Soliton Solutions Of Nonlinear Equations Based On Bilinear Method And Darboux Transformation

Based on symbolic calculation,this paper constructs soliton solutions,soliton molecular solutions and Lump solutions of several nonlinear equations by using the Hirota bilinear method and the Darboux transformation,and conducts the corresponding dynamical analysis.The main innovations are as follows:(1)The bilinear form of the(3+1)dimensional Korteweg-de Vries equation is given by using the Hirota bilinear method,and the single and double soliton solutions of the equation are obtained.Based on the bilinear form,the resonance conditions of the soliton molecular solutions in multiple planes and a new generalized soliton solution are obtained.Finally,the dynamical behavior of soliton solution and soliton molecular solution is analyzed;(2)The single soliton solution,double soliton solution and three soliton solution of the(2+1)dimensional Boiti-Leon-Manna-Pempinelli equation are studied.The solution of the interaction between the Lump and the Kink soliton of the equation is constructed,and it is found that there are different dynamical behaviors under the appropriate regulation of the parameters.Finally,a new Kink soliton solution is obtained under the condition of changing the test solution;(3)The single soliton solution and double soliton solution of the(2+1)dimensional generalized Korteweg-de Vries equation are obtained by using the Hirota bilinear method.Then the Lump solution of the equation and the interaction solution of the Lump and soliton are obtained,and it is found that the different values of the parameters will affect the characteristics of the waveform such as amplitude,velocity and phase.Finally,the bright and dark soliton solution of the equation and the three-dimensional graph are obtained;(4)The integrable three-level coupled Maxwell-Bloch equation with mixed focusing-defocusing case is studied.Based on the linear 3×3 matrix eigenvalue problem,the n-fold Darboux transform is constructed,and the dark-dark soliton solution and the corresponding asymptotic analysis are derived.The elastic collision of double dark-dark soliton molecules or triple dark-dark soliton molecules with ordinary dark-dark solitons and the collision of two double darkdark soliton molecules are proved by the standard asymptotic analysis methods.