Riemann-Hilbert approach and N-soliton formula for a higher-order Chen-Lee-Liu equation

Juan Hu; Jian Xu; Guo-Fu Yu

doi:10.1080/14029251.2018.1503443

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Volume 25, Issue 4, July 2018, Pages 633 - 649

Riemann-Hilbert approach and N-soliton formula for a higher-order Chen-Lee-Liu equation

Authors

Juan Hu

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, P.R. China,hujuan@lsec.cc.ac.cn

Jian Xu

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, P.R. China,jianxu@usst.edu.cn

Guo-Fu Yu^*

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China,gfyu@sjtu.edu.cn

^*Corresponding author.

Corresponding Author

Guo-Fu Yu

Received 5 December 2017, Accepted 18 May 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2018.1503443 How to use a DOI?
Keywords: Higher-order Chen-Lee-Liu equation; Riemann-Hilbert method; N-soliton
Abstract: We consider a higher-order Chen-Lee-Liu (CLL) equation with third order dispersion and quintic nonlinearity terms. In the framework of the Riemann-Hilbert method, we obtain the compact N-soliton formula expressed by determinants. Based on the determinant solution, some properties for single soliton and asymptotic analysis of N-soliton solution are explored. The simple elastic interaction of N solitons is confirmed.
Copyright: © 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

The nonlinear Schrödinger equation (NLS) is an important integrable model that governs weakly nonlinear and dispersive wave packets in one-dimensional physical systems. It plays an important role in wide range of physical subjects, such as nonlinear water waves, nonlinear optics and plasma physics. The NLS equation is low order approximation model of nonlinear effects in optical fibers. To get a more accurate approximation of the higher-order nonlinear effects, a natural approach is to introduce additional higher-order nonlinear terms in the model. Hirota equation, Kundu-Eackhuas equation, and Lakshmanan-Porsezian-Daniel equation are all extension of NLS equation with higher-order dispersion and nonlinear terms.

To study the effect of higher-order perturbations, various modifications and generalizations of the NLS equation have been proposed. Among them, there are three celebrated equations with derivative-type nonlinearities, which are called the derivative NLS (DNLS) equation. The first two DNLS equations are analogues of the NLS equation with second order dispersion and cubic nonlinearity and the third DNLS equation posses second order dispersion and quintic nonlinear term.

The first DNLS equation is the Kaup-Newell (KN) equation [11],

iut+uxx-i(|u|2u)x=0, (1.1)

which is a canonical dispersive equation derived from the Magneto-hydrodynamic equations in the presence of the Hall effect and usually called DNLS (I). Under the gauge transformation,

q(x,t)=u(x,t)exp(i2∫-∞x|u(y,t)|2dy),

the KN equation (1.1) becomes

iqt+qxx+i|q|2qx=0, (1.2)

which appears in optical models of ultrashort pulses and is also referred to as the Chen-Lee-Liu (CLL) equation [2] and called DNLS (II).

The third one takes the form

iqt+qxx-iq2qx*+12|q|4q=0,

which is called the Gerdjikov-Ivanov (GI) equation or DNLS (III) [7]. The unified expression of KN, CLL and GI equations was presented in [4]

Like the NLS equation, the DNLS (II) equation is also a real physical model in optics. In 2007, Moses et al [14] proved optical pulse propagation involving self-steepening without self-phase-modulation. This experiment provide the first experimental evidence of the DNLS (II) equation. The importance of the higher-order nonlinear effects in nonlinear optics and other fields motivates us to consider an integrable model that possesses third dispersion and quintic nonlinearity.

In this paper, we consider the higher-order generalized CLL equation with third dispersion and quintic nonlinear term,

qt+qxxx+32i|q|2qxx-34|q|4qx+32iqx2q*=0. (1.3)

This equation can be derived from the generalized KN hierarchy under n = 2 and proper parameter. The Liouville integrability and multi-Hamiltonian structure for the higher-order CLL equation (1.3) are investigated in [4]. The higher-order CLL equation is also Lax integrable with the linear spectral problem

Yx=UY=(-iσ3λ2+σ3Qλ+14iσ3Q2)Y, (1.4)

Yt=VY=(-4iσ3λ6+4σ3Qλ5+2iσ3Q2λ4+2Z0λ3+Z1λ2+Z2λ+14Z3)Y, (1.5)

where

σ3=(100-1), Q=(0qr0)

Z0=iQx-12σ3Q3, Z1=[Q,Qx]-12iσ3Q4,Z2=Qxxσ3+12iQQxQ-32iQ2Qx+14σ3Q5,Z3=14iσ3Q6+32Q2[Qx,Q]-iσ3(QxxQ+QQxx-Qx2).

