Finite difference method for generalized Zakharov equations

Chang, Qian Shun; Guo, Bo Ling; Jiang, Hong

doi:10.1090/S0025-5718-1995-1284664-5

Finite difference method for generalized Zakharov equations
HTML articles powered by AMS MathViewer

by Qian Shun Chang, Bo Ling Guo and Hong Jiang PDF

Math. Comp. 64 (1995), 537-553 Request permission

Abstract:

A conservative difference scheme is presented for the initial-boundary value problem for generalized Zakharov equations. The scheme can be implicit or semiexplicit depending on the choice of a parameter. On the basis of a priori estimates and an inequality about norms, convergence of the difference solution is proved in order $O({h^2} + {\tau ^2})$, which is better than previous results.

References

Hélène Added and Stéphane Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal. 79 (1988), no. 1, 183–210. MR 950090, DOI 10.1016/0022-1236(88)90036-5
Iwo Białynicki-Birula and Jerzy Mycielski, Gaussons: solitons of the logarithmic Schrödinger equation, Phys. Scripta 20 (1979), no. 3-4, 539–544. Special issue on solitons in physics. MR 544500, DOI 10.1088/0031-8949/20/3-4/033
R. K. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, Solitons in laser physics, Phys. Scripta 20 (1979), no. 3-4, 364–381. Special issue on solitons in physics. MR 544483, DOI 10.1088/0031-8949/20/3-4/011
Qian Shun Chang, Conservative difference scheme for generalized nonlinear Schrödinger equations, Sci. Sinica Ser. A 26 (1983), no. 7, 687–701. MR 721288
Qian Shun Chang and Hong Jiang, A conservative difference scheme for the Zakharov equations, J. Comput. Phys. 113 (1994), no. 2, 309–319. MR 1284857, DOI 10.1006/jcph.1994.1138
Qian Shun Chang and Lin Bao Xu, A numerical method for a system of generalized nonlinear Schrödinger equations, J. Comput. Math. 4 (1986), no. 3, 191–199. MR 860148
Qian Shun Chang and Guo Bin Wang, Multigrid and adaptive algorithm for solving the nonlinear Schrödinger equation, J. Comput. Phys. 88 (1990), no. 2, 362–380. MR 1059375, DOI 10.1016/0021-9991(90)90184-3
Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp. 58 (1992), no. 197, 83–102. MR 1106968, DOI 10.1090/S0025-5718-1992-1106968-6
R. T. Glassey, Approximate solutions to the Zakharov equations via finite differences, J. Comput. Phys. 100 (1992), no. 2, 377–383. MR 1167750, DOI 10.1016/0021-9991(92)90243-R

Self-focussing of laser beam in nonlinear media

20

Milton Lees, Approximate solutions of parabolic equations, J. Soc. Indust. Appl. Math. 7 (1959), 167–183. MR 110212
J. C. López Marcos and J. M. Sanz-Serna, Stability and convergence in numerical analysis. III. Linear investigation of nonlinear stability, IMA J. Numer. Anal. 8 (1988), no. 1, 71–84. MR 967844, DOI 10.1093/imanum/8.1.71
A. Menikoff, The existence of unbounded solutions of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 25 (1972), 407–432. MR 303129, DOI 10.1002/cpa.3160250404
G. L. Payne, D. R. Nicholson, and R. M. Downie, Numerical solution of the Zakharov equations, J. Comput. Phys. 50 (1983), no. 3, 482–498. MR 710406, DOI 10.1016/0021-9991(83)90107-9
T. Ortega and J. M. Sanz-Serna, Nonlinear stability and convergence of finite-difference methods for the “good” Boussinesq equation, Numer. Math. 58 (1990), no. 2, 215–229. MR 1069280, DOI 10.1007/BF01385620
Steven H. Schochet and Michael I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys. 106 (1986), no. 4, 569–580. MR 860310
W. Strauss, Mathematical aspects of classical nonlinear field equations, Nonlinear problems in theoretical physics (Proc. IX G.I.F.T. Internat. Sem. Theoret. Phys., Univ. Zaragoza, Jaca, 1978) Lecture Notes in Phys., vol. 98, Springer, Berlin-New York, 1979, pp. 123–149. MR 542271

Regularity properties for the equations of Langmuir turbulence

289

Collapse of Langmuir waves

35

A class of multidimensional nonlinear Langmuir waves

19