Integrability And Exact Solutions Of Some Nonlinear Evolution Equations

Nonlinear partial differential equations(NLPDEs)are common models to describe the nonlinear physical phenomena in the fields of optics:plasmas physics,condensed matter physics and nonlinear atmosphere dynamics.The study of nonlinear traveling waves(solitary wave,periodic wave,etc.)is vital for the research of immense physical phenomena.The exact solutions of these nonlinear evolution equations are quite useful to understand the mechanism of the complicated practical problems and dynamical processes.In the introduction,different methods for finding exact solutions of the nonlinear evolution equations are briefly described,such as:Lax pair,AKNS method,Painleve analysis method,Hirota bilinear method-Bell polynomial method.Three methods used in the second chapter are mainly introduced,which contains qualitative theory of differential equations?(G'/G)-expansion method and elliptic function expansion method.(1)In charpter 2,the resonant nonlinear Schrodinger equation(RNLSE)with competing weakly nonlocal nonlinearity and fractional temporal evolution,which describes the propagation of optical solitons along the nonlinear optical fibers,is investigated.Dynamic behavior of an equation with the parabolic-type nonlinearity are discussed.The relationship between the orbits of the dynamic system and traveling wave solutions is demonstrated.The characteristics of the orbits imply some possibility to obtain the exact solutions by appropriate methods.The(G'/G)-method and elliptic function method are conducted to get the exact traveling wave solutions,such as periodic wave solutions,shock wave solutions and breaking wave solutions.Meanwhile,qualitative analysis for the high order equilibrium point at(0,0)is conducted to investigate its dynamic behaviors.Particularly,the relation between the phase orbits and the corresponding solutions indicates that a family of the hyperbolic curves near the saddle means the existence of a family of breaking wave solutions(2)In charpter 3,some basic knowledge of optical soliton in the nonlinear Schrodinger equation(NLSE)are briefly introduced,which are common models in many physical phenomena.A special auxiliary equation method-chirp ansatz is mainly introduced and applied to the model describing the propagation of ultrashort optical pulses in optical fibers,which is called cubic-quintic NLSE with variable coefficients.At last,a variety of nontrivial phase chirped soliton solutions for this equation are found,which include(bright,kink,dark solitons etc.)under the constraint conditions.