The Rogue Wave Solution of MNLS Equation Based on Hirota's Bi-linear Derivative Transformation

Tang Yuxuan, Zhou Guoquan,*

School of Physics and Technology, Wuhan University, Wuhan 430072

 基金资助: 国家自然科学基金.  12074295

 Fund supported: The NSFC.  12074295

Abstract

The modified nonlinear Schrodinger (MNLS for brevity) equation and the Derivative nonlinear Schrodinger (DNLS for brevity) equation are two nonlinear differential equations that are closely correlated and fully integrable. Firstly, the spatially periodic breather solution of the MNLS equation has been obtained by method of Hirota's bilinear derivative transform, and then its rogue wave solution is also obtained by a long-wave limit of the Akhmediev-type breather, which can be naturally reduced to a rogue wave solution of the DNLS equation by a simple operation of parameters. The existence of global soliton/rogue wave solutions for the MNLS/DNLS equations is briefly discussed.

Keywords： Rogue wave ; MNLS equation ; DNLS equation ; Hirota's bilinear derivative transform ; Spatially periodic solution ; Breather

Tang Yuxuan, Zhou Guoquan. The Rogue Wave Solution of MNLS Equation Based on Hirota's Bi-linear Derivative Transformation. Acta Mathematica Scientia[J], 2023, 43(1): 132-142 doi:

1 绪论

$$${\rm i}u_t+u_{xx}+{\rm i}(|u|^2u)_x+2\gamma|u|^2u=0. \label{eq:mnls}$$$

$$${\rm i}u'_t+u'_{xx}+{\rm i}(|u'|^2u')_x=0 \label{eq:dnls}$$$

2.1 Hirota双线性导数算符及相关性质

$$$D^m_tD^n_xf\bullet g=(\frac{\partial }{\partial t}- \frac{\partial }{\partial t'})^m(\frac{\partial }{\partial x}- \frac{\partial }{\partial x'})^nf(x,t)g(x',t')|_{t'=t,x'=x}.$$$

$$$(f/g)_x=(D_xf\bullet g)/g^2,$$$
$$$(f/g)_{xx}=(D_x^2f\bullet g)/g^2-f(D_x^2g\bullet g)/g^3.$$$

Hirota双线性导数变换还具有一个重要性质, 设$\eta_j=\omega_jt+k_jx+\eta_j^0$, 其中$j=1, 2$, $\omega_j$$k_j 均为复常数. 则有 $$D^m_tD^n_x{\rm e}^{\eta_1}\bullet {\rm e}^{\eta_2}=(\omega_1-\omega_2)^m(k_1-k_2)^n{\rm e}^{\eta_1+\eta_2},$$ 进而可知, 具有相同系数的线性指数函数的双线性导数为零, 即当\omega_1=\omega_2$$k_1=k_2$ 时,

$$$D^m_tD^n_x{\rm e}^{\eta_1}\bullet {\rm e}^{\eta_2}=0.$$$

$$$F_1(D_t,D_x,\cdots )g_1\bullet f_1+F_2(D_t,D_x,\cdots )g_2\bullet f_2=0,$$$

$$$f=f^{(0)}+\epsilon f^{(1)}+\epsilon^2f^{(2)}, \label{eq:f expand}$$$
$$$g=g^{(0)}+\epsilon g^{(1)}+\epsilon^2g^{(2)}, \label{eq:g expand}$$$

$\epsilon^0$相关方程组

$$$({\rm i}D_t+D^2_x-\lambda)g^{(0)}\bullet f^{(0)}=0, \label{eq:epsilon^0 1st}$$$
$$$({\rm i}D_t+D^2_x-\lambda)f^{(0)}\bullet \overline{f^{(0)}}=2\gamma g^{(0)}\overline{g^{(0)}}, \label{eq:epsilon^0 2nd}$$$
$$$D_xf^{(0)}\bullet \overline{f^{(0)}}=\frac{\rm i}{2}g_0\bar{g_0}. \label{eq:epsilon^0 3rd}$$$

$\epsilon^1$相关方程组

$$$({\rm i}D_t+D^2_x-\lambda)(g^{(1)}\bullet f^{(0)}+g^{(0)}\bullet f^{(1)})=0, \label{eq:epsilon^1 1st}$$$
$$$({\rm i}D_t+D^2_x-\lambda)(f^{(1)}\bullet \overline{f^{(0)}}+f^{(0)}\bullet \overline{f^{(1)}})=2\gamma (g^{(1)}\overline{g^{(0)}}+g^{(0)}\overline{g^{(1)}}), \label{eq:epsilon^1 2nd}$$$
$$$D_x(f^{(1)}\bullet \overline{f^{(0)}}+f^{(0)}\bullet \overline{f^{(1)}})=\frac{\rm i}{2}(g^{(1)}\overline{g^{(0)}}+g^{(0)}\overline{g^{(1)}}). \label{eq:epsilon^1 3rd}$$$

