Existence of generalized solitary wave solutions of the coupled KdV-CKdV system (Q2114404)

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 \[\begin{split} u(\xi,t)&=2I\zeta(\xi-ct)\cos{\left(\left(\sqrt{\frac{\tilde{c}+2\varepsilon}{a}}+r_1\right)(\xi-\tilde{c}t+\theta)\right)}+\mathcal{R}_{00}(\xi-\tilde{c} t,\varepsilon)\\ &+\mathcal{R}_{01}(\xi-\tilde{c} t-\theta,\varepsilon), \end{split}\] \[\begin{split} v(\xi,t)&=-\frac{3}{\tilde{c}}\varepsilon~\mathrm{sech}^2\left(\frac{1}{2}\sqrt{-\frac{\varepsilon}{c}}(\xi-\tilde{c}t)\right)-\frac{2\tilde{c}}{\varepsilon}I^2\zeta(\xi-\tilde{c}t)\\ &+\frac{a}{2c}I^2\zeta(\xi-\tilde{c}t)\cos{\left(2\left(\sqrt{\frac{\tilde{c}+2\varepsilon}{a}}+r_1\right)(\xi-\tilde{c}t+\theta)\right)}+\mathcal{R}_{10}(\xi-\tilde{c} t,\varepsilon)\\ &+\mathcal{R}_{11}(\xi-\tilde{c} t-\theta,\varepsilon), \end{split}\] where the phase shift \(\theta\) is of order \(O(\sqrt{\varepsilon})\), \(\zeta(y)\) is a smooth even cut-off function, \(\mathcal{R}_{k0}\) and \(\mathcal{R}_{k1}~(k=0,1)\) are smooth functions in their arguments, \(\mathcal{R}_{k1}\) is periodic with period \(\frac{2\pi\sqrt{a}}{\sqrt{\tilde{c}+2\varepsilon}+a\sqrt{a}r_1}\) for some constant \(r_1=O(\sqrt{\varepsilon})\), and \(\mathcal{R}_{k0}\) and \(\mathcal{R}_{k1}\) satisfy uniformly with respect to the parameter \(\varepsilon\) for \(\xi-\tilde{c}t\in\mathbb R\), \[\begin{split} |\mathcal{R}_{k0}(\xi-\tilde{c} t,\varepsilon)|\le &M\varepsilon^{\frac{3}{2}}e^{-\frac{3}{4}\sqrt{-\frac{\varepsilon}{c}}|\xi-\tilde{c}t|},\quad |\mathcal{R}_{01}(\xi-\tilde{c}t-\theta,\varepsilon,I)|\le M\varepsilon^2,\\ &|\mathcal{R}_{11}(\xi-\tilde{c}t-\theta,\varepsilon,I)|\le M\varepsilon^{\frac{7}{2}}, \end{split}\] where \(M\) is a generic constant. The proof of Theorem 2 relies on the fixed point theorem, the pertubation methods and the reversibility, which are basically similar to one in [\textit{S. Deng} and \textit{B. Guo}, J. Dyn. Differ. Equations 24, No. 4, 761--776 (2012; Zbl 1257.35155); \textit{Y. Shi} and \textit{S. Deng}, J. Appl. Anal. Comput. 7, No. 2, 392--410 (2017; Zbl 1474.34288); \textit{Y. Shi} and \textit{M. Han}, Discrete Contin. Dyn. Syst., Ser. S 13, No. 11, 3189--3204 (2020; Zbl 1469.34061)].