Lump solutions to nonlinear partial differential equations via Hirota bilinear forms

摘要

Lump solutions are analytical rational function solutions localized in alldirections in space. We analyze a class of lump solutions, generated fromquadratic functions, to nonlinear partial differential equations. The basis ofsuccess is the Hirota bilinear formulation and the primary object is the classof positive multivariate quadratic functions. A complete determination ofquadratic functions positive in space and time is given, and positive quadraticfunctions are characterized as sums of squares of linear functions. Necessaryand sufficient conditions for positive quadratic functions to solve Hirotabilinear equations are presented, and such polynomial solutions yield lumpsolutions to nonlinear partial differential equations under the dependentvariable transformations u=2(ln f)_x and u=2(ln f)_{xx}, where x is one spatialvariable. Applications are made for a few generalized KP and BKP equations.