Numerical Simulations of the Complex Modi ed Korteweg de Vries Equation

In this paper implementations of three numerical schemes for the numerical simulation of the complex modi ed Korteweg de Vries CMKdV equation are reported The rst is an integrable scheme derived by methods related to the Inverse Scattering Transform IST The second is derived from the rst and is called the local IST scheme The third is a standard nite di erence scheme for the CMKdV equation Travelling wave solution as well as a double homoclinic orbit are used as initial conditions Numerical experiments have shown that the standard scheme is subject to instability and the numerical solution becomes unbounded in nite time In contrast the integrable IST scheme does not su er from any instabilities The main di erence among the three schemes is in the discretization of the nonlinear term in the CMKdV equation This demonstrates the importance of proper discretization of nonlinear terms when a numerical method is designed for solving a nonlinear di erential equation Introduction In Herbst et al derived an integrable di erential di erence equation based on the IST that has as its limiting form the CMKdV equation Also they derived analytical expressions for the homoclinic orbits associated with the above equation and investigated the e ect of discretization of the equation in the vicinity of these orbits They showed that a standard nite di erence scheme is subject to an instability On the other hand they showed that the integrable di erential di erence scheme of the CMKdV equation does not su er from any instabilities Recently Taha derived an integrable partial di erence equation based on the IST that has as its limiting form the CMKdV equation qt jqj qx qxxx Here q is a complex valued function and j j denotes the modulus In the present paper this partial di erence equation is used as a numerical scheme for solving Eq This scheme call it the integrable IST scheme as well as its local version are implemented and compared to a standard nite di erence scheme for the numerical simulation of Eq Our numerical experiments have shown that the standard scheme su ers from instability and the numerical solution becomes unbounded in nite time On the other hand our numerical experiments have shown that the integrable partial di erence scheme does not su er from any instabilities This result is in agreement of the result found by for the integrable di erential di erence scheme Also in this paper an integrable partial di erence equation that has as its limiting form the defocusing CMKdV equation qt jqj qx qxxx is derived The method of derivation is similar to the one given in In section the partial di erence equations for and are given The Representation of the CMKdV Equations Using Numer ical Methods i The integrable IST scheme is mQmn Q m n A Q n n D Qmn Sn Q n Pn Q n A Qmn n D Q n S n Qmn P n