Nondegeneracy of the traveling lump solution to the $2+1$ Toda lattice

Abstract

We consider the $2+1$ Toda system \[ \frac{1}{4}\Delta q_{n}=e^{q_{n-1}-q_{n}}-e^{q_{n}-q_{n+1}}\text{ in }\mathbb{R}^{2},\ n\in\mathbb{Z}. \] It has a traveling wave type solution $\left\{ Q_{n}\right\} $ satisfying $Q_{n+1}(x,y)=Q_{n}(x+\frac{1}{2\sqrt{2}},y)$, and is explicitly given by \[ Q_{n}\left( x,y\right) =\ln\frac{\frac{1}{4}+\left( n-1+2\sqrt{2}x\right) ^{2}+4y^{2}}{\frac{1}{4}+\left( n+2\sqrt{2}x\right) ^{2}+4y^{2}}. \] In this paper we prove that \{$Q_{n}$\} is nondegenerate.