Modulation instability, rogue waves and conservation laws in higher-order nonlinear Schr【-逻*辑*与-】oum
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Min-Jie Dong1, Li-Xin Tian,1,21School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China 2Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China
First author contact:*Author to whom any correspondence should be addressed. Received:2020-09-5Revised:2020-11-14Accepted:2020-11-20Online:2021-01-15
Abstract In this paper, the modulation instability (MI), rogue waves (RWs) and conservation laws of the coupled higher-order nonlinear Schrödinger equation are investigated. According to MI and the 2×2 Lax pair, Darboux-dressing transformation with an asymptotic expansion method, the existence and properties of the one-, second-, and third-order RWs for the higher-order nonlinear Schrödinger equation are constructed. In addition, the main characteristics of these solutions are discussed through some graphics, which are draw widespread attention in a variety of complex systems such as optics, Bose-Einstein condensates, capillary flow, superfluidity, fluid dynamics, and finance. In addition, infinitely-many conservation laws are established. Keywords：higher-order nonlinear Schrödinger equation;modulation instability;rogue waves;conservation laws
PDF (1098KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Min-Jie Dong, Li-Xin Tian. Modulation instability, rogue waves and conservation laws in higher-order nonlinear Schrödinger equation. Communications in Theoretical Physics, 2021, 73(2): 025001- doi:10.1088/1572-9494/abcfb6
1. Introduction
Rogue waves (RWs) are regarded as one of the highly unsafe phenomena in the ocean [1], and are usually called monster waves or extreme waves in the ocean. In addition to the ocean, RWs can exist in optical fibers, Bose-Einstein condensates [2], and finance [3, 4]. The peregrine soliton as the classic NLS equation is seen as a possible mathematical explanation in 2010 [2].
Modulation instability (MI) is one of the main reasons for the formation of RWs in nonlinear dispersion systems, and its formation is intimately related to baseband MI. MI is found in the environment of water waves and is also known as the instability of Benjamin Fair [5]. MI can be interpreted as the exponential growth of the initial sine wave perturbation of the plane wave solution. In addition to water waves, MI also exists in plasma, nonlinear optics [6] and Bose-Einstein condensate. However, according to [7-9], we know that not every type of MI leads to the formation of the RW. The baseband MI is defined as a condition where the cw background is unstable relative to a disturbance with an infinitesimal frequency by Baronio et al in 2014 [10]. The passband MI is defined as the interference in which disturbances are experienced in a spectral region where the limit frequency does not include the zero frequency. This shows that in different nonlinear wave models, the existence conditions of RW solutions are consistent with baseband MI conditions. Experiments in optical fibers are designed to provide evidence of the passband and baseband polarization MI in the defocused Manakov system by Frisquet et al in 2015 [11]. Recently, Yang research group studied MI and related localized wave excitations in general high-order NLSE, see [12, 13]. On the other hand, the MI developed from localized perturbations can also studied by the breathers exactly, namely the super-regular breathers, see [14-16].
Conservation laws play an important role in discussing the integrability of soliton equations. Since Dogan discovered the infinite number of conservation law of the KdV equation [17], many methods have been developed. For example, the infinite number of conservation laws or conserved quantities of a continuous system can be obtained through the Bäklund transformation, the couple of Ricatti equations [18], and the scattering problem [19]. Recently, Tsuchida and Wadati proposed a graceful trace identity as a component extension to describe the protection of multi-component cases [20]. In addition, all the above methods have been extended to discrete soliton lattices.
In this paper, we shall use the Darboux-dressing transformation (DDT) to concentrate the rational solutions of the higher-order nonlinear Schrödinger (HONLS) equation [21].$\begin{eqnarray}\begin{array}{l}{\rm{i}}{\psi }_{x}+\displaystyle \frac{1}{2}{\psi }_{{tt}}+| \psi {| }^{2}\psi -{\rm{i}}\alpha ({\psi }_{{ttt}}+6{\psi }_{t}| \psi {| }^{2})\\ \quad +\,\delta ({\psi }_{{tttt}}+6{\psi }_{t}^{2}\overline{\psi }+4\psi | {\psi }_{t}{| }^{2}+8{\psi }_{{tt}}| \psi {| }^{2}\\ \quad +\,2\overline{{\psi }_{{tt}}}{\psi }^{2}+6\psi | \psi {| }^{4})=0,\end{array}\end{eqnarray}$where ψ is the wave function in optics, and t is the time variable, x is the propagation variable. The superscripts denote the complex conjugate and the subscripts represent the partial derivatives. Within our knowledge, the HONLS equation contains three completely integrable equations, which are α=0,δ=0; α≠0,δ=0; α=0,δ≠0. These equations have been studied by many scholars from different perspectives [22-25].
