| Chaos,solitons and fractals,constitute three theoretical bases of nonlinear science,and soliton have attracted more and more scholars’ attention because of its physical and dynamical properties.In this paper,several kinds of nonlinear integrable equations are studied on the basis of Maple.Chapter 1: The relevant background and research status of solitons are briefly introduced.Chapter 2: The main research methods that used in this paper are described,including Hirota bilinear method,Painlevé integrability test,Darboux transformation and the Bell polynomial,etc.Chapter 3: The modified Korteweg-de Vries-sinh-Gordon equation is studied.Painlevéintegrability test is used to verify the integrability of the equation,and the bilinear form of the equation is obtained based on the Hirota bilinear form.The expressions of real soliton solutions and complex soliton solutions are obtained.In addition,the existence of soliton molecules are verified,the existence condition of molecules are given,and we get the three-dimensional graphics of the soliton molecules.Chapter 4: We study the(3+1)-dimensional Hirota bilinear equation and the(3+1)-dimensional Kadomtsev-Petviashvili equation.Based on the bilinear form of the(3+1)-dimensional Hirota bilinear equation,we get some novel solutions of this equation.In addition,we combine the(3+1)-dimensional Hirota bilinear equation with the(3+1)-dimensional Kadomtsev-Petviashvili equation,a new nonlinear integrable equation with bilinear form is obtained.Then,we obtain its bilinear form by using Bell polynomial and calculate the soliton solutions.Chapter 5: We study the variable coefficient KP equation.On the basis of the Painlevéintegrability test,the Lax pair and conjugate Lax pair of this equation are derived.By using the Lax pair and binary Darboux transformation,the 1-lump solution,2-lump solution and N-lump solution of the equation are obtained.Chapter 6: We summarize the work of this paper and look forward to the next step. |
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