Lump solutions to nonlinear partial differential equations via Hirota bilinear forms
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Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to solve Hirota bilinear equations are presented, and such polynomial solutions yield lump solutions to nonlinear partial differential equations under the dependent variable transformations u=2(ln f)_x and u=2(ln f)_{xx}, where x is one spatial variable. Applications are made for a few generalized KP and BKP equations.
Subfields: Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/683456dbc3b2395bc8276ff5ac9f3bdf2942ef4a
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In Geometry and Topology
In Statistical and Nonlinear Physics
Citations in the corpus
- Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs (2018, background): “Other integrable equations, possessing lump solutions, include the three-dimensional threewave resonant interaction [8], the BKP equation [9,10], the Davey–Stewartson equation II [3], the Ishimori-I equation [11] and many others [12,13].”
- Long-Time Asymptotics of a Three-Component Coupled mKdV System (2019, neutral): “incorporating features of other exact solutions, such as lumps [68,69], from the perspective of steepest 204”
- Interaction solutions to Hirota-Satsuma-Ito equation in (2 + 1)-dimensions (2019, background): “, [37] for Hirota bilinear equations and [38] for generalized bilinear equations).”