Darboux Transformations And Explicit Solutions For Three Nonlocal Soliton Equations

Soliton theory,as an vital branch of nonlinear science,researches nonlinear partial d-ifferential equations and seeks their explicit solutions.At present,research methods mainly include Traveling wave method,Darboux transformation,Backlund transformation,Bilinear derivative method,Painleve analysis and others.Soliton refers to a nonlinear waves whose waveforms and velocities remain unchange in equilibrium after the interaction of dispersion and nonlinear effects.It is collision characteristics that the energy of soliton is almost lossless and change slowly,and the ideal state is reached.The nonlocal properties are an effect based on PT symmetry,which satisfies some symmetry conditions.This model depends on space and time variables of functions,and can theoretically obtain solitons which are different from the local.Three kinds of nonlocal model with strong physical significance are studied in this paper:(2+1)-dimensional nonlocal CMKdV-MB equations describing the propagation of ultrashort pulses in optical fiber communications,nonlocal continuous Hirota equation describing vortex motion and laser homoclinic orbits in fluid mechanics and optical fiber com-munications,nonlocal discrete Hirota equation describing nonlinear waves in fluid motion.According to Lax pairs and Darboux transformation,symmetric unbroken and broken explicit solutions of equations are obtained by symbolic calculation.In addition,Mathematics is used to draw the pictures corresponding to the solutions,and the propagation process and char-acteristics in optical fiber communication and fluid mechanics are analyzed.The following is brief explanation about explicit solutions of nonlocal nonlinear evolution equations:1.Symmetric Unbroken Explicit Solutions(1)Periodic solution,is a linear structure which is periodic in space or time,and the structure are also superimposed on each other after multiple iterations.(2)Soliton solution,is a soliton structure that propagation stably in space or time,divided into solid soliton and traveling wave soliton.Solid solitons are parallel along space or time variables.Travelling wave solitons are solitons that propagate stably at an angle to variables.Each soliton has two possible shapes:dark or light.(3)Complex solution,is a nonlinear structure with local periodic oscillation in variables.(4)Oscillating and singular soliton solution,are class of nonlinear structures that periodically oscillate along variables with different amplitude.The amplitude appears singularity under variables vary,and singular soliton is formed.(5)Bound soliton solution,is a periodic structure in which the attraction and repulsion between solitons occur alternately,and the amplitude increases suddenly when solitons attract.In this case,nonlocal equations satisfy PT symmetry condition,and potential functions and self-induced potential are stable symmetric structure.2.Symmetric Broken Explicit Solutions(1)They aren't stable structures in the space and time variables.The nonlocal equations themselves satisfy PT symmetry condition,that is,self-induced potential function is a stable symmetric structure.But it is found that,po-tentials no longer propagate stably although self-induced potential function remains the same when Lax pairs satisfies certain eigenvalues condition.Instead,the amplitude increases or decreases exponentially and energy also change greatly.