Orbital stability of solitary waves of coupled KdV equations (Q1854072)

The authors investigate the following system of coupled KdV equations \[ u_t + u_{xxx} + 6uu_x - 2bvv_x = 0, \qquad v_t + v_{xxx} + 3uv_x = 0, \quad x\in\mathbb{R}, \] where \(b\) is a positive constant. For any positive \(c\) there exists a solitary wave solution of the form \(u(t,x)=\varphi _c(x-ct)\), \(v(t,x)=\psi _c(x-ct)\), where \(\varphi _c(x) = 2c {\operatorname {sech}}^2(\sqrt {c}x)\) and \(\psi _c(x) =c\sqrt {6/b}\operatorname {sech} (\sqrt {c}x)\). It is proved that this solution is orbitally stable; i.e., for every positive \(\varepsilon \) there exists a positive \(\delta \) such that: if any element of \(X=L^2(\mathbb{R})\times H^1(\mathbb{R})\) whose distance from \((\varphi _c, \psi _c)\) in \(X\) does not exceed \(\delta \), is taken as an initial condition, then the system has a solution \((u(t,x), v(t,x))\) which exists for \(t\geq 0\) and satisfies \[ \sup _{0<t<\infty } \inf _{s\in \mathbb{R}} |(u(t,\cdot)- \varphi_c(\cdot -s),\quad v(t,\cdot)- \psi _c(\cdot -s))|_X <\varepsilon. \]