To facilitate calculation, Liu and his colleagues assumed r=iψ and $q={\rm{i}}\overline{\psi }$, the equation (1) can be transformed into following coupled higher-order nonlinear Schrödinger (CHONLS) equation [26]$\begin{eqnarray}\begin{array}{l}{r}_{t}-\displaystyle \frac{{\rm{i}}}{2}{r}_{{xx}}+{\rm{i}}{r}^{2}q+{\rm{i}}\alpha ({\rm{i}}{r}_{{xxx}}-6{\rm{i}}{{rqr}}_{x})\\ \quad +\,\delta (-{\rm{i}}{r}_{{xxxx}}+6{\rm{i}}{{qr}}_{x}{q}_{x}+4{\rm{i}}{{rr}}_{x}{q}_{x}\\ \quad +\,8{\rm{i}}{{rqr}}_{{xx}}+2{\rm{i}}{r}^{2}{q}_{{xx}}-6{\rm{i}}{q}^{2}{r}^{3})=0,\\ {q}_{t}+\displaystyle \frac{1}{2}{\rm{i}}{q}_{{xx}}-{\rm{i}}{q}^{2}r+{\rm{i}}\alpha ({\rm{i}}{q}_{{xxx}}-6{\rm{i}}{{rqq}}_{x})\\ \quad -\,\delta (-{\rm{i}}{q}_{{xxxx}}+6{\rm{i}}{{rq}}_{x}{q}_{x}+4{\rm{i}}{{qr}}_{x}{q}_{x}\\ \quad +\,8{\rm{i}}{{rqq}}_{{xx}}+2{\rm{i}}{q}^{2}{r}_{{xx}}-6{\rm{i}}{r}^{2}{q}^{3})=0.\end{array}\end{eqnarray}$
To our knowledge, the MI, RWs and conservation laws of CHONLS equations have not been reported in the existing literatures. In this paper, the breather solution and the RWs of this equation through DDT are mainly studied. The change of the parameters α and δ will affect the propagation direction of the wave, which will shown in detail with the figure. We also study the relationship between the RWs and the MI of the equation, and the conservation law.
The remainder of our article is constructed as follows. In section 2, according to the modulating instability, the linear stability of nonlinear plane waves with variable coefficients in the presence of small perturbations will be analyzed. In section 3, the DDT of the Lax pair system will be presented. In section 4, the new breather wave and RWs of equation (2) will be systematically derived. The first-order, second-order, and third-order accurate RWs are given and their dynamic characteristics will be analyzed. In section 5, the conservation laws will be constructed. Our conclusions will be drawn in section 6.
2. Modulation instability
In this section, we pay attention to MI on the plane wave state of the equation (2), which is believed to be the cause of the formation of RWs. According to [12-16], the plane wave solutions of equation (2) have the following form$\begin{eqnarray}r=a\exp ({\rm{i}}({bx}+{ct})),\,\,\,\,q=-a\exp (-{\rm{i}}({bx}+{ct})),\end{eqnarray}$where a denotes the amplitude, b denotes the frequency, both of them are real constants, and c is the plane wave number. Substituting equation (3) into (2), it can be obtained that$\begin{eqnarray}c={a}^{2}-\displaystyle \frac{1}{2}{b}^{2}+\alpha (6{a}^{2}b-{b}^{3})+\delta (6{a}^{4}-12{a}^{2}{b}^{2}+{b}^{4}).\end{eqnarray}$
