] are analytic for λ ∈ D − andcontinuous for λ ∈ D − S R S i R , whereas [ J − − ] , [ J − + ] are analytic for λ ∈ D + and continuous for λ ∈ D + S R S i R .Obviously, J − ± satisfy the adjoint scattering equation of Eq.(2.5a): K x = i ( λ −
12 )[ σ, K ] − KU . (2.21)In addition, it is not di ffi cult to find that the inverse matrices J − + and J − − satisfy the following boundary conditions. J − + → I , x → −∞ , (2.22a) J − − → I , x → + ∞ . (2.22b)4aking the similar procedure as above, a matrix function P which is analytic in D − P = [ J − + ] [ J − − ] . (2.23)and P − P − = lim λ → Γ P Γ = R [ i R . (2.24)are expressed. Moreover, we can get that P → I , λ → −∞ , λ ∈ D − . (2.25)and P − = e − i (2 λ − x r r J − + . (2.26)with R ( λ ) ≡ ( r k j ) × = S − ( λ ). Similar to the scattering coe ffi cients s i j above, it is easy to know that r allows ananalytic extension to D + and r is analytically extendible to D − . Generally speaking, r , r can only be defined inthe R S i R . In addition, from (2.14), we get E − J − − = R ( λ ) · E − J − + , λ ∈ R [ i R . (2.27)Summarizing the above the results, we find that two matrix functions P + and P − which are analytic in D + and D − ,respectively, are related by P − ( x , λ ) P + ( x , λ ) = s e i (2 λ − x r e − i (2 λ − x , λ ∈ R [ i R . (2.28)Eq.(2.28) is just the RH problem for the CLL-NLS equation. In order to obtain solution for this RH problem, weassume that the RH problem is non-regular when det P and det P can be zero for λ k ∈ D + and λ k ∈ D − , respectively.1 ≤ k ≤ N . where N is the number of these zeros. Recalling the definitions of P and P , we can see thatdet P ( x , λ ) = s ( λ ) , λ ∈ D + , (2.29)det P ( x , λ ) = r ( λ ) , λ ∈ D − . (2.30)To specify these zeros, we first can take use of a symmetry relation for U U † = σ U σ, where the superscript † means the Hermitian of a matrix. Therefore from Eq. (2.21), we arrive at J †± ( x , t , λ ∗ ) = σ J − ± ( x , t , λ ) σ, (2.31)Then from (2.14), we also gain S † ( λ ∗ ) = σ S − ( λ ) σ, (2.32)which implies the following relations r ( λ ) = s ∗ ( λ ∗ ) λ ∈ D + , (2.33)5 ( λ ) = s ∗ ( λ ∗ ) λ ∈ D − , (2.34) r ( λ ) = − s ∗ ( λ ∗ ) λ ∈ R [ i R , (2.35) r ( λ ) = − s ∗ ( λ ∗ ) λ ∈ R [ i R . (2.36)Moreover from Eq. (2.32) and the definitions of P , P , we point out that the analytic solutions P , P satisfy theinvolution property P † ( x , λ ∗ ) = σ P ( x , λ ) σ, λ ∈ D − (2.37)The similar analysis shows that the potential matrix Q also satisfies another symmetry relations Q = − σ Q σ It follows that J ± ( − λ ) = σ J ± ( λ ) σ. (2.38)and P ( − λ ) = σ P ( λ ) σ. (2.39)thus s ( − λ ) = − s ( λ ) λ ∈ D − , (2.40) s ( − λ ) = − s ( λ ) λ ∈ D + , (2.41) s ( − λ ) = − s ( λ ) λ ∈ R [ i R , (2.42) s ( − λ ) = − s ( λ ) λ ∈ R [ i R . (2.43)Therefore, from (2.34), we find that if λ j is a zero of det P , then ˆ λ j = λ ∗ j is a zero of det P . Moreover, in view of(2.41), we know that − λ is also a zero of det P . Hence we suppose that det P has 2 N simple zeros { λ j } N satisfying λ N + l = − λ l (1 ≤ l ≤ N ), which all lie in D + . From the zeros of det P , we see that det P possesses 2 N simple zeros { ˆ λ j } N satisfying ˆ λ j = λ ∗ j , (1 ≤ j ≤ N ), which are all in D − . Obviously the zeros of det P and det P always appearin quadruples. To solve the RH problem, we need the scattering data including the continuous scattering data { s , s } and the discrete scattering data { λ j , ˆ λ j , v j , ˆ v j } which a single column vector v j and row vector ˆ v j satisfying P ( λ j ) v j = , ˆ v j P ( ˆ λ j ) = . (2.44)On the one hand, by taking the Hermitian conjugate of P ( λ j ) v j =
