Construction and characterization of solutions converging to solitons for supercritical gKdV equations

Vianney Combet

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arXiv · 0907.5486

Construction and characterization of solutions converging to solitons for supercritical gKdV equations

Vianney Combet

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Abstract

We consider the generalized Korteweg-de Vries equation in the supercritical case, and we are interested in solutions which converge to a soliton in large time in H^1. In the subcritical case, such solutions are forced to be exactly solitons by variational characterization, but no such result exists in the supercritical case. In this paper, we first construct a "special solution" in this case by a compactness argument, i.e. a solution which converges to a soliton without being a soliton. Secondly, using a description of the spectrum of the linearized operator around a soliton due to Pego and Weinstein, we construct a one parameter family of special solutions which characterizes all such special solutions.

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Vianney Combet. 2009-08-04. Construction and characterization of solutions converging to solitons for supercritical gKdV equations. https://arxiv.org/abs/0907.5486

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