In order to perform the linear stability analysis, we add a small perturbation term in the the plane wave solution$\begin{eqnarray}\begin{array}{rcl}r & = & (a+\epsilon R)\exp ({\rm{i}}({bx}+{ct})),\\ q & = & -(a-\epsilon Q)\exp (-{\rm{i}}({bx}+{ct})),\end{array}\end{eqnarray}$where ϵ is a small parameter, Q and R are the function of x and t. Substituting the above solution (5) into the (2), then collecting the terms in the first order of ϵ, we can get the following perturbation equation for Q and R$\begin{eqnarray}\begin{array}{l}2{R}_{t}+2{{bR}}_{x}-{\rm{i}}{R}_{{xx}}+2{\rm{i}}(Q-R){a}^{2}\\ \quad +\,\delta (24{\rm{i}}(Q-R){a}^{4}-24{\rm{i}}(Q-R){a}^{2}{b}^{2}\\ \quad +\,\alpha (12{\rm{i}}(Q-R){a}^{2}b-6{\rm{i}}{R}_{{xx}}b\\ \quad -\,12{R}_{x}{a}^{2}+6{R}_{x}{b}_{2}-2{R}_{{xxx}})-16{\rm{i}}{R}_{{xx}}{a}^{2}\\ \quad +\,12{\rm{i}}{R}_{{xx}}{b}^{2}+4{\rm{i}}{Q}_{{xx}}{a}^{2}+48{R}_{x}{a}^{2}b\\ \quad -\,8{R}_{x}{b}^{3}-2{\rm{i}}{R}_{{xxxx}}+8{R}_{{xxx}}b),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}2{{Q}}_{{t}}+2{{bQ}}_{{x}}+{\rm{i}}{{Q}}_{{xx}}+2{\rm{i}}({Q}-{R}){{a}}^{2}\\ \quad +\,\delta (24{\rm{i}}({Q}-{R}){{a}}^{4}-24{\rm{i}}({Q}-{R}){{a}}^{2}{{b}}^{2}\\ \quad +\,\alpha (12{\rm{i}}({Q}-{R}){{a}}^{2}{b}+6{\rm{i}}{{Q}}_{{xx}}{b}\\ \quad -\,12{{Q}}_{{x}}{{a}}^{2}+6{{Q}}_{{x}}{{b}}_{2}-2{{Q}}_{{xxx}})+16{\rm{i}}{{Q}}_{{xx}}{{a}}^{2}\\ \quad -\,12\ {\rm{i}}{{Q}}_{{xx}}{{b}}^{2}-4{\rm{i}}{{R}}_{{xx}}{{a}}^{2}\\ \quad +\,48{{Q}}_{{x}}{{a}}^{2}{b}-8{{Q}}_{{x}}{{b}}^{3}+2{\rm{i}}{{Q}}_{{xxxx}}+8{{Q}}_{{xxx}}{b}).\end{array}\end{eqnarray}$
Noting the linearity of the above equation (6) with respect to R and Q, we assume R and Q as$\begin{eqnarray}\begin{array}{rcl}R & = & U\exp ({\rm{i}}({\rm{\Lambda }}x-{\rm{\Omega }}t))+V\exp (-{\rm{i}}({\rm{\Lambda }}x-{\rm{\Omega }}t)),\\ Q & = & U\exp (-{\rm{i}}({\rm{\Lambda }}x-{\rm{\Omega }}t))+V\exp ({\rm{i}}({\rm{\Lambda }}x-{\rm{\Omega }}t)),\end{array}\end{eqnarray}$where Λ is the wave number, Ω is the modulation frequency, while U and V are small parameters. Putting equation (8) into (6), the following dispersion relation for the perturbations are obtained$\begin{eqnarray}\begin{array}{rcl} & & {f}_{11}U+{f}_{12}V=0,\\ & & {f}_{21}U+{f}_{22}V=0,\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}{f}_{11} & = & 