Advances in Differential Equations

Nonlinear travelling waves on complete Riemannian manifolds

Mayukh Mukherjee

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We study travelling wave solutions to nonlinear Schrödinger and Klein-Gordon equations on complete Riemannian manifolds, which have a bounded Killing field $X$. For a natural class of power-type nonlinearities, we use standard variational techniques to demonstrate the existence of travelling waves on complete weakly homogeneous manifolds. If the manifolds in question are weakly isotropic, we prove that they have genuine subsonic travelling waves, at least for a non-empty set of parameters. Finally we establish that a slight perturbation of the Killing field $X$ will result in a controlled perturbation of the travelling wave solutions (in appropriate $L^p$-norms).

Article information

Adv. Differential Equations Volume 23, Number 1/2 (2018), 65-88.

First available in Project Euclid: 26 October 2017

Permanent link to this document

Primary: 35J61: Semilinear elliptic equations 35H20: Subelliptic equations


Mukherjee, Mayukh. Nonlinear travelling waves on complete Riemannian manifolds. Adv. Differential Equations 23 (2018), no. 1/2, 65--88.

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