0, we can construct the relationship betweeneach pair of v j and ˆ v j . v j = σ v j − N N + ≤ j ≤ N , ˆ v j = v † j σ ≤ j ≤ N . (2.45)On the other hand, in order to obtain the spatial evolutions for vectors v j ( x ), taking the x -derivative to equation P v j = v j = e i ( λ j − ) σ x v j ≤ j ≤ N , (2.46)6here v j = v j | x = ffi cient s = s = P ( λ ) = I − N X k = N X j = v k ( M − ) k j ˆ v j λ − ˆ λ j , (2.47a) P ( λ ) = I + N X k = N X j = v k ( M − ) k j ˆ v j λ − λ k . (2.47b)where M = ( m k j ) N × N is a matrix whose entries are m k j = ˆ v k v j λ j − ˆ λ k , ≤ k , j ≤ N . (2.48)
3. Inverse scattering transform
In this section, with the help of P in (2.47a), we can write out explicitly the potential Q . Owing to P ( λ ) is thesolution of spectral problem (2.5a), we assume that the asymptotic expansion of P ( λ ) at large λ as P = I + P (1)1 λ + P (2)1 λ + O ( λ − ) λ → ∞ , (3.1)Then by substituting the above expansion into (2.5a) and comparing O ( λ ) terms,we obtains Q = − i [ σ, P (1)1 ] = − i ( P (1)1 ) i ( P (1)1 ) . (3.2)which implies that r can be reconstructed as r = − i ( P (1)1 ) . (3.3)where P (1)1 = ( P (1)1 ) × and ( P (1)1 ) i j is the ( i ; j )-entry of P (1)1 , i , j = ,
2. Here, the matrix function P (1)1 can be foundfrom (2.47a) P (1)1 = N X k = N X j = v k ( M − ) k j ˆ v j . (3.4)
4. The soliton solutions
To derive the solutions for the CLL-NLS equation, we also need the scattering data at time t , which need investigatethe time evolution of scattering data. In fact, by using (2.5b) and (2.14), making the limit x → + ∞ , and taking intoaccount the boundary condition (2.9a) for J + as well as V → x → ±∞ , we arrive at s | t = s | t = , s | t = − i ( λ −
12 ) s , s | t = − i ( λ −
12 ) s , d λ j dt = , v j | t = − i ( λ −
12 ) v j . (4.1)Combining (2.45) with (2.46), we can derive the column vectors v j and the row vector ˆ v j explicitly, v j = e θ j σ v j , ≤ j ≤ N σ e θ j − N σ v j − N , , N + ≤ j ≤ N (4.2)7nd ˆ v j = v † j e θ ∗ j σ σ ≤ j ≤ Nv † j − N , e θ ∗ j − N σ , N + ≤ j ≤ N (4.3)where θ j = i ( λ j − ) x − i ( λ j − ) t ( λ j ∈ D + ), v j is a constant vector.we have chosen v j = ( c j , T , it follows from (3.4) that the N-soliton solutions for the CLL-NLS equation reads r = i N X k = N X j = c k e θ k − θ ∗ j ( M − ) k j + i N X k = N + N X j = c k − N e θ k − N − θ ∗ j ( M − ) k j − i N X k = N X j = N + c k e θ k − θ ∗ j − N ( M − ) k j − i N X k = N + N X j = N + c k − N e θ k − N − θ ∗ j − N ( M − ) k j . (4.4)and M = ( m k j ) N × N is given by m k j = c j c ∗ k e θ ∗ k + θ j − e − ( θ ∗ k + θ j ) λ ∗ k − λ j , ≤ k , j ≤ N , c j − N c ∗ k e θ ∗ k + θ j − N + e − ( θ ∗ k + θ j − N ) λ ∗ k − λ j , ≤ k ≤ N , N + ≤ j ≤ N , c j c ∗ k − N e θ ∗ k − N + θ j + e − ( θ ∗ k − N + θ j ) − λ ∗ k − N − λ j , ≤ j ≤ N , N + ≤ k ≤ N , c j − N c ∗ k − N e θ ∗ k − N + θ j − N − e − ( θ ∗ k − N + θ j − N ) − λ ∗ k − N + λ j − N , N + ≤ k , j ≤ N . (4.5)with θ j = i ( λ j − ) x − i ( λ j − ) t ( λ j ∈ D + )The simplest situation occurs when N = r = − ic e θ − θ ∗ ( m + m − m − m ) detM (4.6)where M = ( m k j ) × is given by m = | c | (cid:16) e θ ∗ + θ + ξ − e − θ ∗ − θ − ξ (cid:17) λ ∗ − λ , m = | c | (cid:16) e θ ∗ + θ + ξ + e − θ ∗ − θ − ξ (cid:17) λ ∗ + λ , m = | c | (cid:16) e θ ∗ + θ + ξ + e − θ ∗ − θ − ξ (cid:17) − λ ∗ − λ , m = | c | (cid:16) e θ ∗ + θ + ξ − e − θ ∗ − θ − ξ (cid:17) − λ ∗ + λ . Letting λ = λ + i λ , | c | = e ξ , then the single-soliton solution can be written as r = λ λ e iY ( λ sinhX + i λ coshX ) λ cosh X + λ sinh X . (4.7)with X = λ λ ( λ − λ ) t − λ λ x − λ λ t + ln | c | , Y = λ − λ ) x − λ − λ λ + λ − λ − λ ) t . In order to understand the properties of the resulting soliton solution, we can take the model | r | = | λ λ | q λ cosh X + λ sinh X . (4.8)8ow we investigate the case for N = r = − i X k = X j = c k e θ k − θ ∗ j ( M − ) k j − i X k = X j = c k − e θ k − − θ ∗ j ( M − ) k j + i X k = X j = c k e θ k − θ ∗ j − ( M − ) k j + i X k = X j = c k − e θ k − − θ ∗ j − ( M − ) k j . (4.9)and M = ( m k j ) × is given by m = c c ∗ e θ ∗ + θ − e − θ ∗ − θ λ ∗ − λ , m = c c ∗ e θ ∗ + θ − e − θ ∗ − θ λ ∗ − λ , m = c c ∗ e θ ∗ + θ + e − θ ∗ − θ λ ∗ + λ , m = c ∗ c e θ ∗ + θ + e − θ ∗ − θ λ ∗ + λ , m = c c ∗ e θ ∗ + θ − e − θ ∗ − θ λ ∗ − λ , m = c c ∗ e θ ∗ + θ − e − θ ∗ − θ λ ∗ − λ , m = c c ∗ e θ ∗ + θ + e − θ ∗ − θ λ ∗ + λ , m = c c ∗ e θ ∗ + θ + e − θ ∗ − θ λ ∗ + λ , m = c c ∗ e θ ∗ + θ + e − θ ∗ − θ − λ ∗ − λ , m = c c ∗ e θ ∗ + θ + e − θ ∗ − θ − λ ∗ − λ , m = c c ∗ e θ ∗ + θ − e − θ ∗ − θ − λ ∗ + λ , m = c c ∗ e θ ∗ + θ − e − θ ∗ − θ − λ ∗ + λ , m = c c ∗ e θ ∗ + θ + e − θ ∗ − θ − λ ∗ − λ , m = c c ∗ e θ ∗ + θ + e − θ ∗ − θ − λ ∗ − λ , m = c c ∗ e θ ∗ + θ − e − θ ∗ − θ − λ ∗ + λ , m = c c ∗ e θ ∗ + θ + e − θ ∗ − θ − λ ∗ + λ . (4.10)Further, we can also write the obtained N-soliton solutions (4.4) into the form of the determinant ratio r = i det e M detM . (4.11)where e M is e M = f ˆ g M . (4.12)where the matrix M is defined by Eq.(4.5), and f = ( v , v · · · v N , ) , ˆ g = (ˆ v , ˆ v · · · ˆ v N , ) T
5. Conclusions
In this work, by applying RH method, we establish the N-soliton solutions for the CLL-NLS equation. First ofall, the Lax pair of the coupled CLL-NLS equation are transformed to obtain the corresponding Jost solution. Thenwe study spectrum analysis and construct the particular RH problem, which is non-regular case. Then we derive thescattering data including the continuous scattering data { s , s } and the discrete scattering data { λ j , ˆ λ j , v j , ˆ v j } . Withaid of reconstructing the potential, we solve the N-soliton solutions for the CLL-NLS. Finally we obtain a simple andcompact N-soliton solutions formula. 9 cknowledgements This work is supported by the National Natural Science Foundation of China under the grant No. 11601055,Natural Science Foundation of Anhui Province under the grant No. 1408085QA06, Natural Science Research Projectsof Anhui Province under Grant Nos. KJ2019A0637 and gxyq2019096.
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