Ground states of some coupled nonlocal fractional dispersive PDEs

TL;DR

This paper proves the existence of positive, radially symmetric ground state solutions for coupled fractional dispersive PDEs, including nonlinear fractional Schrödinger and Korteweg-de Vries equations, across various dimensions.

Contribution

It establishes the existence of ground states for a class of coupled fractional PDEs with positive symmetry, extending previous results to fractional and coupled systems.

Findings

Existence of positive radially symmetric ground states for all positive coupling parameters.

Ground states exist in dimensions 1 to 3 for fractional order s between n/4 and 1.

Results apply to coupled fractional Schrödinger and KdV equations.

Abstract

We show the existence of ground state solutions to the following stationary system coming from some coupled fractional dispersive equations such as: nonlinear fractional Schr\"odinger (NLFS) equations (for dimension $n=1,\, 2,\, 3$) or NLFS and fractional Korteweg-de Vries equations (for $n=1$), $$ \left \{ \begin{array}{ll} (-\Delta)^{s} u+ \lambda_1 u &= u_1^{3}+\beta uv,\quad u\in W^{s,2}(\mathbb{R}^n), (-\Delta)^{s} v + \lambda_2 v &= \frac 12 v^{2}+\frac 12 \beta u^2,\quad v\in W^{s,2}(\mathbb{R}^n), \end{array} \right. $$ where $\lambda_j>0$, $j=1,2$, $\beta\in \mathbb{R}$, $n=1,\, 2,\, 3$, and $\frac n4< s<1$. Precisely, we prove the existence of a positive radially symmetric ground state for any $\beta>0$.