Here λ is a spectral parameter. The superscript “*” represents complex conjugate and [A, B] = A B − B A, i.e., commutator. When r = q^*, the compatibility condition yields the zero-curvature equation, U_t−V_x−[U, V] = 0, which generates equation (1.3). The rogue wave solutions of the higher-order CLL equation were studied in [21] by using Darboux transformation method.

The topic of the singular Riemann-Hilbert problem was very well researched between 1979 and 1984. There are numerous papers on this topic especially in connection with integrable systems. First, there are papers by Zakharov and his group including the paper by Zakharov and Mikhailov who introduced this technique to study integrable relativistic models [20]. Then Harnad and his collaborators contributed several papers on this topic, see [9]. All these papers by Zakharov and Mikhailov as well as much more general set up of Harnad et al., focus on Lax pairs with rational dependence on the spectral parameter, rather than on the polynomial case. Hirota’s τ function method and then the breakthrough by the Kyoto school of interpreting the τ function in representation theoretic theorems [3,10] achieve success in the polynomial case. The NLS hierarchy and many more are covered by the theory of free two-components fermions. The main advantage of the representation theoretic construction is that the soliton τ function is constructed for the whole hierarchy, and the soliton solution field u is known for the whole hierarchy.

In this paper, based on the Riemann-Hilbert method for the higher-order CLL equation (1.3), we will present its N-soliton solutions with vanishing boundary condition in the compact determinant form and give an asymptotic analysis for these solutions. The Riemann-Hilbert method streamlines the inverse scattering transformation (IST) method and could be regarded as simpler version of IST [1,15,18]. Recently, the Riemann-Hilbert method have been widely adopted to solve nonlinear integrable models [5,6,8,12,16,17,19,22]

The paper is organized as follows. In Section 2, we present the construction of Riemann-Hilbert problem for the higher-order CLL equation (1.3). In Section 3, we solve the non-regular and regular Riemann-Hilbert problems. In Section 4, we construct the N-soliton solution to the higher-order CLL equation in the determinant form and discuss the asymptotic behaviour of N-soliton interactions. The Section 5 is devoted to conclusion and discussion.

2. The Riemann-Hilbert problem for higher-order CLL

We first assume the vanishing boundary condition,

Q→0, as x→∞.

Note that when x → ∞, from the spectral problem (1.4), we have asymptotic behaviour Y∝e-iλ2σ3x-4iλ6σ3t . Thus it will be convenient to express Y as

Y=Φe-iλ2σ3x-4iλ6σ3t, (2.1)

so that the new matrix function Φ is x–independent at infinity. Inserting (2.1) into the Lax pair (1.4)-(1.5), we can rewrite the Lax pair in the form

Φx+iλ2[σ3,Φ]=(σ3Qλ+14iσ3Q2)Φ, (2.2)

Φt+4iλ6[σ3,Φ]=(4σ3Qλ5+2iσ3Q2λ4+2Z0λ3+Z1λ2+Z2λ+14Z3)Φ. (2.3)

In order to formulate a Riemann-Hilbert problem for the solution of the inverse spectral problem, we seek solutions of the spectral problem which approach the 2 × 2 identity matrix as λ → ∞

Consider a solution of (2.2)-(2.3) of the form

Φ=D+Φ1λ+Φ2λ2+Φ3λ3+𝒪(1λ4),

where D and Φ_k(k = 1,2,3) are independent of the spectral parameter λ. Substituting the above expansion into the the Lax pair (2.2)-(2.3) and comparing the same order of λ, we find that D is diagonal and satisfies

Dx=-i4qrσ3D, (2.4)

Dt=-14Z3D, (2.5)

with Z3=(i4q3r3+32qr(qxr-qrx)+iqxrx-iqxxr-iqrxx)σ3 .