6{\rm{i}}{\rm{\Lambda }}\alpha {b}^{2}-2{\rm{i}}{\rm{\Omega }}+2{\rm{i}}{\rm{\Lambda }}b-6{\rm{i}}{{\rm{\Lambda }}}^{2}\alpha b\\ & & -12{\rm{i}}{\rm{\Lambda }}{a}^{2}\alpha -{\rm{i}}{{\rm{\Lambda }}}^{2}+24{\rm{i}}{a}^{4}\delta \\ & & +12{\rm{i}}{{\rm{\Lambda }}}^{2}{b}^{2}\delta +48{\rm{i}}{\rm{\Lambda }}{a}^{2}b\delta -8{\rm{i}}{{\rm{\Lambda }}}^{3}b\delta \\ & & +12{\rm{i}}{a}^{2}\alpha b-16{\rm{i}}{{\rm{\Lambda }}}^{2}{a}^{2}\delta \\ & & +2{\rm{i}}{{\rm{\Lambda }}}^{3}\alpha -24{\rm{i}}{a}^{2}{b}^{2}\delta +2{\rm{i}}{a}^{2}+2{\rm{i}}{{\rm{\Lambda }}}^{4}\delta -8{\rm{i}}{\rm{\Lambda }}{b}^{3}\delta ,\end{array}\end{eqnarray}$$\begin{eqnarray*}\begin{array}{rcl}{f}_{12} & = & {f}_{21}=4{\rm{i}}{{\rm{\Lambda }}}^{2}{a}^{2}\delta -24{\rm{i}}{a}^{4}\delta \\ & & +24{\rm{i}}{a}^{2}{b}^{2}\delta -12{\rm{i}}{a}^{2}b\alpha -2{\rm{i}}{a}^{2},\\ {f}_{22} & = & -6{\rm{i}}{\rm{\Lambda }}\alpha {b}^{2}+2{\rm{i}}{\rm{\Omega }}-2{\rm{i}}{\rm{\Lambda }}b-6{\rm{i}}{{\rm{\Lambda }}}^{2}\alpha b\\ & & +12{\rm{i}}{\rm{\Lambda }}{a}^{2}\alpha -{\rm{i}}{{\rm{\Lambda }}}^{2}+24{\rm{i}}{a}^{4}\delta \\ & & +12{\rm{i}}{{\rm{\Lambda }}}^{2}{b}^{2}\delta -48{\rm{i}}{\rm{\Lambda }}{a}^{2}b\delta +8{\rm{i}}{{\rm{\Lambda }}}^{3}b\delta \\ & & +12{\rm{i}}{a}^{2}\alpha b-16{\rm{i}}{{\rm{\Lambda }}}^{2}{a}^{2}\delta \\ & & -2{\rm{i}}{{\rm{\Lambda }}}^{3}\alpha -24{\rm{i}}{a}^{2}{b}^{2}\delta +2{\rm{i}}{a}^{2}+2{\rm{i}}{{\rm{\Lambda }}}^{4}\delta +8{\rm{i}}{\rm{\Lambda }}{b}^{3}\delta ,\end{array}\end{eqnarray*}$the existence conditions of the solutions for U and V (i.e. UV≠0) can be obtained$\begin{eqnarray}\left(\begin{array}{cc}{f}_{11} & {f}_{12}\\ {f}_{21} & {f}_{22}\end{array}\right)=0.\end{eqnarray}$
The power gain at any frequency Ω is obtained from equation (12) and is given by$\begin{eqnarray}G=\left|\mathrm{Im}({\rm{\Omega }})\right|=| {\rm{\Lambda }}| {\left(-({{\rm{\Lambda }}}^{2}-4{a}^{2})\left(\displaystyle \frac{1}{2}+3b\alpha -{{\rm{\Upsilon }}}^{2}\right)\right)}^{\tfrac{1}{2}},\end{eqnarray}$in which G stands for the gain with $({{\rm{\Lambda }}}^{2}-4{a}^{2})\left(\tfrac{1}{2}+3b\alpha -{{\rm{\Upsilon }}}^{2}\right)\lt 0$. Figure 1 shows the gain spectra at three power levels.
Figure 1.
New window|Download| PPT slide Figure 1.Gain spectra of modulation instability for the parameter values: (a) α=1, δ=1; (b) α=0, δ=0; (c) α=1, δ=0; (d) α=0, δ=1 , and for the different power levels as legend.
