Soliton Solutions And Their Dynamical Properties Of Some Nonlinear Integrable Equations

The researches of the soliton solutions and their dynamic properties for non-linear integrable equations are the most important topics in the theoretical study of integrable systems.In this dissertation,we mainly investigate the coupled focusing-defocusing complex short pulse equation?spatial discrete Hirota equation?the spatial discrete nonlocal complex modified Korteweg-de Vries(mKdV)equation and a nonlocal complex coupled dispersionless equation.We study the integrability of these nonlinear integrable equations and obtain the different types of exact solutions for these non-linear integrable equations including the soliton solutions,breathers and rogue wave solutions.We do research the relations of the integrability between a spatial discrete nonlocal complex mKdV equation and a corresponding continuous equation.The re-sults obtained in this paper undoubtedly add the new and valuable contents for people recognizing and understanding these nonlinear integrable systems.The main contents are summarized as follows:In chapter 1,we briefly elaborate some classic solving methods for the nonlinear integrable systems and the concepts of the rogue wave solution.Finally,we state the main achievements and the innovative point of this dissertation.In chapter 2,we investigate a coupled focusing-defocusing complex short pulse equation.As well known,The nonlinear Schrodinger(NLS)equation can describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear dispersive media,such as nonlinear optics.However,in the range of ultra-short pulses which the width of optical pulse is in the order of femtosecond(10-15s),the NLS e-quation becomes less accurate.Then T.Schafer and C.E.Wayne[Phys.D.196(2004)]proposed the short pulse(SP)equation to describe the propagation of ultra-short op-tical pulses in nonlinear media.Its appearance immediately attracted people's great interests.A.Sakovich and S.Sakovich[J.Phys.Soc.Japan 74(2005)]gave the Lax pair of the SP equation and indicated SP equation can convert to the sine-Gordon equation through the hodograph transformation.Bao-Feng Feng[Phys.D.297(2015)]proposed the complex short pulse(CSP)equation and coupled CSP equation from the Maxwell e-quations.He constructed the multi-soliton for the CSP equation and focusing-focusing CSP equation by Hirota's bilinear method.After that,L.M.Ling,B.F.Feng and Z.N.Zhu[Phys.D 327(2016)]obtained the multi-soliton and multi-beather and higher-order rogue wave(RW)solutions in CSP equation by Darboux transformation(DT)method.They[Phys.Rev.E 93(2016)]put forward the defocusing CSP equation from the Maxwell equations and constructed multi-dark soliton solutions by the same method.The relevant works of the coupled NLS equations(focusing-focusing,focusing-defocusing,defocusing-defocusing)inspire us to study the coupled focusing-defocusing CSP equation.By using the Hirota's bilinear method,we contruct the bright-bright,bright-dark,dark-dark soliton solutions,breathers and rogue wave solutions in the cou-pled focusing-defocusing CSP equation and also point out that there are three types of solitons in these soliton solutions:smooth soliton,cuspon soliton and loop soliton.We observe inelastic collisions between the bright-bright solitons and the periodic phe-nomenon of the band states in the bright-dark solitons.When a smoothed dark soliton collides with a cusponed dark soliton,the singularity of the cusponed dark soliton could still maintain.These never appear in the CSP equation and focusing-defocusing CCSP equation.In chapter 3,we consider a spatial discrete Hirota equation.A.Pickering,H.Q.Zhao and Z.N.Zhu[Proc.R.Soc.A 427(2016)]has shown that the integrability of the spatial discrete Hirota equation converge to the integrability of Hirota equation.L.Draper[OceanuslO(1964)]first proposed the concept about the rogue wave(RW).After that,many natural phenomenons about the RW have been recorded in the differ-ent fields.V.E.Zakharov[J.Appl.Mech.Tech.Phys.9(1968)]pointed out that the rational solutions of the NLS equation could well simulate the occurrence of the RW in the deep ocean.Recently,N.Akhmediev,J.M.Soto-Crespo and A.Ankiewicz[Phys.Lett.A 373(2009)]indicated that he higher-order RWs are nothing other than non-linear superpositions of the first-order rational solutions.B.L.Guo,L.M.Ling and Q.P.Liu[Phys.Rev.E 85(2012)]constructed a generalized Darboux transformation for the NLS equation and obtained the Nth-order rogue wave solutions of the focusing NLS equation and Hirota equation.J.He[Phys.Rev.E 85(2012)]solved