Note that the CLL equation admits the conservation law

(qr)t=(qxrx-32iqr(qxr-qrx)+14q3r3-qrxx-qrxx)x

Thus, the two eqs. (2.4) and (2.5) for D are consistent and are both satisfied if we define

D=exp(-i4∫-∞xq(x′,t)r(x′,t)dx′σ3).

We introduce a new function J by DJ = Φ. According to the asymptotic behaviour of Φ, it’s easy to see that

J=𝕀+𝒪(1λ), λ→∞

The Lax pair of eq. (2.2)-(2.3) becomes

Jx+iλ2[σ3,J]=U^J, (2.6)

Jt+4iλ6[σ3,J]=V^J, (2.7)

where

U^=exp(i4∫-∞xq(x′,t)r(x′,t)dx′σ^3)(λσ3Q+i2σ3Q2),

V^=exp(i4∫-∞xq(x′,t)r(x′,t)dx′σ^3)(4λ5σ3Q+2iλ4σ3Q2+2λ3Z0+λ2Z1+λZ2+Z32)

Here the notation σ^3 is defined by σ^3A=[σ^3,A] , and thus exp(σ^3)A=expσ3Aexp(-σ3) .

The Jost solutions J_± of the spectral equation (2.6) obey the constant asymptotic condition

J±→J+0, x→±∞, (2.8)

respectively, and they satisfy the following integral equation

J+(x;λ)=J+0-∫x∞e-iλ2(x-y)σ3U^(y)J+(y)eiλ2(x-y)σ3dy,J-(x;λ)=J-0+∫-∞xe-iλ2(x-y)σ3U^(y)J-(y)eiλ2(x-y)σ3dy,

where J-0=𝕀 and J+0=exp(i4∫-∞∞qrdxσ3) . From the above Volterra type integral equations, we can prove the existence and uniqueness of the Jost solutions through standard iteration method. We partition J_± into columns as J±=(J±(1),J±(2)) , then J-(1),J+(2) are analytic for λ∈𝔺+ and continuous for λ∈𝔺+∪𝕉∪i𝕉 , while the columns J+(1),J-(2) are analytic for λ∈𝔺- and continuous for λ∈𝔺-∪𝕉∪i𝕉 , where

𝔺+={λ∣argλ∈(0,π2)∪(π,32π)}, 𝔺-={λ∣argλ∈(π2,π)∪(32π,2π)}

The quarter 𝔺+ and 𝔺- are displayed in Fig. 1.

We define E=e-iλ2σ3x , then J₊E and J₋E are both solutions to linear equation (1.4). They are dependent and linearly related by a scattering matrix S(λ) as

J-E=J+ES(λ), λ∈𝕉∪iR. (2.9)

From the Abel’s identity and since the trace of Q satisfies tr(Q) = 0, the determinants of J_± are constants for all x. Considering the boundary conditions (2.8), we have

detJ±=1.

Thus we can derive det S(λ) = 1 according to the relation (2.9). Furthermore, from (2.9), we have

S(λ)=(sij)2×2=limx→+∞E-1J-E=I+∫-∞∞eiλ2σ3yU^J-e-iλ2σ3ydy, λ∈𝕉∪i𝕉.

Based on the analytic property of J₋, s₁₁ is analytic extension to 𝔺+ and s₂₂ is analytic in 𝔺- . We define a new Jost solution P⁺ is

P+=(J-(1),J+(2))=J-H1+J+H2=J+E(s110s211)E-1, (2.10)

with H₁ = diag{1,0} and H₂ = diag{0,1}. P ⁺ is analytic in 𝔺+ with the asymptotic behavior

P+(x,λ)→I, λ∈𝔺+→∞. (2.11)

In order to obtain the behavior of Jost solution P ⁺ for large λ, we use the following expansion

P+=I+P1+λ+P2+λ2+𝒪(λ-3)

Substituting the expansion into the spectral problem (2.6) and compare the coefficients of λ, we have

i[σ3,P1+]=exp(i4∫-∞xqrdx′σ3)σ3Qexp(-i4∫-∞xqrdx′σ3),

which leads to

q=2i(P1+)12exp(-i2∫-∞xqrdx′), r=2i(P1+)21exp(i2∫-∞xqrdx′). (2.12)

We consider the adjoint scattering problem of (2.2)

