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摘要
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(epsilon x)u(m))(x) = 0, in R-t x R-x, m = 2, 3, and 4, with 0 <= lambda < 1, 1 < a(.) < 2 a strictly increasing, positive, and asymptotically flat potential, and epsilon small enough. In [C. Munoz, "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 ->-infinity) parallel to u(t) - Q(. - (1 - lambda)t)parallel to(H1(R)) = 0, 0 <= lambda < 1, provided epsilon 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 <= 5-m/m+3, sup(t >> 1/epsilon) parallel to u(t) - 2(-1/(m-1))Q(c infinity)(. - rho(t))parallel to(H1(R)) <= K epsilon(1/2), for suitable scaling and translation parameters c(infinity)(lambda) >= 1 and rho '(t) similar to (c(infinity) - 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 5-m/m+3 < lambda < 1. Surprisingly, there exists a fixed, positive number (lambda) over tilde is an element of (5-m/m+3, 1), independent of epsilon, such that the following alternative holds: (1) Refraction. For all 5-m/m+3 < lambda < (lambda) over tilde, the soliton solution behaves as in [C. Munoz, "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(infinity) < 1 and rho '(t) similar to c(infinity) - lambda > 0. (2) Reflection. If (lambda) over tilde < lambda < 1, then the soliton-like solution is reflected by the potential and satisfies sup(t >> 1/epsilon) parallel to u(t) - Q(c infinity) (. - rho(t))parallel to(H1(R)) <= K epsilon(1/2), with 0 < c(infinity) < lambda and rho '(t) similar to c(infinity) - lambda < 0. This last is a completely new type of soliton-like solution for gKdV equations, also present in the nonlinear Schrodinger case [C. Munoz, "On the soliton dynamics under slowly varying medium for generalized nonlinear Schrodinger equations," Math. Ann., to appear]. Moreover, for any 0 < lambda < 1, with <(lambda)over tilde> (sic) lambda, the solution is not pure as t -> +infinity, in the sense that limsup(t ->+infinity) parallel to u(t) - kappa(lambda)Q(c infinity)(. - rho(t))parallel to(H1(R)) > 0, with kappa(lambda) depending on lambda.
- 出版日期2012
