A (2 + 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation that possesses a Hirota bilinear form is considered. Starting with its Hirota bilinear form, a class of explicit lump solutions is computed through conducting symbolic computations with Maple, and a few plots of a specicpresented lump solution are made to shed light on the characteristics of lumps. The result provides a new example of (2 + 1)-dimensional nonlinear partial differential equations which possess lump solutions.
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Lump solutions to nonlinear partial differential equations via Hirota bilinear forms
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 logarithmic derivative transformations. Applications are made for a few generalized KP and BKP equations.
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- Award ID(s):
- 1664561
- NSF-PAR ID:
- 10079108
- Date Published:
- Journal Name:
- Journal of differential equations
- Volume:
- 264
- Issue:
- 4
- ISSN:
- 1090-2732
- Page Range / eLocation ID:
- 2633-2659
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- The DOI is not currently available.
