Initial value problems and solutions of the Kadomtsev-Petviashvili equation. (English) Zbl 1064.35160

Ablowitz, M. J.; Villarroel, J.

Shabat, A.B.(ed.) et al., New trends in integrability and partial solvability. Proceedings of the NATO Advanced Research Workshop, Cadiz, Spain, June 12–16, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1835-5/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 132, 1-47 (2004).

Summary: Initial value problems and solutions associated with the Kadomtsev-Petviashvili equation \[ (u_t+u_{xxx}+6uu_x)_x+3 \varepsilon^2u_{yy}=0, \] are analyzed. The discussion includes the inverse scattering transform for suitably decaying data, solutions decaying off a background line multi-pole lump soliton solutions and solutions which are slowly decaying. Existence and uniqueness of the associated eigenfunctions are discussed in terms of natural functional norms.
For the entire collection see [Zbl 1050.35003].

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Keywords:

discrete spectrum; decaying data; lump soliton solutions; Existence; uniqueness; eigenfunctions