3. Lax pair and the DDT
Due to complete integrability [26], equation (2) can be cast into the following 2×2 linear eigenvalue problem$\begin{eqnarray}{{\rm{\Phi }}}_{x}=M{\rm{\Phi }},\,\,\,\,\,\,{{\rm{\Phi }}}_{t}=N{\rm{\Phi }},\end{eqnarray}$where ${\rm{\Phi }}={({\phi }_{1},{\phi }_{2})}^{{\prime} }$ (′ mean a matrix transpose) is the vector eigenfunction, φ1 and φ2 are the complex functions of (x, t), M and N are the 2×2 square matrices, and$\begin{eqnarray}M=\lambda J+{M}_{1},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}N & = & {\lambda }^{2}J+\lambda {M}_{1}+\displaystyle \frac{1}{2}{N}_{1}\\ & & -\alpha (4{\lambda }^{3}J+4{\lambda }^{2}{M}_{1}+2\lambda {N}_{1}+{N}_{2})+\delta {N}_{3},\end{array}\end{eqnarray}$with i2=−1, J=diag(i,−i) and$\begin{eqnarray}{M}_{1}=\left(\begin{array}{cc}0 & q\\ r & 0\end{array}\right),\,\,\,\,\,\,\,\,{N}_{1}=\left(\begin{array}{cc}{\rm{i}}{qr} & -{\rm{i}}{q}_{x}\\ {\rm{i}}{r}_{x} & -{\rm{i}}{qr}\end{array}\right),\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{N}_{2} & = & \left(\begin{array}{cc}\ -\ {q}\ {{r}}_{{x}}+{{rq}}_{{x}} & \ \ 2\ {r}\ {{q}}^{2}-{{q}}_{{xx}}\\ \ 2\ {{r}}^{2}{q}-{{r}}_{{xx}} & \ \ {q}\ {{r}}_{{x}}-{{rq}}_{{x}}\end{array}\right),\\ {N}_{3} & = & \left(\begin{array}{cc}{\rm{i}}{{n}}_{11} & \ \ {{n}}_{12}\\ {{n}}_{21} & -{\rm{i}}{{n}}_{11}\end{array}\right),\end{array}\end{eqnarray}$$\begin{eqnarray*}\begin{array}{rcl}{n}_{11} & = & -8{\lambda }^{4}-4{rq}{\lambda }^{2}-2{\rm{i}}({{qr}}_{x}-{{rq}}_{x})\lambda \\ & & +{{qr}}_{{xx}}+{{rq}}_{{xx}}-3{q}^{2}{r}^{2}-{r}_{x}{q}_{x},\\ {n}_{12} & = & 4{\rm{i}}{q}_{x}{\lambda }^{2}-4{{rq}}^{2}\lambda +2{q}_{{xx}}\lambda \\ & & -{\rm{i}}{q}_{{xxx}}-2{\rm{i}}q(-3{{rq}}_{x}-4{\rm{i}}{\lambda }^{3}),\\ {n}_{21} & = & 4{\rm{i}}{r}_{x}{\lambda }^{2}+4{{qr}}^{2}\lambda -2{r}_{{xx}}\lambda \\ & & -{\rm{i}}{r}_{{xxx}}-2{\rm{i}}r(-3{r}_{x}q+4{\rm{i}}{\lambda }^{3}),\end{array}\end{eqnarray*}$the compatibility condition Mt−Nx+MN−NM=0.
The following unified DDT yields$\begin{eqnarray}{{\rm{\Phi }}}_{[1]}=K{\rm{\Phi }},\,\,\,\,\,\,K={I}_{2}-\displaystyle \frac{{\lambda }_{1}-\overline{{\lambda }_{1}}}{\lambda -\overline{{\lambda }_{1}}{P}_{1}},\end{eqnarray}$where$\begin{eqnarray}{P}_{1}=\displaystyle \frac{{{\rm{\Lambda }}}_{0}\overline{{{\rm{\Lambda }}}_{0}}}{\overline{{{\rm{\Lambda }}}_{0}}{{\rm{\Lambda }}}_{0}},\,\,\,\,\,\,{{\rm{\Lambda }}}_{0}={\rm{\Phi }}(x,t,{\lambda }_{1}){Z}_{0}=\left(\begin{array}{c}\phi \\ \varphi \end{array}\right),\end{eqnarray}$and I2=diag(1,1),Φ is a special vector for the lax pair with λ=λ1. It is necessary to show that the linear system can be rewritten in next form$\begin{eqnarray}{{\rm{\Phi }}}_{[1]x}={M}_{[1]}{{\rm{\Phi }}}_{[1]},\,\,\,\,\,\,{{\rm{\Phi }}}_{[1]t}={N}_{[1]}{{\rm{\Phi }}}_{[1]},\end{eqnarray}$and transformation between potential functions reads as$\begin{eqnarray}{r}_{[1]}={r}_{[0]}+{\rm{i}}(\overline{{\lambda }_{1}}-{\lambda }_{1})[{r}_{1},{\varrho }_{2}],\,\,\,\,\,\,\,{q}_{[1]}=-\overline{{r}_{[1]}},\end{eqnarray}$where commutator [A, B]=AB−BA.
4. Breather wave solutions and RWs
In what follows, we will construct breathing waves and RWs of the equation (2) based on the DDT.
