Multi-solitary waves for the one-dimensional Zakharov system

TL;DR

This work proves the existence of multi-soliton solutions for the one-dimensional Zakharov system by constructing a backward-in-time sequence of near-multi-soliton solutions and employing modulation theory to align pulsations, translations, and phases. A Weinstein functional G(t) is crafted to serve as a Lyapunov-type quantity, with careful coercivity estimates ensuring the residuals decay exponentially. By combining local-quantity control, coercivity of the quadratic forms, and a bootstrap argument, the authors obtain exponential convergence to a sum of $K$ solitary waves with distinct speeds $c_k$. The analysis also introduces modified energies to gain higher-regularity control, enabling a compactness argument that yields a limiting solution that asymptotically decomposes into the prescribed multi-soliton configuration. The result extends the multi-soliton framework to the Zakharov system in dimension 1 and provides a structured approach for higher regularity via modified energies.

Abstract

Given different speeds $c_1$ , ... , $c_K$, in the present paper we establish the existence of a solution to the Zakharov system in dimension 1 that behaves asymptotically like a $K$-solitary wave, each wave travelling with speed $c_k$. The proof is adapted from previous results for the NLS and gKdV equations.