Kx+iλ2[σ3,K]=-KU^

It is easy to verify that J±-1 satisfy the above adjoint equation together with the boundary condition J±-1→I when x → ±∞, respectively. Denote the k–th row vector of J±-1 as (J±-1)(k) for convenience and define

P-=((J--1)(1)(J+-1)(2))

One can check that P⁻ is analytic in 𝔺- and tends to identity I when λ → ∞. If we use the notation R(λ) = S⁻¹(λ), then

J--1=ERE-1J+-1

and

P-=H1J--+H2J+-1=E(s22-s1201)E-1J+-1. (2.13)

P⁻ is analytic in 𝔺- with the asymptotic behavior

P-(x,λ)→I, λ∈𝔺-→∞. (2.14)

Summarizing the above results, we have constructed two matrix functions P ⁺ and P^−, that are analytic in the complex region 𝔺+ and 𝔺- , respectively. Thus we have the Riemann-Hilbert problem by P ⁺, P⁻ as

P-P+=G(x,λ)=E(1-s12s211)E-1, λ∈𝕉∪i𝕉. (2.15)

We now investigate the evolutions of the scattering coefficients s_ij. The relation (2.9) and (2.8) leads to

St=limx→+∞E-1J-,tE

Then according to the evolution property (2.3) and Q → 0 as |x| → ∞, we have

St+4iλ6[σ3,S]=0,

and the evolutions of the entries of the scattering matrix S satisfy

s11,t=s22,t=0s12,t+8iλ6s12=0, s21t-8iλ6s12=0.

Thus the elements s₁₁, s₂₂ are time-independent and

s12(λ,t)=s12(λ,0)e-8iλ6t, s21(λ,t)=s21(λ,0)e8iλ6t.

3. Solutions for the Riemann-Hilbert problem

Note that the transpose and conjugate of U^ satisfies the relation U^†=-U^, we can verify that

J†(x,λ*)=J-1(x,λ), λ∈𝕉∪𝕉

Here † denotes the operation of transpose and complex conjugate. From the definition of P ⁺ and P^−, we have the relation

(P+)†(λ*)=P-(λ) λ∈𝕉∪i𝕉. (3.1)

It follows from the relation (2.9) that

S†(λ*)=S-1(λ), λ∈𝕉∪iR.

which implies the following relations

s11*(λ*)=s22(λ), s22*(λ*)=s11(λ), s12*(λ*)=-s21(λ), s21*(λ*)=-s12(λ), λ∈𝕉∪iR

Furthermore, from the symmetric property σ₃Q σ₃ = −Q and σ₃Q² σ₃ = Q², we conclude that

J(x,t,-λ)=σ3J(x,t,λ)σ3

Thus from the definition of P^±(λ), we have the symmetry relation

P±(x,t,-λ)=σ3P±(x,t,λ)σ3. (3.2)

Applying this reduction to (2.9), then it readily implies

S(-λ)=σ3S(λ)σ3

Thus s₁₁(λ) is an odd function, and each zero λ_k of s₁₁ is accompanied with zero −λ_k. For simplicity, we assume all zeros are simple and then the kernels of P+(λk) and P-(λ¯k) contain only a single column vector |v_k〉 and row vector 〈v_k|, respectively,

P+(λk)|vk〉=0, 〈vk|P-(λ¯k)=0, k=1,2,…,N. (3.3)

Here |v_k〉 = 〈. v_k|^†. From the relation (3.1), we have

λ¯k=λk*. (3.4)

Differentiating both sides of the first equation of (3.3) with respect to x and t, and recalling the Lax pair (1.4)-(1.5), we have

P+(λk;x)(d|vk〉dx+iλ2σ3|vk〉)=0, P+(λk;x)(d|vk〉dt+4iλ6σ3|vk〉)=0

It concludes that

|vk〉=e-iλk2σ3x-4iλk6σ3tt|vk,0〉e∫x0xak(s)ds+∫t0tbk(s)ds, (3.5)

where a_k(x) and b_k(t) are two scalar functions.

Based on the above analysis, we have the following theorem for the solution to the non-regular Riemann-Hilbert problem.

Theorem 1.

