The Study On The Exact Solution And Integrability Of Derivative Nonlinear Schr(o|¨)dinger Equation With A Parameter

It is well known that the nonlinear Schr(o|¨)dinger equation(NLSE)is one of the most generic soliton equations.It arises from many fields,such as quantum field theory,weakly nonlinear dispersive water waves and nonlinear optics.To study the effect of higher-order perturbations,various modifications and generations of the NLSE have been proposed and studied.Among them,there are three celebrated equations which are the Chen-Lee-Liu(CLL),the Kaup-Newell(KN)and Gerjikov-Ivanov(GI)equation.In this thesis,the generalized isospectral and nonisospectral derivative nonlinear Schr(o|¨)dinger hierarchies are deduced from a spectral problem with a parameter by using the zero curvature equation.As far as we known,the nonisospectral equation is a new integrable hierarchy.The equation is CLL,KN,GI equation respectively by selecting the different parameter.Through a variable transformation,the isospectral equation can be transformed into the bilinear equations.Base on these equations,the exact solutions are discussed through the Hirota's method and Wronskian technique.Furthermore,its infinite conservation laws are found and Hamiltonian structure are constructed.That means that the equation is integrable in Liouville sense.Finally,the recursion oper-ator of the generalized derivative nonlinear Schr(o|¨)dinger hierarchy is proved to be a hereditary symmetry.It can be decomposed into the product of a symplectic operator and a inverse-symplectic operator.The infinite Hamiltonian functionals are founded by Fokas' Method.We conclude that the equation is integrable in Liouville sense.