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

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 $\displaystyle{u(x,z)\sim \om(x-f(z))+ \sum_{i=1}^{\infty}\om_{0}((x, z)-ξ_i\vec{e}_{1})}$ as $L\to +\infty$ where $\om$ is a unique positive homoclinic solution of $\om"-\om+\om^{p}=0$ in $\mathbb{R}$ ; $\om_{0}$ 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 $j\ge 1.$ 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.

5 figures, American Journal of Math (to appear)

References in corpus (1)

• Constant mean curvature surfaces with three ends