The solution to a non-regular Riemann-Hilbert problem (2.15) with simple zeros under the canonical normalized condition (2.11) and (2.14) is

P+(λ)=P+(λ)T(λ), P-(λ)=T-1(λ)P-(λ),

where

T(λ)=∏k=1NTk(λk)=∏k=1N(I+Akλ-λk*-σ3Akσ3λ+λk*),T-1(λ)=∏k=1NTk-1(λk)=∏k=1N(I+Ak†λ-λk-σ3Ak†σ3λ+λk),Ak=λk-λk*2(αk00αk*)|wk〉〈wk|, λk-1=〈wk|(λk*00λk)|wk〉,

P-(λ)P+(λ)=T(λ)G(λ)T-1(λ), λ∈𝕉∪i𝕉, (3.6)

where P_± are analytic in 𝔺±, respectively, and P_± → I as λ → ∞.

Proof. From the symmetry relation (3.2), we can suppose that simple zeros of det P ⁺ (λ) are {±λk∈𝔺+,1≤k≤N} . The symmetry relation (3.4) implies that {±λk*} are zeros of detP⁻(λ). Both ker(P+(±λk)) and ker(P-(±λk*)) are one-dimensional and the kernel space is spanned by single column vector |v_k〉 and single row vector 〈v_k|, respectively, i.e,

P+(λk)|vk〉=0, 〈vk|P-(λk)=0.

From the definition of A₁, we construct a meromorphic matrix function

T1(λ)=I+A1λ-λ1*-σ3A1σ3λ+λ1*

with simple poles at λ=±λ1*∈𝔺- . One can check through direct computation that

T1-1(λ)=I+A1†λ-λ1-σ3A1†σ3λ+λ1, detT1(λ)=λ2-λ12λ2-λ1*2

Since ±λ₁ are simple zeros of detP^±(λ), the matrix P⁺(λ)T₁⁻¹(λ) is non-singular at λ = ±λ₁, and T₁(λ)P⁻(λ) is non-singular at λ = ±λ^*. In general, near the point λ_k, we have det P⁺(λ) ∼ λ − λ_k and det P⁺(λ) ∼ λ + λ_k near the point − λ_k. Similarly, detP-(λ)~λ±λk* near the point ±λk* by the involution relations (3.3) and (3.4). Let T_k(λ) be a matrix with determinant

detTk(λ)=λ2-λk2λ2-λ*2,

then detP+(λ)Tk-1(λ)≠0 at points ±λ_k and detT_k(λ)P⁻(λ) ≠ 0 at points ±λk* . We introduce

T(λ)=TN(λ)TN-1(λ)⋯T1(λ), T-1(λ)=T1-1(λ)T2-1(λ)⋯TN-1(λ),

that accumulates all zeros of the Riemann-Hilbert problem. We can cancel all zeros ±λ_j and ±λj* , (j = 1,2, …, N) of detP^±(λ) by

P+(λ)=P+(λ)T-1(λ), P-(λ)=T(λ)P-(λ). (3.7)

Substituting P^±(λ) into (2.15), we obtain a normalized regular Riemann-Hilbert problem (3.6).

From the above properties of T_k(λ), we could readily obtain the explicit expression for the matrix T_k(λ) (cf. Ref. [13])

Tk(λ)=I+Akλ-λk*-σ3Akσ3λ+λk*, Tk-1(λ)=I+Ak†λ-λk-σ3Ak†σ3λ+λk, k=1,2,…,N.

4. Soliton solutions

Based on the above analysis, we are ready to construct N-soliton solutions for the higher-order CLL equation (1.3). From the asymptotic expansion of Jost solutions as λ → ∞, the potential q can be expressed as (2.12). From the expression of Tk-1(λ) and A_k, there exists some column vector |z_k〉 such that

Resλ=λkT-1(λ)=limλ→λk(λ-λk)T-1(λ)=|vk〉〈zk|, 〈zk|=|zk〉†.

