Existence of traveling waves for a fourth order Schr\" odinger equation with mixed dispersion in the Helmholtz regime

TL;DR

This paper proves the existence of traveling wave solutions for a fourth order Schrödinger equation with mixed dispersion in the Helmholtz regime, using dual methods and providing resolvent estimates.

Contribution

It introduces a novel existence proof for traveling waves in this specific PDE setting and derives resolvent estimates applicable to similar operators.

Findings

Existence of traveling wave solutions under certain conditions.

Derived Green function estimates for the operator.

Established $L^p - L^q$ resolvent bounds.

Abstract

In this paper, we study the existence of traveling waves for a fourth order Schr\" odinger equations with mixed dispersion, that is, solutions to $$\Delta^2 u +\beta \Delta u +i V \nabla u +\alpha u =|u|^{p-2} u,\ in\ \R^N ,\ N\geq 2.$$ We consider this equation in the Helmholtz regime, when the Fourier symbol $P$ of our operator is strictly negative at some point. Under suitable assumptions, we prove the existence of solution using the dual method of Evequoz and Weth provided that $p\in (p_1 , 2N/(N-4)_+)$. The real number $p_1$ depends on the number of principal curvature of $M$ staying bounded away from $0$, where $M$ is the hypersurface defined by the roots of $P$. We also obtained estimates on the Green function of our operator and a $L^p - L^q$ resolvent estimate which can be of independent interest and can be applied to other operators.