Lump solutions to nonlinear partial differential equations via Hirota bilinear forms |
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Authors: | Wen-Xiu Ma Yuan Zhou |
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Affiliation: | 1. Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China;2. Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA;3. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China;4. College of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, China;5. International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa |
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Abstract: | 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 and , where x is one spatial variable. Applications are made for a few generalized KP and BKP equations. |
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Keywords: | 35Q51 37K40 35Q53 Soliton Integrable equation Hirota bilinear form Lump solution |
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