Meanwhile, considering the symmetry relation T_k(−λ) = σ₃T_k(λ)σ₃, T(λ) and T⁻¹(λ) have compact form

T(λ)=I+∑k=1N(Bkλ-λk*-σ3Bkσ3λ+λk*), (4.1)

T-1(λ)=I+∑k=1N(Bk†λ-λk-σ3Bk†σ3λ+λk), (4.2)

with B_k = |z_k〉〈z_k|. According to the identity T(λ)T⁻¹(λ) = T⁻¹(λ)T(λ) = I, we have

Resλ=λjT(λ)T-1(λ)=0

We arrive at

T(λj)Bj†=0,

and it yields

[I+∑k=1N(Bkλj-λk*-σ3Bkσ3λj+λk*)]|vj〉=0, j=1,2,…,N,

or equivalently

Solving these linear algebraic equations, we obtain

|zl〉1=∑j=1NM-1(l,j)|vj〉1,|zl〉2=∑j=1NM^-1(l,j)|vj〉2,

where |z_l〉_k denotes the k – th element of |z_l〉, and the entries of matrices M and M^ are defined as

Mjk=〈vk|σ3|vj〉λj+λk*-〈vk∣vj〉λj-λk*, M^jk=-〈vk|σ3|vj〉λj+λk*-〈vk∣vj〉λj-λk*, j,k=1,2,…,N.

According to the Plemelj formula, the solution to the Riemann-Hilbert problem (3.6) is

(P+(λ))-1=I+12πi∫ΓT(ξ)(I-G)T-1(ξ)(P+(ξ))-1ξ-λdξ, λ∈𝔺+

and thus when λ → ∞, we have

P+(λ)→I+12πiλ∫ΓT(ξ)(I-G)T-1(ξ)(P+(ξ))-1dξ

From (4.1), we find T(λ) has expansion

T(λ)→I+1λ∑k=1N(|zk〉〈vk|-σ3|zk〉〈vk|σ3), λ→∞

Thus, as λ → ∞, P ⁺ (λ) = P ₊ (λ)T(λ) has the expansion formula

P+=I+P1+λ+P2+λ2+𝒪(1λ3)

with

P1+=∑k=1N(|zk〉〈vk|-σ3|zk〉〈vk|σ3)+12πi∫ΓT(ξ)(I-G)T-1(ξ)(P+(ξ))-1dξ

In the reflection-less case, that is G = I, we obtain N-soliton solutions from (2.12). The formula for N-soliton solution is

q=2i(P1+)12exp(2i∫-∞x(P1+)12(P1+)21dx′), (4.3)

r=2i(P1+)21exp(-2i∫-∞x(P1+)12(P1+)21dx′), (4.4)

(P1+)12=2∑k=1N|zk〉1vk|2=-2detM1detM〉, (4.5)

(P1+)21=2∑k=1N|zk〉2vk|1=-2detM2detM^〉, (4.6)

where the matrix M₁ and M₂ have the form

M1=(M11⋯M1N|v1〉1⋮⋯⋮⋮MN1⋯MNN〈vN1v1|2〉⋯vN|2〉0), M2=(M^11⋯M^1N〈v12⋮⋯⋮⋮M^N1⋯M^NN〈vN2v1|1〉⋯vN|1〉0).

To obtain the explicit formulae for N-soliton solutions, we take

|vk〉=(cke-θkeθk),

where θk=i(λk2x+4λk6t) , and λ_k, c_k are arbitrary constants. Then the general N-soliton solution to the higher-order CLL equation (1.3) is represented by

q=-4idetM^1detMexp(8i∫-∞xdet(M^1M^2)det(MM^)dx′)

with

M^1=(M11⋯M1Nc1e-θ1⋮⋯⋮⋮MN1⋯MNNcNe-θNeθ1*⋯eθN*0), M^2=(M^11⋯M^1Neθ1⋮⋯⋮⋮M^N1⋯M^NNeθNc1*e-θ1*⋯cN*e-θN*0)

and

Mj,k=-2λk*λj2-λk*2cjck*e-θj-θk*-2λjλj2-λk*2eθj+θk*, M^jk=-2λjλj2-λk*2cjck*e-θj-θk*-2λk*λj2-λk*2eθj+θk*.

In what follows, we shall investigate the properties of the single, two- and N-soliton solutions.

4.1. One and two-soliton solution

To obtain one-soliton solution, we set N=1,θk=iλk2x+4iλk6t and |v_k_{, 0}〉 = (c_k, 1)^T in formula (3.5). Then according to (4.3), one-soliton reads as

q=-2ic1(λ12-λ1*2)exp(-θ1+θ1*)|c1|2λ1*exp(-θ1-θ1*)+λ1exp(θ1+θ1*)exp(-32i∫-∞xλ1,R2λ1,I2||c1|2λ1*exp(-θ1-θ1*)+|λ1exp(θ1+θ1*)dx′).

