Nondegeneracy, Morse index and Orbital stability of the lump solution to the KP-I equation
Abstract：The KP-I equation is an important dispersive equation appearing in many physical contexts. It is also a classical 2+1 dimensional integrable system. Its one dimensional reduction is the well-known KdV equation. In this talk, we discuss the nonlinear stability of the classical lump solution to the KP-I equation. We investigate the Backlund transformation associated to this lump. Based on this, we show that the lump solution is nondegenerated in the sense that the corresponding linearized KP-I operator does not have nontrivial kernel. Generalizing this argument, we also prove that a particular family of $y$-periodic soliton solutions connected to lump is nondegenerated in suitable sense. Finally, using these results, we prove that the Morse index of the lump solution is equal to one and the lump is orbitally stable. The question of asymptotical stability remains open. This is joint work with Juncheng Wei from UBC.