The Study Of Analytical Solution To Some Nonlinear Models

In this Ph D thesis,the Hirota bilinear method,Riemann-Hilbert(RH)method and symbolic computation are utilized to investigate some significant nonlinear integrable equations in mathematical physics and obtain explicit analytic solutions including soliton solutions,breathers,Lumps and rogue waves and interaction solutions consisted of them.By analyzing the structure of expressions,the rich dynamic behaviors of obtained various solutions are further studied by means of various graphs.Based on symbolic computation software Maple,this paper develops the construction of analytic solutions of nonlinear evolution equation,including the following two sections:In the first section,the basic theory of bilinear method combined with the long-wave limit method,the generalized homoclinic wave test method and the hypothesis function method are employed to explore the generalized Boiti-Leon-Manna-Pempinelli(BLMP)equation,(3+1)-dimensional generalized Jimbo-Miwa(JM)equation,(3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama(YTSF)equation to constructs the bilinear forma and many different types of analytic solutions.In the second section,the N-soliton solutions are formulated with the aid of RH method in the research of the Lakshmanan-Porsezian-Daniel(LPD)equation,the higher-order coupled nonlinear Schrodinger(NLS)equation and the four-order coupled NLS equation.The specific contents are as follows:The first chapter is the introduction part.Firstly,the research background and current situation of the development for soliton theory are introduced,and the application of some classical methods in the field is also elaborated.I focus on introduction of the symbolic computation,RH method,Hirota bilinear method and related long-wave limit method,generalized homoclinic wave test method,hypothesis function method,and elaborate the selected topic and the main work of this paper.In the second chapter,the generalized BLMP equation is first presented.On the basis of bilinear formula and hypothesis function method,the general expression of analytic solution is obtained.In addition,the Lump,as well as inelastic interaction solution consisted of Lump and soliton are generated by selecting appropriate parameter values.Finally,the propagation orbit,velocity and extremum of the Lump solutions on(x,y)plane are studied in detail.Under investigation in the third chapter is an generalized(3+1)-dimensional JM equation,which can be used to describe many nonlinear phenomena in mathematical physics.With the aid of Hirota bilinear method and long-wave limit method,M-order lumps which describe multiple collisions of Lumps are derived.The propagation orbit,velocity and extremum of the 1-order Lump solutions on(x,y)plane are investigated in detail.Resorting to the extended homoclinic test technique,we obtain the breather-kink solutions,rational breather solutions and rogue wave solutions for the generalized JM equation.Meanwhile,through analysis and calculation,the amplitude and period of breather-kink solutions increase with p increasing and the extremum of rational breather solution and rogue waves are also derived.T-order breathers are obtained by means of choosing appropriate complex conjugate parameters on N-soliton solutions.It has been proven that periods of the 1-order breather solutions on the(x,y)plane are determined by k12 and k12p11+k11p12,while locations are determined by k11 and k11p11-k12p12.Furthermore,hybrid solutions composed of the kink solitons,breathers and Lumps for the generalized JM equation are worked out.Some figures are given to display the dynamical characteristics of these solutions at the end of this chapter.In the fourth chapter,with the aid of bilinear method,the formula of N-soliton solution to the generalized(3+1)-dimensional YTSF equation is succinctly obtained.By means of long-wave limit method on 2M-soliton solutions under special parameter constraints,Morder lumps can be successfully constructed.Furthermore,the propagation orbit,velocity and extremum of the 1-order lump solutions on(x,y)plane are studied in detail.Finally,we investigate three types of hybrid solutions,which describe interaction between breathers and solitons,or between lumps and solitons or breathers.These collisions are elastic,which do not lead to any changes of amplitudes,velocities and shapes of the solitons,breathers and Lumps after interaction.In the fifth chapter,the integrable LPD equation originating in nonlinear fiber is studied via the RH approach.Firstly we give the spectral analysis of the Lax pair,from which a RH problem is formulated.Afterwards,by solving the special RH problem with reflectionless under the conditions of irregularity,the formula of general N-soliton solutions can be obtained.In addition,the localized structures and dynamic behaviors of the breathers and solitons corresponding to the real part,imaginary part and modulus of the resulting solutions are shown graphically and discussed in detail.Unlike 1-or 2-order breathers and solitons,3-order breathers and soliton solutions rapidly collapse when they interact with each other.This phenomenon results in unbounded amplitudes which imply higher-order solitons are not a simple nonlinear superposition of basic soliton solutions.In the sixth chapter,the high-order coupled NLS equation in an optical fiber is investigated.Under the condition of zero boundary,the spectral problem and time development formula are transformed into a concise form.Then,by potential reconstruction and solving the matrix RH problem under the condition of no reflection,the expressions of the multisoliton solutions of the high-order coupled NLS equation are given.The propagation and collision dynamic behaviors of these resulting breathers and soliton solutions are presented by selecting appropriate parameters with some graphics.The innovation and highlights of this chapter are shown through obtained interesting results.The one is that the higherorder linear and nonlinear term ? has important impact on the velocity,phase,period,and wavewidth of wave dynamics.The other is that collisions for the second-order breathers and soliton solutions are elastic interaction which imply they remain bounded all the time.Nevertheless,third-order breathers and soliton solutions are inelastic interaction and the amplitude decrease rapidly with time when collisions occur.In the seventh chapter,the multi-soliton solutions and breathers to the coupled fourthorder NLS equation in the birefringent or two-mode fiber are derived by the RH approach.Firstly,by assuming the property that the potential function decayed rapidly at infinity,a new transformation is introduced to change the given spectral problem into a concise form.Secondly,through analyzing the new spectral problem and related properties,then a matrix RH problem on the real axis is strictly formulated.Then,by solving the special RH problem with no reflection and reconstructing potential functions,the general formula of N-soliton solutions can be computed for the coupled four-order NLS equation.Similarly,the impact of higher-order linear and nonlinear term r on the velocity,phase,period,and wavewidth of breathers and solitons have been studied in detail.Interestingly,three-solitons display different dynamics which demonstrate amplitudes of the right-direction waves gradually becomes larger during the propagation process.The eighth chapter is the summary and prospect.The methods used in this paper,the research equations and the main results are summarized.Based on the current research results,the application scope of the method will be broadened,some important mathematical physical equations are further studied,and the content of the future research is expected to be further advanced.