Integrable Nonisospectral Equations And Localized Waves

In the recent years,rogue waves which are space-time localized have been one of prominent topic in integrable systems.Integrable nonisospectral equations can be to describe solitary waves in nonuniform medium.In this dissertation,we focus on the investigations of nonisospectral effects on generating space-time localized waves of different integrable systems.Firstly,we focus on investigating the dynamics of a family of three nonisospectral focusing nonlinear Schr?dinger equations(NNLSEs),that each equation follows a specific time dependency of the spectral parameter.We employ the Wronskian technique as our tool of investigation,as well,the different sorts of solutions are given.In order to interpret the nonisospectral effects on the waves behavior,dynamics of the solutions are illustrated in detail,where the investigation of the first two nonisospectral counterparts(NNLSE-I and NNLSE-II)is based on their gauge connection to the classical nonlinear Schr?dinger equation.;on the other hand,and in particular,the third equation,NNLSE-III admits spacetime localized waves generated from its soliton solutions.Since the NNLSE-III shows that localized rogue waves can be generated from soliton solutions with respect to the nonisospectral effects,the investigation is extended to a semidiscrete nonlinear Schr?dinger equation,which is a direct integrable discretization of the NNLSE-III.Solutions are derived by means of bilinear approach and the double Casoratian structure of solutions is different from the continuous counterpart of the NNLSE-III.The analysis of the solutions in double Casoration shows that both solitons and multiple-pole solutions admit space-time localized rogue wave behavior.And more interestingly,the solutions allow blow-up at finite time,which is a different property compared to the continuous counterpart.Finally,we introduce an idea about the nonisospectral effects in order to generate space-time localized solitary waves.For instance,three nonlinear integrable equations are investigated,namely,the nonisospectral counterparts of the Korteweg-de Vries,modified Korteweg-de Vries and the Hirota equation,respectively.Different types of solutions for these equations are obtained through the employment of the Wronskian technique.Nonisospectral effects forcing solutions to these equations to have space-time localized wave behavior is also illustrated.The research provides both mathematical and physical insight to understand the connection between nonisospectral effects and generating localized solitary waves.