$\epsilon^2$相关方程组

$$$({\rm i}D_t+D^2_x-\lambda)(g^{(2)}\bullet f^{(0)}+g^{(1)}\bullet f^{(1)}+g^{(0)}\bullet f^{(2)})=0, \label{eq:epsilon^2 1st}$$$
$$$({\rm i}D_t+D^2_x-\lambda)(f^{(2)}\bullet \overline{f^{(2)}}+f^{(1)}\bullet \overline{f^{(1)}}+f^{(0)}\bullet \overline{f^{(2)}})=2\gamma (g^{(2)}\overline{g^{(0)}}+g^{(1)}\overline{g^{(1)}}+g^{(0)}\overline{g^{(2)}}), \label{eq:epsilon^2 2nd}$$$
$$$D_x(f^{(2)}\bullet \overline{f^{(2)}}+f^{(1)}\bullet \overline{f^{(1)}}+f^{(0)}\bullet \overline{f^{(2)}})=\frac{\rm i}{2}(g^{(2)}\overline{g^{(0)}}+g^{(1)}\overline{g^{(1)}}+g^{(0)}\overline{g^{(2)}}). \label{eq:epsilon^2 3rd}$$$

$\epsilon^3$相关方程组

$$$({\rm i}D_t+D^2_x-\lambda)(g^{(2)}\bullet f^{(1)}+g^{(1)}\bullet f^{(2)})=0, \label{eq:epsilon^3 1st}$$$
$$$({\rm i}D_t+D^2_x-\lambda)(f^{(2)}\bullet \overline{f^{(1)}}+f^{(1)}\bullet \overline{f^{(2)}})=2\gamma (g^{(2)}\overline{g^{(1)}}+g^{(1)}\overline{g^{(2)}}), \label{eq:epsilon^3 2nd}$$$
$$$D_x(f^{(2)}\bullet \overline{f^{(1)}}+f^{(1)}\bullet \overline{f^{(2)}})=\frac{\rm i}{2}(g^{(2)}\overline{g^{(1)}}+g^{(1)}\overline{g^{(2)}}). \label{eq:epsilon^3 3rd}$$$

$\epsilon^4$相关方程组

$$$({\rm i}D_t+D^2_x-\lambda)g^{(2)}\bullet f^{(2)}=0, \label{eq:epsilon^4 1st}$$$
$$$({\rm i}D_t+D^2_x-\lambda)f^{(2)}\bullet \overline{f^{(2)}}=2\gamma g^{(2)}\overline{g^{(2)}}, \label{eq:epsilon^4 2nd}$$$
$$$D_xf^{(2)}\bullet \overline{f^{(2)}}=\frac{\rm i}{2}g^{(2)}\overline{g^{(2)}}. \label{eq:epsilon^4 3rd}$$$

3 MNLS和DNLS方程的一阶空间周期解

$$$g^{(0)}=\rho {\rm e}^{{\rm i}\omega t}, \label{eq:g0}$$$
$$$f^{(0)}={\rm e}^{{\rm i}\beta x}. \label{eq:f0}$$$

$\begin{matrix} && \beta= \rho^2/4, \\ && \omega=2\gamma \rho^2+3\rho^4/16,\\ && \lambda=-\omega-\beta^2=-2\gamma \rho^2-\rho^4/4, \label{eq:Dispersion relationship} \end{matrix}$

$$$g\rightarrow \rho \exp({\rm i}\omega t),$$$
$$$f\rightarrow \exp({\rm i}\beta t),$$$

$$$u\rightarrow \rho\exp({\rm i}(\omega t-3\beta x)), \label{eq:t -infty}$$$

$$$u\rightarrow \rho\exp({\rm i}(\omega t-3\beta x+\phi)). \label{eq:t +infty}$$$

