New entire positive solution for the nonlinear Schrodinger equation: Coexistence of fronts and bumps

TL;DR

This paper constructs new positive solutions for a nonlinear Schrödinger equation in two dimensions, demonstrating the coexistence of front-like and bump-like structures, a novel phenomenon in the field.

Contribution

It introduces the first known solutions exhibiting coexistence of fronts and bumps for the nonlinear Schrödinger equation, linking to geometric concepts like triunduloids.

Findings

Existence of solutions with combined front and bump structures.

Asymptotic behavior characterized by a sum of homoclinic and localized solutions.

First demonstration of coexistence phenomena in this context.

Abstract

In this paper we construct a new kind of positive solutions of $$\De u-u+u^{p}=0 \text{on} \R^2$$ when $p> 2.$ These solutions $\displaystyle{u(x,z)\sim \om(x-f(z))+ \sum_{i=1}^{\infty}\om_{0}((x, z)-\xi_i\vec{e}_{1})}$ as $L\rightarrow +\infty$ where $\om$ is a unique positive homoclinic solution of $\om"-\om+\om^{p}=0$ in $\R$ ; $\om_{0}$ is the two dimensional positive solution and $\vec{e}_{1}= (1, 0)$ and $\xi_{j}$ are points such that $\xi_{j}= jL+ \mathcal{O}(1)$ for all $j\geq 1.$ This represents a first result on the {\em coexistence} of fronts and bumps. Geometrically, our new solutions correspond to {\em triunduloid} in the theory of CMC surface.