the higher-order RW for the Hirota equation by improved DT method and X.Wang,Y.Q.Li,Y.Chen[Wave Motion 51(2014)]solved the higher-order RW for the coupled Hirota equa-tion by generalized DT method.For the discrete Hirota equation,N.Akhmediev,A.Ankiewicz and J.M.Soto-Crespon[Phys.Rev.E 80(2009)]obtained the higher-order RW for the Ablowitz-Ladik(AL)equation as well as the discrete Hirota equation by bi-linear's Hirota method.Later on,Yasuhiro,Ohta and J.K.Yang[J.Phys.A 47(2014)]also derived the higher-order RW for AL equation and the discrete Hirota equation byHirota s bilinear method.In this chapter,we construct the higher-order RW solutions for the spatial discrete Hirota equation by the generalized DT.We analyze the dynam-ical property for the first,second and third order RWs.The higher-order RWs exhibit the rich structures for the spatial discrete Hirota equation.The time evolution and dynamical behaviors of the one and two order RWs are studied by means of numerical simulations,which demonstrates that strong and weak RWs are,respectively,nearly stable and strongly unstable solutions.We apply the contour line method to obtain an-alytical formulae of the length and width for the first-order RW solution of the spatial discrete Hirota equation.The modulational instability of the RWs is also analyzed.In chapter 4,we investigate the relations of integrability between the spatial dis-crete nonlocal complex modified Korteweg-de Vries(mKdV)equation and the nonlocal complex mKdV equation.Z.N.Zhu,H-Q Zhao,X.N.Wu J.Math.Phys.52(2011)]in-vestigated the continuous limits of a new coupled spatial discrete mKdV system.They pointed out that the coupled discrete mKdV system including the Lax pairs,the Dar-boux transformation,soliton solutions,and conservation converge to the corresponding results in the coupled mKdV system.Recently,Ablowitz and Musslimani Phys.Rev.Lett.110(2013)]proposed a new nonlinear integrable equation iqt(x,t)+qxx(x,t)±2q2(x,t)q*(-x,t)=0,(2)which is called the nonlocal NLS equation,and solved it's Cauchy problem by inverse scattering transformation method.In the document Phys.Rev.E 90(2014)],they popularized the PT symmetry to the discrete NLS equation and proposed the discrete nonlocal NLS equation.A discrete soliton solution was obtained by using a left-right Riemann-Hilbert formulation.M.J.Ablowitz and Z.H.Musslimani proposed a series of nonlocal nonlinear evolution equation in[Nonlinearity 29(2016)]and[Stud.Appl.Math.,139(2016)],which included the nonlocal real and complex mKdV equation.L.Y.Ma,S.F.She and Z.N.Zhu J.Math.Phys.58(2017)]proved that the integrable nonlocal mKdV equation is gauge equivalent to a spin-like model.In this chapter,we investigate the relations of integrability between a spatial discrete nonlocal complex mKdV equation and nonlocal complex mKdV equation.We construct the Lax pair and N-fold DT of the spatial discrete nonlocal complex mKdV equation,and get a series of exact solutions are derived including antidark soliton,M-type soliton,breather soliton,kink,periodic wave and localized rational soliton solutions.We show that the spatial discrete nonlocal complex mKdV equation yields the corresponding results of the nonlocal complex mKdV equation when the space discrete step tends to zero.In chapter 5,we do research on a nonlocal complex coupled dispersionless equation.M.J.Ablowitz,Z.H.Musslimani[Nonlinearity 29(2016)]also proposed a nonlocal Sine-Gordon equation.J.L.Ji,Z.L,Huang and Z.N.Zhu[Ann.Math.sci.appl.2(2017)]solved it by the DT mothod and they also proposed a new nonlocal complex coupled dispersionless equation.K.Chen,X.Deng,S.Y Lou,D.J.Zhang[Stud.in Appl.Math.141(2018)]derived the solutions for the nonlocal complex coupled dispersionless equation reduced from the AKNS hierarchy by considering suitable constraints on the elements in double Wronskians.In this chapter,we derive the exact solutions for the nonlocal complex coupled dispersionless equation by DT method and analyze the properties of the solutions.From the zero seed solution,we obtain the antidark soliton,W-type soliton,growing soliton,decaying soliton and periodic solution;From the nonzero seed solution,we obtain the antidark-antidark,antidark-dark,dark-dark solitons by the 1-fold DT.By the two-fold DT,we derive antidark-W-shape soliton and antidark-antidark-W-shape soliton solutions from the zero seed solution.We also investigate the interaction between the two solitons and three solitons.Comparing with the nonlocal real coupled dispersionless equation,the exact solutions of nonlocal complex coupled dispersionless exhibit richer properties.