When we take c₁ = 1, one-soliton solution yields the form

q=-2i(λ12-λ1*2)exp(-2iθ1,I)λ1*exp(-2θ1,R)+λ1exp(2θ1,R)exp(-32i∫-∞xλ1,R2λ1,I2|λ1*exp(-2θ1,R)+λ1exp(2θ1,R)|2dx′).

Here the subscripts R and I denote the real and imaginary part of θ₁, respectively. We can separate the real and imaginary parts of θ₁ as

θ1,R=-2λ1,Rλ1,I(x+4(3λ1,R4-10λ1,R2λ1,I2+3λ1,I4)t),θ1,I=(λ1,R2-λ1,I2)(x+4(λ1,R2-4λ1,Rλ1,I+λ1,I2)(λ1,R2+4λ1,Rλ1,I+λ1,I2)t),

Thus the velocity for the single soliton is v1=4(3λ1,R4-10λ1,R2λ1,I2+3λ1,I4) and the center position for |q| locates on the line

x+4(3λ1.R4-10λ1,R2λ1J2+3λ1J4)t=0,

with the amplitude 4|λ_1, _I|. The profile to one-soliton solution with parameter λ=1+i2 is displayed in Fig. 2.

The two-soliton solution with parameters λ1=1-i4,λ2=1+i2 and c₁ = c₂ = 1 is depicted in Fig 3. In this case, the two left-travelling solitons with amplitude 1 and 2, velocity 9.6875 and 2.75, respectively, pass through each other and keep each amplitude and velocity.

4.2. Interactions between N -solitons

In order to analyze the N-soliton solution interactions with the variations of corresponding positions and amplitudes, we assume v₁<v₂<⋯<v_N and keep x−v_kt = constant with vj=4(3λj,R4-10λj,R2λj,I2+3λj,I4) , j = 1,2,....,N. In the following we study the asymptotic behavior of the N solitons.

Theorem 2.

Assuming Im(λk2)<0(k=1,2,…,N) and v₁<v₂<⋯<v_N, then N-soliton solution has the asymptotic behavior

q^≡-4idetM^1detM~∑k=1N-2i(λk2-λk*2)exp(-2iθk,I∓arg(Φk)∓argΨk)λk*exp(-2θk,R∓ln|ΦkΨk|)+λkexp(2θk,R±ln|ΨkΦk|) (4.7)

as t →±∞ with

Φk=∏j=1k-1λj*2-λk2λk2-λj2, Ψk=∏j=k+1Nλj2-λk2λj*2-λk2. (4.8)

Proof. When Imλk2<0 , the asymptotic behaviour of exp (θ_k_, _R) = −2 λ_k_,_Rλ_kI(x + v_kt) is mainly determined by x + v_kt. In the vicinity Ω_k: x = −v_kt, when t → + ∞, we have asymptotic behaviour

x+vjt=(vj-vk)t→+∞,j>k,x+vjt=(vj-vk)t→-∞,j<k.

Thus, when t → + ∞

exp(-θj)→0, j>k and exp(θj)→0, j<k

and the matrix elements m_jl have the asymptotic expression

{j<kl<k mjl~ajle-θj-θl*{j<kl>k mjl~e-θj+θl*(ajle-2θl*+bjle2θj),{j>kl>k mjl~bjleθj+θl*{j>kl<k mjl~eθj-θl*(ajle-2θj+bjle2θl*),

and

mjk~{ajke-θk*-θjj<kbjkeθj+θk*j>k, mkl~{akle-θk-θi*l<kbkleθk+θil>k,

where

ajl=-2λl*λj2-λl*2, bjl=-2λjλj2-λl*2.

