Existence of generalized solitary wave solutions of the coupled KdV-CKdV system
type: Item
instance of: scholarly article
title: Existence of generalized solitary wave solutions of the coupled KdV-CKdV system
zbMATH Open document ID: 1495.34066
author: Haijing Li
author: Shengfu Deng
published in: Qualitative Theory of Dynamical Systems
publication date: 2022-03-15T00:00:00Z
review text: The paper under review studies the existence of generalized solitary wave solutions to the following coupled KdV-CKdV system \[ \begin{cases} u_t+2bu_\xi+au_{\xi\xi\xi}=-2b(uv)_\xi,\\ v_t+bv_\xi+bvv_\xi+cv_{\xi\xi\xi}=-b(|u|^2)_\xi, \end{cases}\tag{1} \] where \(a,b,c\in\mathbb R\) are constants. The authors consider the traveling wave solution of (1) with the form \(\xi=x-\tilde{c}t\) for a traveling wave speed \(\tilde{c}\). By setting \[ u(x)=u(\xi,t),\qquad v(x)=v(\xi,t), \] and inegrating it once, with integral constant equal to zero, then the existence of generalized solitary wave solutions of the system (1) is changed into the existence of generalized homoclinic solutions to the following ordinary differential system \[ \begin{cases} u_x=u_1,\\ u_{1x}=\frac{1}{a}\left(\left(\tilde{c}-2b\right)u-2buv\right),\\ v_x=v_1,\\ v_{1x}=\frac{1}{c}\left(\left(\tilde{c}-2b\right)v-bu^2-\frac{b}{2}v^2\right) \end{cases} \] for \(ac\neq 0\). The main result of the paper can be formulated as follows. Theorem 2. Assume that \(c<0\), \(\tilde{c}a>0\), \(b=\tilde{c}+\varepsilon\) and \(\varepsilon>0\) is a small parameter where \(\tilde{c}\) is the traveling speed. There exist constants \(\varepsilon_0>0\) and \(I_0>0\) such that for each \(\varepsilon\in(0,\varepsilon_0)\) and \(I=I_0\varepsilon^{\frac{3}{2}}\), system (1) has an even reversible generalized solitary wave solution exponentially approaching a periodic solution. Moreover, the expressions of the even functions \(u\) and \(v\) are given by \[\begin{split} u(\xi,t)&=2I\zeta(\xi-ct)\cos{\left(\left(\sqrt{\frac{\tilde{c}+2\varepsilon}{a}}+r_1\right)(\xi-\tilde{c}t+\theta)\right)}+\mathcal{R}_{00}(\xi-\tilde{c} t,\varepsilon)\\ &+\mathcal{R}_{01}(\xi-\tilde{c} t-\theta,\varepsilon), \end{split}\] \[\begin{split} v(\xi,t)&=-\frac{3}{\tilde{c}}\varepsilon~\mathrm{sech}^2\left(\frac{1}{2}\sqrt{-\frac{\varepsilon}{c}}(\xi-\tilde{c}t)\right)-\frac{2\tilde{c}}{\varepsilon}I^2\zeta(\xi-\tilde{c}t)\\ &+\frac{a}{2c}I^2\zeta(\…
reviewed by: Xingchen Yu
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 7489703
zbMATH Keywords: coupled KdV-CKdV system
zbMATH Keywords: generalized solitary wave solutions
zbMATH Keywords: generalized homoclinic solutions
zbMATH Keywords: reversibility
MaRDI profile type: Publication
full work available at URL: s12346-022-00570-5
OpenAlex ID: W4212965358
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DOI: 10.1007/S12346-022-00570-5
description: scientific article; zbMATH DE number 7489703