A multiplicity result for the generalized Kadomtsev-Petviashvili equation
The authors consider the problem of uniqueness of weak solitary-wave solutions to the (generally, nonintegrable) equation \[ u_t+u_{xxx} + f'(u)\cdot u_x=D_x^{-1}u_{yy}. \] The solitary wave is sought for in the form \(u(x,y,t) = u(x - ct,y)\), \(c\) being the velocity of its motion along the \(x\) axis. The solutions of this type are looked for as stationary points of the corresponding Hamiltonian. The main result is a proof of the fact that, under certain conditions imposed on the function \(f(u)\) in the equation (essentially, a class of power functions \(f(u) \sim u^p\) with \(p\) = const is considered), the equation has, at least, \textit{two} geometrically distinct weak solitary-wave solutions. The proof is based on the use of the techniques of the Lyusternik - Schnirelman theory for stationary points of the Hamiltonian.
scientific article; zbMATH DE number 1127673