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Nondegeneracy of the lump solution to the KP-I equation Liu, Yong


The KP-I equation \[ \partial_{x}\left( \partial_{t}u+\partial_{x}^{3}u+3\partial_{x}\left( u^{2}\right) \right) -\partial_{y}^{2}u=0 \] has a lump solution of the form $Q\left( x-t,y\right) ,$ where \[ Q\left( x,y\right) =Q\left( x,y\right) =4\frac{y^{2}-x^{2}+3}{\left( x^{2}+y^{2}+3\right)^{2}}. \] We show that $Q$ is nondegenerate in the following sense: Suppose $\phi$ is a smooth solution to the equation \[ \partial_{x}^{2}\left( \partial_{x}^{2}\phi-\phi+6Q\phi\right) -\partial _{y}^{2}\phi=0. \] Assume \[ \phi\left( x,y\right) \rightarrow0,\text{ as }x^{2}+y^{2}\rightarrow+\infty. \] Then $\phi=c_{1}\partial_{x}Q+c_{2}\partial_{y}Q,$ for certain constants $c_{1},c_{2}.$

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