Inelastic Character of Solitons of Slowly Varying gKdV Equations

Abstract

In this paper we study soliton-like solutions of the variable coefficients, the subcritical gKdV equation

$$u_t + (u_{xx} -\lambda u + a(\varepsilon x) u^m )_x =0,\quad {\rm in} \quad \mathbb{R}_t\times\mathbb{R}_x, \quad m=2,3\,\, { \rm and }\,\, 4,$$

with ${\lambda\geq 0, a(\cdot ) \in (1,2)}$ a strictly increasing, positive and asymptotically flat potential, and ${\varepsilon}$ small enough. In previous works (Muñoz in Anal PDE 4:573–638, 2011; On the soliton dynamics under slowly varying medium for generalized KdV equations: refraction vs. reflection, SIAM J. Math. Anal. 44(1):1–60, 2012) the existence of a pure, global in time, soliton u(t) of the above equation was proved, satisfying

$$\lim_{t\to -\infty}\|u(t) - Q_1(\cdot -(1-\lambda)t) \|_{H^1(\mathbb{R})} =0,\quad 0\leq \lambda<1,$$

provided ${\varepsilon}$ is small enough. Here R(t, x) := Q _c (x − (c − λ)t) is the soliton of R _t +  (R _xx −λ R + R ^m) _x = 0. In addition, there exists ${\tilde \lambda \in (0,1)}$ such that, for all 0 < λ < 1 with ${\lambda\neq \tilde \lambda}$ , the solution u(t) satisfies

$$\sup_{t\gg \frac{1}{\varepsilon}}\|u(t) - \kappa(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}\lesssim \varepsilon^{1/2}.$$

Here ${{\rho'(t) \sim (c_\infty(\lambda) -\lambda)}}$ , with ${{\kappa(\lambda)=2^{-1/(m-1)}}}$ and ${{c_\infty(\lambda)>\lambda}}$ in the case ${0<\lambda<\tilde\lambda}$ (refraction), and ${\kappa(\lambda) =1}$ and c _∞(λ) < λ in the case ${\tilde \lambda<\lambda<1}$ (reflection).

In this paper we improve our preceding results by proving that the soliton is far from being pure as t → + ∞. Indeed, we give a lower bound on the defect induced by the potential a(·), for all ${{0<\lambda<1, \lambda\neq \tilde \lambda}}$ . More precisely, one has

$$\liminf_{t\to +\infty}\| u(t) - \kappa_m(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}>rsim \varepsilon^{1 +\delta},$$

for any ${{\delta>0}}$ fixed. This bound clarifies the existence of a dispersive tail and the difference with the standard solitons of the constant coefficients, gKdV equation.