Partially concentrating solutions for systems with Lotka-Volterra type interactions

TL;DR

This paper establishes the existence of standing wave solutions with partial concentration in systems of equations with Lotka-Volterra interactions, highlighting novel behaviors in multi-component coupled systems.

Contribution

It proves the existence of partially concentrating standing waves for multi-component Lotka-Volterra systems, especially analyzing the three-equation case and comparing different coupling types.

Findings

Existence of standing wave solutions with specific asymptotic profiles.

Last components exhibit concentration while the first remains quantum.

Comparison of Lotka-Volterra and other coupling systems.

Abstract

In this paper we consider the existence of standing waves for a coupled system of $k$ equations with Lotka-Volterra type interaction. We prove the existence of a standing wave solution with all nontrivial components satisfying a prescribed asymptotic profile. In particular, the $k-1$-last components of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature. We analyze first in detail the result with three equations since this is the first case in which the coupling has a role contrary to what happens when only two densities appear. We also discuss the existence of solutions of this form for systems with other kind of couplings making a comparison with Lotka-Volterra type systems.