(3.18)式中$\phi$ 相应于当$t$ 从负无穷至正无穷时波的相移

$$$\exp({\rm i}\phi )=a_1 a_2 \overline{b_1} \overline{b_2}/(b_1 b_2)^2, \label{eq:phi relationship}$$$

$$$\lim_{t \to + \infty} u\rightarrow \rho\exp({\rm i}(\omega t-3\beta x)),$$$
$$$\lim_{t \to - \infty} u\rightarrow \rho\exp({\rm i}(\omega t-3\beta x+\phi)).$$$

$$$\Omega_{\pm}=\frac{q\rho^2}{2}(\sqrt{2}\sqrt{1+8\gamma/\rho^2}-{\rm i})+O(q^3), \label{eq:Omega}$$$

$\Omega$ 保留到$q$ 的二阶项, 并引入辅助参变量函数$\sigma(\gamma,\rho)$ 以化简角频率$\Omega$ 的函数形式

$$$\Omega=q\rho^2(\sigma-{\rm i})/2, \label{eq:Omega sigma}$$$
$$$\sigma(\gamma,\rho)=\pm \sqrt{2}\sqrt{1+8\gamma/\rho^2}. \label{eq:sigma}$$$

$$$f={\rm e}^{{\rm i}\beta x}Fq^2+O(q^3), \label{eq:f rouge wave solution}$$$
$$$g=\rho {\rm e}^{{\rm i}\omega t}Gq^2+O(q^3). \label{eq:g rouge wave solution}$$$

$$${{u}_{{\rm RW}}}=\lim_{q \to 0} g \overline{f}/f^2 =\rho {\rm e}^{{\rm i}(-3\beta x+\omega t)}G\cdot \overline{F}/F^2, \label{eq:uRW mnls}$$$

$\begin{matrix} \label{eq:f uRW} F&=&\frac{1}{4{{\rho }^{4}}{{\sigma }^{2}}\left( 8\gamma +{\rm i}{{\rho }^{2}}\sigma +{{\rho }^{2}} \right)}{{\rm e}^{{\rm i}\beta x}}(128\gamma +16{\rm i}{{\rho }^{2}}\sigma +16{{\rho }^{2}}+8\gamma {{\rho }^{8}}{{\sigma }^{4}}{{t}^{2}} \\ &&8\gamma {{\rho }^{8}}{{\sigma }^{2}}{{t}^{2}}+{\rm i}{{\rho }^{10}}{{\sigma }^{5}}{{t}^{2}}+{{\rho }^{10}}{{\sigma }^{4}}{{t}^{2}}+ {\rm i}{{\rho }^{10}}{{\sigma }^{3}}{{t}^{2}}+{{\rho }^{10}}{{\sigma }^{2}}{{t}^{2}}+8{{\rho }^{6}}{{\sigma }^{3}}t \\ && - 8{\rm i}{{\rho }^{6}}{{\sigma }^{2}}t-32\gamma {{\rho }^{6}}{{\sigma }^{2}}tx- 4{\rm i}{{\rho }^{8}}{{\sigma }^{3}}tx-4{{\rho }^{8}}{{\sigma }^{2}}tx+32\gamma {{\rho }^{4}}{{\sigma }^{2}}{{x}^{2}} \\ &&+ 4{\rm i}{{\rho }^{6}}{{\sigma }^{3}}{{x}^{2}}+4{{\rho }^{6}}{{\sigma }^{2}}{{x}^{2}}+16{\rm i}{{\rho }^{4}}{{\sigma }^{2}}x), \end{matrix}$
$\begin{matrix} \label{eq:g uRW} G&=&\frac{1}{4{{\rho }^{4}}{{\sigma }^{2}}\left( \sigma -2{\rm i} \right)\left( \sigma +2{\rm i} \right)\left( -8{\rm i}\gamma +{{\rho }^{2}}\sigma -{\rm i}{{\rho }^{2}} \right)}(384{\rm i}\gamma {{\sigma }^{2}}-512{\rm i}\gamma \\ & &- 112{{\rho }^{2}}{{\sigma }^{3}}-80{\rm i}{{\rho }^{2}}{{\sigma }^{2}}+64{{\rho }^{2}}\sigma -64{\rm i}{{\rho }^{2}}-8{\rm i}\gamma {{\rho }^{8}}{{\sigma }^{6}}{{t}^{2}}-40{\rm i}\gamma {{\rho }^{8}}{{\sigma }^{4}}{{t}^{2}} \\ & & -32{\rm i}\gamma {{\rho }^{8}}{{\sigma }^{2}}{{t}^{2}}+{{\rho }^{10}}{{\sigma }^{7}}{{t}^{2}}-{\rm i}{{\rho }^{10}}{{\sigma }^{6}}{{t}^{2}}+5{{\rho }^{10}}{{\sigma }^{5}}{{t}^{2}}-5{\rm i}{{\rho }^{10}}{{\sigma }^{4}}{{t}^{2}} \\ & &+ 4{{\rho }^{10}}{{\sigma }^{3}}{{t}^{2}}-4{\rm i}{{\rho }^{10}}{{\sigma }^{2}}{{t}^{2}}-128\gamma {{\rho }^{4}}{{\sigma }^{4}}t+256\gamma {{\rho }^{4}}{{\sigma }^{2}}t-24{\rm i}{{\rho }^{6}}{{\sigma }^{5}}t \\ & & -24{{\rho }^{6}}{{\sigma }^{4}}t+32{\rm i}\gamma {{\rho }^{6}}{{\sigma }^{4}}tx+128{\rm i}\gamma {{\rho }^{6}}{{\sigma }^{2}}tx-4{{\rho }^{8}}{{\sigma }^{5}}tx+4{\rm i}{{\rho }^{8}}{{\sigma }^{4}}tx \\ & & -16{{\rho }^{8}}{{\sigma }^{3}}tx+16{\rm i}{{\rho }^{8}}{{\sigma }^{2}}tx-32{\rm i}\gamma {{\rho }^{4}}{{\sigma }^{4}}{{x}^{2}}-128{\rm i}\gamma {{\rho }^{4}}{{\sigma }^{2}}{{x}^{2}}+4{{\rho }^{6}}{{\sigma }^{5}}{{x}^{2}} \\ & & -4{\rm i}{{\rho }^{6}}{{\sigma }^{4}}{{x}^{2}}+16{{\rho }^{6}}{{\sigma }^{3}}{{x}^{2}}-16{\rm i}{{\rho }^{6}}{{\sigma }^{2}}{{x}^{2}}-512\gamma {{\rho }^{2}}{{\sigma }^{2}}x+16{{\rho }^{4}}{{\sigma }^{4}}x-64{\rm i}{{\rho }^{4}}{{\sigma }^{3}}x), \end{matrix}$

