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

2010/08/13 by Sanjiban Santra, Juncheng Wei, Santra, Sanjiban +1 · 1 citation
Computer Science · Mathematics · #35J10 #35J65 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1008.2248

openalex publication_date 2010/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we construct a new kind of positive solutions of \De u-u+u^p=0 on \R² when p> 2. These solutions u(x,z)∼ \om(x-f(z))+ ∑_i=1^∞\om₀((x, z)-ξ_ie₁) as L→ +∞ where \om is a unique positive homoclinic solution of \om"-\om+\om^p=0 in \R ; \om₀ is the two dimensional positive solution and e₁= (1, 0) and ξ_j are points such that ξ_j= jL+ O(1) for all j≥ 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.

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• From nonlinear Schrödinger equation to interacting particle system: 1 < p < 2

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