Nondegeneracy, Morse Index and Orbital Stability of the Lump Solution to the KP-I Equation

Abstract

Let $Q(x,y)=4\frac{y^2-x^2+3}{(x^2+y^2+3{)}^2}$ be the lump solution of the KP-I equation $${\partial }_x^2({\partial }_x^{2}u-u+3u^2)-{\partial }_y^{2}u=0.$$ We show that this solution is rigid in the following senses: the only decaying solutions to the linearized operator $${\partial }_x^2({\partial }_x^2\varphi -\varphi +6Q\varphi )-{\partial }_y^2\varphi =0$$ consist of the linear combinations of ${\partial }_{x}Q$ and ${\partial }_{y}Q$. Furthermore we show that the Morse index is exactly one and that it is orbital stable.

37 pages; Morse index and orbital stability added;comments welcome

References in corpus (1)

• Stability of the line soliton of the Kadomtsev--Petviashvili-I equation with the critical traveling speed

Cited by in corpus (2)

• Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
• Nondegeneracy of the traveling lump solution to the $2+1$ Toda lattice