In the vicinity Ω_k, we have

detM~|-2λ1*λ12-λ1*2⋯-2λk-1*λ12-λk-1*2-2λk*λ12-λk*20⋯0⋮⋮⋮⋮⋮-2λ1*λk-12-λ1*2⋯-2λk-1*λk-12-λk-1*2-2λk*λk-12-λk*20⋯0-2λ1*λk2-λ1*2⋯-2λk-1*λk2-λk-1*2-2λk*-2λke2θk+2θk*λk2-λk*2-2λke2θkλk2-λk+1*2⋯-2λke2θkλk2-λN*20⋯0-2λk+1e2θk*λk+12-λk*2-2λk+1λk+12-λk+1*2⋯-2λk+1λk+12-λN*2⋮⋮⋮⋮⋮0⋯0-2λNe2θk*λN2-λk*2-2λNλN2-λk+1*2⋯-2λNλN2-λN*2|×exp(-∑j=1k(θj+θj*)+∑j=k+1N(θj+θj*)),

and

detM1~|-2λ1*λ12-λ1*2⋯-2λk-1*λ12-λk-1*2-2λk*λ12-λk*20⋯01⋮⋮⋮⋮⋮⋮-2λ1*λk-12-λ1*2⋯-2λk-1*λk-12-λk-1*2-2λk*λk-12-λk*20⋯01-2λ1*λk2-λ1*2⋯-2λk-1*λk2-λk-1*2-2λk*-2λke2θk+2θk*λk2-λk*2-2λke2θkλk2-λk+1*2⋯-2λke2θkλk2-λN*210⋯0-2λk+1e2θk*λk+12-λk*2-2λk+1λk+12-λk+1*2⋯-2λk+1λk+12-λN*20⋮⋮⋮⋮⋮⋮0⋯0-2λNe2θk*λN2-λk*2-2λNλN2-λk+1*2⋯-2λNλN2-λN*200⋯0e2θk*1⋯10|×exp(-∑j=1k(θj+θj*)+∑j=k+1N(θj+θj*)).

By use of the Laplace expansion and the determinant formula of Cauchy matrix, we conclude that in the vicinity Ω_k, q has the asymptotic behaviour as

q^=-4idetM1detM~-2i(λk2-λk*2)exp(2iθk,I-arg(Φk)-argΨk)λk*exp(-2θk,R-ln|ΦkΨk|)+λkexp(2θk,R+ln|ΨkΦk|)

Similarly, when t → −∞, we can prove

q^=-4idetM1detM~-2i(λk2-λk*2)exp(2iθk,l+arg(Φk)+argΨk)λk*exp(-2θk,R+ln|ΦkΨk|)+λkexp(2θk,R-ln|ΨkΦk|)

Thus we have the result that on the whole plane q^ has the asymptotic behaviour as (4.7)

From the asymptotic behaviour of N-soliton solution, we know that the interaction of N solitons is elastic and only phase shifts and displacements happen.

5. Conclusion and discussion

The inverse scattering method has been applied to the higher-order CLL equation and by considering the associated Riemann-Hilbert problem, we successfully give a simple representation for the N-soliton in the determinant form. In the context we only consider the simple zeros for of the scattering matrix. The much more general case with multiple zeros would lead to more solutions. Here we only consider solutions with vanishing boundary conditions.

Because {±λ_j} appear simultaneously as zeros of det P ⁺, we can assume that detP ⁺ has 2N simple zeros {λj}12N satisfying λ_N ₊_j = λ_j, 1 ≤ j ≤ N, which all lie in 𝔺+ . From the symmetry property (3.2), we can choose the particular column vector relation

|vj〉=σ3|vj-N〉, N+1≤j≤2N

In this paper, we give explicit soliton solutions to a higher-order CLL equation in the determinant form. We know that the whole NLS hierarchy could be reduced from the (extended) KP hierarchy and soliton solutions can be obtained by using the reduction method. It is deserved to compare soliton formulas obtained here with the tau function formalism.

A third problem is how to adjust the analysis and seek the Jost solutions of the spectral problem so that solutions with non-vanishing boundary conditions can be obtained. The global well-posedness, long-time behavior and asymptotic stability of solitons are left for future studies.

6. Acknowledgements

Authors appreciate the referees for their valuable suggestions. The work is supported by National Natural Science Foundation of China (Grant no. 11371251, 11501365, 11771395).

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 25 - 4
Pages: 633 - 649
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2018.1503443 How to use a DOI?
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TY  - JOUR
AU  - Juan Hu
AU  - Jian Xu
AU  - Guo-Fu Yu
PY  - 2021
DA  - 2021/01/06
TI  - Riemann-Hilbert approach and N-soliton formula for a higher-order Chen-Lee-Liu equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 633
EP  - 649
VL  - 25
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1503443
DO  - 10.1080/14029251.2018.1503443
ID  - Hu2021
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