Mixed dimensional infinite soliton trains for nonlinear Schr\"odinger equations

TL;DR

This paper constructs complex solutions to nonlinear Schrödinger equations called mixed dimensional infinite soliton trains, which consist of infinitely many solitons of different dimensions, extending previous work on single-dimensional trains.

Contribution

It introduces the construction of mixed dimensional infinite soliton trains for nonlinear Schrödinger equations, expanding the understanding of multi-dimensional soliton solutions.

Findings

Successfully constructed mixed dimensional infinite soliton trains.

Extended previous single-dimensional train results to multiple dimensions.

Utilized spatial $L^ Infty$ bounds for lower dimensional trains.

Abstract

In this note we construct mixed dimensional infinite soliton trains, which are solutions of nonlinear Schr\"odinger equations whose asymptotic profiles at time infinity consist of infinitely many solitons of multiple dimensions. For example infinite line-point soliton trains in 2D space, and infinite plane-line-point soliton trains in 3D space. This note extends the works of Le Coz, Li and Tsai [5, 6], where single dimensional trains are considered. In our approach, spatial $L^\infty$ bounds for lower dimensional trains play an essential role.