4.1. Breather wave solutions
Based on the [27-34], the corresponding solution of the Lax pair can be sought in a new form$\begin{eqnarray*}\begin{array}{rcl}{\rm{\Phi }} & = & \left(\begin{array}{c}\phi (x,t)\\ \varphi (x,t)\end{array}\right)={\rm{\Delta }}{FGZ},\,\,\,\,\,\,\,Z=\left(\begin{array}{c}{\vartheta }_{1}\\ {\vartheta }_{2}\end{array}\right),\\ F & = & \exp ({\rm{i}}{Rx}),\,\,\,\,\,\,{G}=\exp ({\rm{i}}{St}),\\ {\rm{\Delta }} & = & \mathrm{diag}(1,\exp ({\rm{i}}({bx}+{ct}))),\end{array}\end{eqnarray*}$where Z is an arbitrary complex vector. It is easy to obtain that R and S meet$\begin{eqnarray}R=\left(\begin{array}{cc}\lambda & {\rm{i}}a\\ -{\rm{i}}a & -b-\lambda \end{array}\right),\,\,\,\,\,\,S=\left(\begin{array}{cc}{S}_{11} & {S}_{12}\\ {S}_{21} & {S}_{22}\end{array}\right),\end{eqnarray}$where$\begin{eqnarray*}\begin{array}{rcl}{S}_{11} & = & {\lambda }^{2}-\displaystyle \frac{1}{2}{a}^{2}+(-2{a}^{2}b+2{a}^{2}\lambda -4{\lambda }^{3})\alpha \\ & & +(-3{a}^{4}+3{a}^{2}{b}^{2}-4{a}^{2}b\lambda +4{a}^{2}{\lambda }^{2}-8{\lambda }^{4})\delta ,\\ {S}_{12} & = & -{S}_{21}=\displaystyle \frac{{\rm{i}}a}{2}((4{a}^{2}-2{b}^{2}+4b\lambda -8{\lambda }^{2})\alpha \\ & & +(-12{a}^{2}b+8{a}^{2}\lambda +2{b}^{3}-4{b}^{2}\lambda \\ & & +8b{\lambda }^{2}-16{\lambda }^{3})\delta +2\lambda -b),\\ {S}_{22} & = & -{\lambda }^{2}-\displaystyle \frac{1}{2}{a}^{2}+\displaystyle \frac{1}{2}{b}^{2}+(-4{a}^{2}b-2{a}^{2}\lambda +{b}^{3}+4{\lambda }^{3})\alpha \\ & & +(-3{a}^{4}+9{a}^{2}{b}^{2}+4{a}^{2}b\lambda -4{a}^{2}{\lambda }^{2}-{b}^{4}-8{\lambda }^{4})\delta .\end{array}\end{eqnarray*}$
Through complex calculations, the exponential matrices F can be written as$\begin{eqnarray}F={\tau }^{-1}\exp \left(-\displaystyle \frac{{\rm{i}}{xb}}{2}\right)\left(\begin{array}{cc}{F}_{1} & -2a\sinh \left(\tfrac{{\text{}}x\tau }{2}\right)\\ 2a\sinh \left(\tfrac{x\tau }{2}\right) & {F}_{4}\end{array}\right),\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}{F}_{1} & = & {\rm{i}}\sinh \left(\displaystyle \frac{x\tau }{2}\right)(b+2\lambda )+\tau \cosh \left(\displaystyle \frac{x\tau }{2}\right),\\ {F}_{4} & = & -{\rm{i}}\sinh \left(\displaystyle \frac{x\tau }{2}\right)(b+2\lambda )+\tau \cosh \left(\displaystyle \frac{x\tau }{2}\right),\\ \tau & = & \sqrt{-4{a}^{2}-{b}^{2}-4b\lambda -4{\lambda }^{2}}.\end{array}\end{eqnarray}$
According the Theorem, the first-order RWs arrive at$\begin{eqnarray}{R}_{1}={r}_{0}+\displaystyle \frac{4\overline{{\phi }_{1}[0]}{\phi }_{2}[0]}{{\phi }_{1}[0]\overline{{\phi }_{1}[0]}+{\phi }_{2}[0]\overline{{\phi }_{2}[0]}},\end{eqnarray}$where$\begin{eqnarray}{F}_{0}=\left(\begin{array}{cc}1-{ax} & -{ax}\\ {ax} & 1+{ax}\end{array}\right),\end{eqnarray}$$\begin{eqnarray}{G}_{0}=\left(\begin{array}{cc}1-{a}^{2}{\rm{i}}t-6{a}^{3}\alpha t-12{\rm{i}}{a}^{4}\delta t & -{\rm{i}}{a}^{2}t-6{a}^{3}\alpha t-12{\rm{i}}{a}^{4}\delta t\\ {\rm{i}}{a}^{2}t+6{a}^{3}\alpha t+12{\rm{i}}{a}^{4}\delta t & 1+{a}^{2}{\rm{i}}t+6{a}^{3}\alpha t+12{\rm{i}}{a}^{4}\delta t\end{array}\right),\end{eqnarray}$with$\begin{eqnarray}\left(\begin{array}{c}{\phi }_{1}[0]\\ {\phi }_{2}[0]\end{array}\right)={\rm{\Phi }}[0]={{AF}}_{0}{G}_{0}{Z}_{0}.\end{eqnarray}$
It is easy to observe that the expression of R1 contains two arbitrary parameters α and δ, which are the third-order dispersion coefficient and the four-order dispersion coefficient respectively. With the increase of α and δ, the crest of the RWs deflect clockwise and its width decreases. Figure 3 illustrates the above dynamic characteristics.
