On the nonexistence of pure multi-solitons for the quartic gKdV equation

TL;DR

This paper proves that pure multi-soliton solutions do not exist for the quartic gKdV equation within a certain range of speeds, extending previous results to more general cases without requiring detailed solution descriptions.

Contribution

It establishes the nonexistence of pure multi-solitons for the quartic gKdV equation for a range of speeds using a new approach that avoids detailed solution analysis.

Findings

Pure multi-solitons do not exist for certain speed ranges.

The nonexistence applies to any number of solitons greater than one.

The proof extends previous results to a broader set of conditions.

Abstract

We consider the quartic (nonintegrable) (gKdV) equation. Let u(t) be an outgoing 2-soliton of the equation, i.e. a solution behaving exactly as the sum of two solitons (of speeds c1 and c2) for large positive time. In arXiv:0910.3204, for nearly equal solitons, the solution u(t) is computed up to some order of epsilon=1-c2/c1, everywhere in time and space. In particular, it is deduced that u(t) is not a multi-soliton for large negative time, proving the nonexistence of pure multi-soliton in this context. In the present paper, we prove the same result for an explicit range of speeds: 3/4 c1< c2< c1, by a different approach, which does not longer require a precise description of the solution. In fact, the nonexistence result holds for outgoing N-solitons, for any N>1, under an explicit assumption on the speeds, which is a natural generalization of the condition for N=2.