图4

$$${{{u}'}_{{\rm RW}}}=\lim_{q \to 0} g \overline{f}/f^2 =\rho {\rm e}^{{\rm i}(-3\beta x+\omega t)}G'\cdot \overline{F'}/F'^2, \label{eq:uRW dnls}$$$

$\begin{matrix} \label{eq:f uRW dnls} F'&=&\frac{1}{4{{\rho }^{4}}{{\sigma }^{2}}(\sigma -{\rm i})}(16\sigma +{{\rho }^{8}} {{\sigma }^{5}}{{t}^{2}}-{\rm i}{{\rho }^{8}}{{\sigma }^{4}}{{t}^{2}}+ {{\rho }^{8}}{{\sigma }^{3}}{{t}^{2}}-{\rm i}{{\rho }^{8}}{{\sigma }^{2}}{{t}^{2}}-8{\rm i}{{\rho }^{4}}{{\sigma }^{3}}t \\ &&- 8{{\rho }^{4}}{{\sigma }^{2}}t-4{{\rho }^{6}}{{\sigma }^{3}}tx+4{\rm i}{{\rho }^{6}} {{\sigma }^{2}}tx+4{{\rho }^{4}}{{\sigma }^{3}}{{x}^{2}}-4{\rm i}{{\rho }^{4}}{{\sigma }^{2}}{{x}^{2}} +16{{\rho }^{2}}{{\sigma }^{2}}x-16{\rm i}), \end{matrix}$
$\begin{matrix} \label{eq:g uRW dnls} G'&=&\frac{1}{4{{\rho }^{4}}{{\sigma }^{2}}(\sigma -{\rm i})(\sigma -2{\rm i})(\sigma +2{\rm i})} (-112{{\sigma }^{3}}-80{\rm i}{{\sigma }^{2}}+64\sigma +{{\rho }^{8}}{{\sigma }^{7}}{{t}^{2}} \\ & &-{\rm i}{{\rho }^{8}}{{\sigma }^{6}}{{t}^{2}}+5{{\rho }^{8}}{{\sigma }^{5}}{{t}^{2}} -5{\rm i}{{\rho }^{8}}{{\sigma }^{4}}{{t}^{2}}+4{{\rho }^{8}}{{\sigma }^{3}}{{t}^{2}} -4{\rm i}{{\rho }^{8}}{{\sigma }^{2}}{{t}^{2}}-24{\rm i}{{\rho }^{4}}{{\sigma }^{5}}t \\ &&- 24{{\rho }^{4}}{{\sigma }^{4}}t-4{{\rho }^{6}}{{\sigma }^{5}}tx +4{\rm i}{{\rho }^{6}}{{\sigma }^{4}}tx-16{{\rho }^{6}}{{\sigma }^{3}}tx+16{\rm i}{{\rho }^{6}}{{\sigma }^{2}}tx +4{{\rho }^{4}}{{\sigma }^{5}}{{x}^{2}} \\ && -4{\rm i}{{\rho }^{4}}{{\sigma }^{4}}{{x}^{2}}+16{{\rho }^{4}}{{\sigma }^{3}}{{x}^{2}} -16{\rm i}{{\rho }^{4}}{{\sigma }^{2}}{{x}^{2}}+16{{\rho }^{2}}{{\sigma }^{4}}x -64{\rm i}{{\rho }^{2}}{{\sigma }^{3}}x-64{\rm i}). \end{matrix}$