As shown in figure 4, a three-dimensional diagram of the second-order RWs are plotted, according to which we observe that the main peak is surrounded by four lower peaks when s1=0. However, when amplify s1, the single peak can split into three peaks, which are symmetric about the straight line t=0 in the (x,t)-plane of figure 5. The solution is called the ‘three sisters’ or a ‘RW triplet’.
Due to the complex expressions showing of the higher-order solutions, the third-order RWs are plotted on here. Figures 7 and 8 illustrate two different spatial and temporal distribution patterns of the third-order solution under the same parameter values and different sj(j=1,2,3), from which, we can see that the wave presents a ring pattern with the internal second-order fundamental mode. The distance between a ring pattern and inner second-order fundamental pattern increases with the increase of s2. Figure 7 is the ring-triangle distribution, figure 8 shows a five-pass distribution including six first-order basic patterns.
5. Conservation laws
In this section, according to [35-37] we present infinitely many independent conservation laws as a further support of the integrability for equation (2), from its Lax pairs. By introducing the function ${\rm{\Gamma }}=\tfrac{{\phi }_{2}}{{\phi }_{1}}$, the Ricatti equation obtained$\begin{eqnarray}{{\rm{\Gamma }}}_{x}=r-2{\rm{i}}\lambda {\rm{\Gamma }}-q{{\rm{\Gamma }}}^{2}.\end{eqnarray}$
Supposing Γ=Γ−1λ+Σn=0Γnλ−n , where Γn are the functions of x and t to be determined, substituting it into expression (39) and equating the coefficients of the same power of Γ to zero, the recurrence relations are obtained,$\begin{eqnarray*}\begin{array}{c}{\lambda }^{0}:{{\rm{\Gamma }}}_{1}=-\displaystyle \frac{1}{2}{\rm{i}}{r},\\ {\lambda }^{-1}:{{\rm{\Gamma }}}_{2}=\displaystyle \frac{1}{2}{\rm{i}}({q}{{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{1{x}})=\displaystyle \frac{1}{4}{qr},\\ {\lambda }^{-2}:{{\rm{\Gamma }}}_{3}=\displaystyle \frac{1}{2}{\rm{i}}({q}{{\rm{\Gamma }}}_{2}+{{\rm{\Gamma }}}_{2{x}})=\displaystyle \frac{1}{8}{{q}}^{2}{r},\\ {\lambda }^{-{n}}:{{\rm{\Gamma }}}_{{n}}=\displaystyle \frac{1}{2}{\rm{i}}({q}{{\rm{\Gamma }}}_{{n}-1}+{{\rm{\Gamma }}}_{{n}-1{x}})=\displaystyle \frac{1}{8}{{q}}^{{n}}{r}.\end{array}\end{eqnarray*}$From the compatibility condition ${(\mathrm{ln}{\phi }_{1})}_{{\text{}}{xt}}={(\mathrm{ln}{\phi }_{1})}_{{\text{}}{tx}}$, one can get$\begin{eqnarray}\begin{array}{rcl}{\left(q{{\rm{\Gamma }}}_{1}\right)}_{t} & = & (6{\rm{i}}{{\rm{\Gamma }}}_{1}{{qrq}}_{x}\delta -4{{\rm{\Gamma }}}_{2}{q}^{2}r\delta -2{{\rm{\Gamma }}}_{1}\alpha {q}^{2}r+4{\rm{i}}{{\rm{\Gamma }}}_{3}{q}_{x}\delta \\ & & {\left.-{\rm{i}}{{\rm{\Gamma }}}_{1}{q}_{{xxx}}\delta +2{{\rm{\Gamma }}}_{2}{q}_{{xx}}\delta -4{{\rm{\Gamma }}}_{3}\alpha q-8q{{\rm{\Gamma }}}_{4}\delta +q{{\rm{\Gamma }}}_{2}\right)}_{x}.\end{array}\end{eqnarray}$
