Numerical Simulation Of Rouge Wave Solutions Of A Kind Of Nonlinear Schr?dinger Equation

In recent years,a very important phenomenon known as the giant ocean wave has attracted people's attention.The probability of such waves is very low,once they occur,they can cause serious damage to ships or fixed Marine structures and pose a fatal threat to maritime workers [8].This giant sea wave was later known as the “rouge wave”,also known as the “strange wave”,“extreme wave” or “abnormal wave”.Due to the extremely destructive nature of rouge waves,it is particularly important to study them.Scholars have also conducted in-depth research on them through numerical experiments and theoretical methods,and later found that the rational solutions of some nonlinear partial differential equations also have the properties of rouge waves.For example,the nonlinear Schr ¨odinger equation can be used to describe the formation of rouge waves in deep water [1].Nonlinear schrodinger(NLS)equation is the basis of quantum mechanics,and a kind of solution of it has rouge wave properties,so it becomes more important to study.Because before and after a certain time,the amplitude of the rouge wave will be more than twice the average amplitude of its surroundings,so the numerical simulation of the strange wave solution is difficult.The commonly used numerical simulation methods include finite difference method,finite element method,finite volume method,spectral method,etc [18].However,due to the particularity of rouge wave solutions,in this paper,the local discontinuous finite element method(LDG)method [7] is used to simulate the strange wave solution of nonlinear Schr ¨odinger equation,that is because LDG method can be very sensitive to the impact [16],is very suitable for solving the rapidly changing solution.In addition,because the nonlinear Schr¨0dinger equation is a development equation that depends on both space and time,so on the basis of spatial dispersion,we should also choose an appropriate time method for time dispersion.We choose the third-order TVD-RK method [12] to discretize the time in this paper.The third-order TVD-RK method is an explicit and strong stability preserving method for discretization in time.Due to the particularity of the rouge wave,this method is very suitable for its dispersion.In addition,the ideas and discrete methods in this paper are also generalized and applied to a class of NLS equations,including NLS equations with self-steepness term,Kundu-NLS equations,and coupled NLS equations.We give the full discrete format of each equation,and get the rouge wave solution of these equations by numerical simulation.