| Volume 7, Number 2, 2017, Pages 392-410 DOI：10.11948/2017025 |
| Existence of generalized homoclinic solutions of a coupled KdV-type Boussinesq system under a small perturbation |
| Yixia Shi,Shengfu Deng |
| Keywords:Generalized homoclinic solution, coupled KdV-type Boussinesq system, reversibility. |
| Abstract: |
| This paper considers the coupled KdV-type Boussinesq system with a small perturbation $u_{xx}=6cv-6u-6uv+\varepsilon f(\varepsilon,u,u_{x},v,v_{x}),$ $ v_{xx}=6cu-6v-3u^{2}+\varepsilon g(\varepsilon,u,u_{x},v,v_{x}),$ where $c=1+\mu$, $\mu>0$ and $\varepsilon$ are small parameters. The linear operator has a pair of real eigenvalues and a pair of purely imaginary eigenvalues. We first change this system into an equivalent system with dimension 4, and then show that its dominant system has a homoclinic solution and the whole system has a periodic solution if the perturbation functions $g$ and $h$ satisfy some conditions. By using the contraction mapping theorem, the perturbation theorem, and the reversibility, we theoretically prove that this homoclinic solution, when higher order terms are added, will persist and exponentially approach to the obtained periodic solution (called generalized homoclinic solution) for small $\varepsilon$ and $\mu>0$. |
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