$$$iv_t+v_xx+{\rm i}|v|^2v_x=0. \label{eq:CLL}$$$

$$${v}_{{\rm RW}}(x,t)={{u}'}_{{\rm RW}}\exp(\frac{\rm i}{2}\int|{{u}'}_{{\rm RW}}|^2 \,{\rm d}x ), \label{eq:CLL transform}$$$

参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Akhmediev N, Ankiewicz A, Taki M.

Waves that appear from nowhere and disappear without a trace

Physics Letters A, 2009, 373(6): 675-678

Akhmediev N, Soto-Crespo J M, Ankiewicz A.

Extreme waves that appear from nowhere: on the nature of rogue waves

Physics Letters A, 2009, 373(25): 2137-2145

Wen X Y, Yang Y, Yan Z.

Generalized perturbation $(n, M)$-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation

Physical Review E, 2015, 92(1): 012917

Pan C, Bu L, Chen S, et al.

General rogue wave solutions under SU (2) transformation in the vector Chen-Lee-Liu nonlinear Schrödinger equation

Physica D: Nonlinear Phenomena, 2022, 434: 133204

He J, Xu S, Cheng Y.

The rational solutions of the mixed nonlinear Schrödinger equation

Zhao L C, Guo B, Ling L.

High-order rogue wave solutions for the coupled nonlinear Schrödinger equations-II

Journal of Mathematical Physics, 2016, 57(4): 043508

Ankiewicz A, Akhmediev N.

Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy

Physical Review E, 2017, 96(1): 012219

Xu S, He J, Wang L.

The Darboux transformation of the derivative nonlinear Schrödinger equation

Journal of Physics A: Mathematical and Theoretical, 2011, 44(30): 305203

Guo B, Ling L, Liu Q P.

High-order solutions and generalized Darboux transformations of derivative nonlinear Schrödinger equations

Studies in Applied Mathematics, 2013, 130(4): 317-344

Zhang Y, Guo L, Xu S, et al.

The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation

Communications in Nonlinear Science and Numerical Simulation, 2014, 19(6): 1706-1722

Wang L H, Porsezian K, He J S.

Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation

Physical Review E, 2013, 87(5): 053202

Lou S, Lin J.

Rogue waves in nonintegrable KdV-type systems

Chinese Physics Letters, 2018, 35(5): 050202

Wu H, Fei J, Ma W.

Lump and rational solutions for weakly coupled generalized Kadomtsev-Petviashvili equation

Modern Physics Letters B, 2021, 35(26): 2150449

Zhou G, Li X.

Space periodic solutions and rogue wave solution of the derivative nonlinear Schrödinger equation

Wuhan University Journal of Natural Sciences, 2017, 22(5): 373-379

Ma W X.

Nonlocal PT-symmetric integrable equations and related Riemann-Hilbert problems

Partial Differential Equations in Applied Mathematics, 2021, 4: 100190

Ablowitz M J, Musslimani Z H.

Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation

Nonlinearity, 2016, 29(3): 915

Yang B, Yang J.

Rogue waves in the nonlocal ${\mathcal {PT}}$ PT-symmetric nonlinear Schrödinger equation

Letters in Mathematical Physics, 2019, 109(4): 945-973

Rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger (NLS) equation are studied by Darboux transformation. Three types of rogue waves are derived, and their explicit expressions in terms of Schur polynomials are presented. These rogue waves show a much wider variety than those in the local NLS equation. For instance, the polynomial degrees of their denominators can be not only n(n+1), but also n(n-1)+1 and n2, where n is an arbitrary positive integer. Dynamics of these rogue waves is also examined. It is shown that these rogue waves can be bounded for all space and time or develop collapsing singularities, depending on their types as well as values of their free parameters. In addition, the solution dynamics exhibits rich patterns, most of which have no counterparts in the local NLS equation.

