Dynamics of Soliton-like Solutions for Slowly Varying, Generalized KdV Equations: Refraction versus Reflection

Abstract

In this work we continue our study of the description of the soliton-like solutions of the variable coefficients, subcritical gKdV equation $u_t + (u_{xx} -\lambda u + a(\varepsilon x) u^m )_x =0$, in $\mathbb{R}_t\times \mathbb{R}_x$, $m=2,3$, and 4, with $0\leq \lambda<1$, $1<a(\cdot )<2$ a strictly increasing, positive, and asymptotically flat potential, and $\varepsilon$ small enough. In [C. Muñoz, “On the soliton dynamics under a slowly varying medium for generalized KdV equations,” Anal. PDE, to appear] we proved the existence (and uniqueness in most cases) of a pure soliton-like solution $u(t)$ satisfying $\lim_{t\to -\infty}\|u(t) - Q(\cdot -(1-\lambda)t) \|_{H^1(\mathbb{R})} =0$, $0\leq \lambda<1$, provided $\varepsilon$ is small enough. Here $R(t,x) := Q_c(x-(c-\lambda)t)$ is the standard $H^1$-soliton solution of $R_t + (R_{xx} -\lambda R + R^m)_x =0$. In addition, this solution is global in time and satisfies (i) for all $0<\lambda\leq\frac{5-m}{m+3}$, $\sup_{t\gg \frac 1\varepsilon }\|u(t) - 2^{-1/(m-1)}Q_{c_\infty}(\cdot -\rho(t)) \|_{H^1(\mathbb{R})} \leq K\varepsilon^{1/2}$, for suitable scaling and translation parameters $c_\infty(\lambda)\geq 1$ and $\rho'(t) \sim (c_\infty-\lambda)$, and for $K>0$. In the cubic case, $m=3$, this result also holds for $\lambda=0$. The purpose of this paper is the following: We give an almost complete description of the remaining case $\frac{5-m}{m+3}<\lambda<1$. Surprisingly, there exists a fixed, positive number $\tilde \lambda \in (\frac{5-m}{m+3} ,1)$, independent of $\varepsilon$, such that the following alternative holds: (1) Refraction. For all $\frac{5-m}{m+3}<\lambda<\tilde \lambda$, the soliton solution behaves as in [C. Muñoz, “On the soliton dynamics under a slowly varying medium for generalized KdV equations,” Anal. PDE, to appear] and satisfies (i) above, but now $\lambda <c_\infty<1$ and $\rho'(t) \sim c_\infty -\lambda >0$. (2) Reflection. If $\tilde \lambda <\lambda<1$, then the soliton-like solution is reflected by the potential and satisfies $\sup_{t\gg \frac 1\varepsilon }\|u(t) - Q_{c_\infty}(\cdot -\rho(t)) \|_{H^1(\mathbb{R})} \leq K\varepsilon^{1/2},$ with $0<c_\infty <\lambda$ and $\rho'(t) \sim c_\infty-\lambda <0$. This last is a completely new type of soliton-like solution for gKdV equations, also present in the nonlinear Schrödinger case [C. Muñoz, “On the soliton dynamics under slowly varying medium for generalized nonlinear Schrödinger equations,” Math. Ann., to appear]. Moreover, for any $0<\lambda<1$, with $\tilde\lambda\neq \lambda$, the solution is not pure as $t\to +\infty$, in the sense that $\limsup_{t\to +\infty}\|u(t) - \kappa(\lambda)Q_{c_\infty}(\cdot -\rho(t)) \|_{H^1(\mathbb{R})}>0,$ with $\kappa(\lambda) $ depending on $\lambda$.