A split-step Fourier method for the complex modified Korteweg-de Vries equation. (English) Zbl 1035.65111

Summary: The complex modified Korteweg-de Vries (CMKdV) equation is solved numerically by three different split-step Fourier schemes. The main difference among the three schemes is in the order of the splitting approximation used to factorize the exponential operator. The space variable is discretized by means of a Fourier method for both linear and nonlinear subproblems. A fourth-order Runge-Kutta scheme is used for the time integration of the nonlinear subproblem.
Classical problems concerning the motion of a single solitary wave with a constant polarization angle are used to compare the schemes in terms of the accuracy and the computational cost. Furthermore, the interaction of two solitary waves with orthogonal polarizations is investigated and particular attention is paid to the conserved quantities as an indicator of the accuracy. Numerical tests show that the split-step Fourier method provides highly accurate solutions for the CMKdV equation.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Karney, C. F.F.; Sen, A.; Chu, F. Y.F., Nonlinear evolution of lower hybrid waves, Phys. Fluids, 22, 940-952 (1979) · Zbl 0396.76014
[2] Gorbacheva, O. B.; Ostrovsky, L. A., Nonlinear vector waves in a mechanical model of a molecular chain, Physica D, 8, 223-228 (1983)
[3] Erbay, S.; Suhubi, E. S., Nonlinear wave propagation in micropolar media II. Special cases, solitary waves and Painlevé analysis, Int. J. Engng. Sci., 27, 915-919 (1989) · Zbl 0694.73007
[4] Erbay, H. A., Nonlinear transverse waves in a generalized elastic solid and the complex modified Korteweg-de Vries equation, Physica Scripta, 58, 9-14 (1998) · Zbl 0978.74043
[5] Tappert, F., Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method, Lect. Appl. Math. Amer. Math. Soc., 15, 215 (1974)
[6] Taha, T. R.; Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys., 55, 203-230 (1984) · Zbl 0541.65082
[7] Weideman, J. A.C.; Herbst, B. M., Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 23, 485-507 (1986) · Zbl 0597.76012
[8] Pathria, D.; Morris, J. L., Pseudo-spectral solution of nonlinear Schrödinger equations, J. Comput. Phys., 87, 108-125 (1990) · Zbl 0691.65090
[9] Fornberg, B.; Driscoll, T. A., A fast spectral algorithm for nonlinear wave equations with linear dispersion, J. Comput. Phys., 155, 456-467 (1999) · Zbl 0937.65109
[10] Taha, T. R.; Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations. IV. Numerical, Korteweg-de Vries equation, J. Comput. Phys., 55, 231-253 (1984) · Zbl 0541.65083
[11] Taha, T. R.; Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations. IV. Numerical, modified Korteweg-de Vries equation, J. Comput. Phys., 77, 540-548 (1988) · Zbl 0646.65087
[12] Herbst, B. M.; Ablowitz, M. J.; Ryan, E., Numerical homoclinic instabilities and the complex modified Korteweg-de Vries equation, Computer Physics Communications, 65, 137-142 (1991) · Zbl 0900.65350
[13] Taha, T. R., Numerical simulations of the complex modified Korteweg-de Vries equation, Mathematics and Computers in Simulation, 37, 461-467 (1994) · Zbl 0811.65117
[14] Sanz-Serna, J. M.; Calvo, M. P., Numerical Hamiltonian Problems (1994), Chapman and Hall: Chapman and Hall London · Zbl 0816.65042
[15] Yoshida, H., Construction of higher order symplectic integrators, Phys. Lett. A, 150, 262-268 (1990)
[16] Suzuki, M., General theory of higher-order decomposition of exponential operators and symplectic operators, Phys. Lett. A, 165, 387-395 (1992)
[17] McLachlan, R., Symplectic integration of Hamiltonian wave equation, Numer. Math., 66, 465-492 (1994) · Zbl 0831.65099
[18] Weideman, J. A.C.; Cloot, A., Spectral methods and mappings for the evolution equations on the infinite line, Comp. Meth. App. Mech. Engng., 80, 467-481 (1990) · Zbl 0732.65095
[19] Fornberg, B.; Whitham, G. B., A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. Roy. Soc. London A, 289, 373-404 (1978) · Zbl 0384.65049
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