Lin J, Jin X W, Gao X L, et al.

Solitons on a periodic wave background of the modified KdV-Sine-Gordon equation

Communications in Theoretical Physics, 2018, 70(2): 119-126

Li L, Duan C, Yu F.

An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation

Physics Letters A, 2019, 383(14): 1578-1582

The Hirota bilinear method has been studied in a lot of local equations, but there are few of works to solve nonlocal equations by Hirota bilinear method. In this letter, we show that the nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation admits multiple complex soliton solutions. A variety of exact solutions including the single bright soliton solutions and two bright soliton solutions are derived via constructing an improved Hirota bilinear method for nonlocal complex MKdV equation. From the gauge equivalence, we can see the difference between the solution of nonlocal integrable complex MKdV equation and the solution of local complex MKdV equation. (C) 2019 Elsevier B.V.

Zhong W P, Yang Z, Beli${\rm\acute{c}}$ M, et al.

Breather solutions of the nonlocal nonlinear self-focusing Schrödinger equation

Physics Letters A, 2021, 395: 127228

Ma W X, Zhou R.

A coupled AKNS-Kaup-Newell soliton hierarchy

Journal of Mathematical Physics, 1999, 40(9): 4419-4428

Chen Z Y, Huang N N.

Explicit N-soliton solution of the modified nonlinear Schrödinger equation

Physical Review A, 1990, 41(7): 4066-4069

Huang N N, Chen Z Y.

Alfven solitons

Journal of Physics A: Mathematical and General, 1990, 23(4): 439-453

Chen H H, Lee Y C, Liu C S.

Integrability of nonlinear Hamiltonian systems by inverse scattering method

Physica Scripta, 1979, 20(3/4): 490-492

Kaup D J, Newell A C.

An exact solution for a derivative nonlinear Schrödinger equation

Journal of Mathematical Physics, 1978, 19(4): 798-801

Kawata T, Inoue H.

Exact solution of derivative nonlinear Schrödinger equation under nonvanishing conditions

J Phys Soc Jpn, 1978, 44(6): 1968-1976

Steudel H.

The hierarchy of multi-soliton solutions of derivative nonlinear Schrödinger equation

J Phys A, 2003, 36: 1931-1946

Zhou G Q, Huang N N.

An $N$-soliton solution to the DNLS equation based on revised inverse scattering transform

Journal of Physics A: Mathematical and Theoretical, 2007, 40(45): 13607-13623

Zhou G.

A multi-soliton solution of the DNLS equation based on pure Marchenko formalism

Wuhan University Journal of Natural Sciences, 2010, 15(1): 36-42

Zhou G, Bi X.

Soliton solution of the DNLS equation based on Hirota's bilinear derivative transform

Wuhan University Journal of Natural Sciences, 2009, 14(6): 505-510

Cai H.

Research about MNLS Equation and DNLS Equation

Wuhan Univ, 2005

Chen X J, Lam W K.

Inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

Physical Review E, 2004, 69(6): 066604

Chen X J, Yang J K, Lam W K.

$N$-soliton solution for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

Journal of Physics A: Math Gen, 2006, 39(13): 3263-3274

Lashkin V M.

$N$-soliton solutions and perturbation theory for DNLS with nonvanishing condition

J Phys A, 2007, 40: 6119-6132

Zhou G.

A newly revised inverse scattering transform for DNLS+ equation under nonvanishing boundary condition

Wuhan University Journal of Natural Sciences, 2012, 17(2): 144-150

Zhou G.

Explicit breather-type and pure N-Soliton solution of DNLS+ equation with nonvanishing boundary condition

Wuhan University Journal of Natural Sciences, 2013, 18(2): 147-155

Osman M S, Almusawa H, Tariq K U, et al.

On global behavior for complex soliton solutions of the perturbed nonlinear Schrödinger equation in nonlinear optical fibers

Journal of Ocean Engineering and Science, 2022, 7(5): 431-443

Bourgain J.

Global Solutions of Nonlinear Schrödinger Equations

Providence, RI: American Mathematical Society, 1999

/

 〈 〉