Fractal Solitons, Arbitrary Function Solutions, Exact Periodic Wave and Breathers for a Nonlinear Partial Differential E
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Fractal Solitons, Arbitrary Function Solutions, Exact Periodic Wave and Breathers for a Nonlinear Partial Differential Equation by Using Bilinear Neural Network Method∗ ZHANG Runfa · BILIGE Sudao · CHAOLU Temuer
DOI: 10.1007/s11424-020-9392-5 Received: 17 September 2019 / Revised: 1 March 2020 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract This paper extends a method, called bilinear neural network method (BNNM), to solve exact solutions to nonlinear partial differential equation. New, test functions are constructed by using this method. These test functions are composed of specific activation functions of single-layer model, specific activation functions of “2-2” model and arbitrary functions of “2-2-3” model. By means of the BNNM, nineteen sets of exact analytical solutions and twenty-four arbitrary function solutions of the dimensionally reduced p-gBKP equation are obtained via symbolic computation with the help of Maple. The fractal solitons waves are obtained by choosing appropriate values and the self-similar characteristics of these waves are observed by reducing the observation range and amplifying the partial picture. By giving a specific activation function in the single layer neural network model, exact periodic waves and breathers are obtained. Via various three-dimensional plots, contour plots and density plots, the evolution characteristic of these waves are exhibited. Keywords Arbitrary function solutions, bilinear neural network method, breather, Lump solitons waves, solitons.
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Introduction
Nonlinear evolution equations (NLEEs), which can be used to model nonlinear phenomena, is an effective tool to describe the real world. Although it is more difficult to solve than the ZHANG Runfa · BILIGE Sudao (Corresponding author) Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China. Email: [email protected]. CHAOLU Temuer College of Art and Sciences, Shanghai Maritime University, Shanghai 201306, China. ∗ This research was supported by the National Natural Science Foundation of China under Grant Nos. 11661060, 11571008, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant No. NJYT-20-A06 and the Natural Science Foundation of Inner Mongolia Autonomous Region of China under Grant No. 2018LH01013. This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.
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ZHANG RUNFA · BILIGE SUDAO · CHAOLU TEMUER
linear equations, with the development of computer technology, researchers have successfully dealt with a large number of equations via symbolic computation[1] . A new algebro-geometric approach was first introduced by Feng and Gao. In [2–4] they provided a polynomial time algorithm for computing rational general solutions of first-order autonomous AODEs[5] . WINKLER presents the algebro-geometric method for computing explicit formula solutions for algebraic differential equations[5] . Recently, we put forward a new unified method, called bilinear neural network metho
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