Substituting expressions (5) into (40), the infinitely-many conservation laws for System (2) derived as$\begin{eqnarray}\displaystyle \frac{\partial {R}_{n}}{\partial t}=\displaystyle \frac{\partial {S}_{n}}{\partial x},\end{eqnarray}$with$\begin{eqnarray}\begin{array}{l}{R}_{n}=q{{\rm{\Gamma }}}_{n},\\ {S}_{n}=6{\rm{i}}{{\rm{\Gamma }}}_{n}{{qrq}}_{x}\delta -4{{\rm{\Gamma }}}_{n+1}{q}^{2}r\delta -2{{\rm{\Gamma }}}_{n}\alpha {q}^{2}r+4{\rm{i}}{{\rm{\Gamma }}}_{n+2}{q}_{x}\delta \\ -\,{\rm{i}}{{\rm{\Gamma }}}_{n}{q}_{{xxx}}\delta +2{{\rm{\Gamma }}}_{n+1}{q}_{{xx}}\delta -4{{\rm{\Gamma }}}_{n+2}\alpha q-8q{{\rm{\Gamma }}}_{n+3}\delta +q{{\rm{\Gamma }}}_{n+1},\end{array}\end{eqnarray}$where Rn and Sn represent the conserved fluxes and conserved densities, respectively.
6. Conclusions
In this paper, we studied the MI of equation (2), which described MI for the possible generation mechanism of RWs. Based on the DDT method, we studied the first-, second- and third-order RW solutions under different forms of equation (2). In order to help the readers better understand the solutions, figures 2-7 give the breather wave and RWs by looking for the appropriate parameters, respectively. Based on Lax pair (15), an infinite number of conservation laws (41) was constructed to prove the integrability of equation (2).
Figure 2.
New window|Download| PPT slide Figure 2.Breather solutions in equation (2) for parameters: $\alpha =\tfrac{1}{100}$, $\delta =\tfrac{1}{100}$ (a), (b), (c) $\lambda =\tfrac{3}{4}{\rm{i}}$ ; (d), (e), (f) $\lambda =\tfrac{5}{4}{\rm{i}}$.
Figure 3.
New window|Download| PPT slide Figure 3.One-order RWs in equation (2) for parameters (a), (d): $\alpha =\tfrac{1}{100}$, $\delta =\tfrac{1}{100}$; (b), (e): $\alpha =\tfrac{1}{50}$, $\delta =\tfrac{1}{50}$; (c), (f): $\alpha =\tfrac{1}{10}$, $\delta =\tfrac{1}{10}$.
Figure 4.
New window|Download| PPT slide Figure 4.Second-order RWs in equation (2) for parameters (a), (d): $\alpha =\tfrac{1}{100}$, $\delta =\tfrac{1}{100}$; (b), (e) $\alpha =\tfrac{1}{50}$, $\delta =\tfrac{1}{50}$; (c), (f): $\alpha =\tfrac{1}{10}$, $\delta =\tfrac{1}{10}$.
Figure 5.
New window|Download| PPT slide Figure 5.Second-order RWs in equation (2) for parameters $\alpha =\tfrac{1}{100}$, $\delta =\tfrac{1}{100}$ (a): s0=0s1=1, (b): s0=0s1=10, (c): s0=0s1=100.
New window|Download| PPT slide Figure 7.The third-order RWs in equation (2) for parameters $\alpha =\tfrac{1}{100}$, $\delta =\tfrac{1}{100}.$ (a) s0=0s1=1s2=1 , (b) s0=0s1=10s2=10, (c) s0=0s1=100s2=100.
Figure 8.
New window|Download| PPT slide Figure 8.The third-order RWs in equation (2) for parameters $\alpha =\tfrac{1}{100}$, $\delta =\tfrac{1}{100}\ {s}_{0}=0$, s1=0, (a) s2=10, (b) s2=100, (c) s2=1000.
Acknowledgments
We express our sincere thanks to the editor and the referees for their valuable comments. This work is supported by the National Natural Science Foundation of China (Grant No.71690242, No. 11731014, No. 12001241) and the Basic Research Program of Jiangsu Province (Grant No. BK20200885).
,1,21School of Mathematical Sciences, 
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