Numerical simulation of the stochastic Korteweg-de Vries equation

Physica D: Nonlinear Phenomena

Volumn 134, Issue 2, 1999, Pages 200-226

  Debussche, Arnaud ^a   Printems, Jacques ^b  

^a UNIVERSITÉ PARIS SACLAY   (France)

^b University Sorbonne Paris Nord   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0346207389     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(99)00072-X     Document Type: Article

References (38)

1. M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, PA, 1981.
2. T.R. Akylas, On the excitation of long nonlinear water waves by a moving pressure distribution, J. Fluid. Mech. 141 (1984) 455-466.
3. J.L. Bona, V.A. Dougalis, O.A. Karakashian, W.R. McKinney, Conservative, high order numerical schemes for the generalized Korteweg-de Vries equation, Phil. Trans. R. Soc. London A 351 (1995) 107-164.
4. J.L. Bona, B.-Y. Zhang, The initial value problem for the forced Korteweg-de Vries equation, Proc. R. Soc. Edinburgh 126A (1996) 571-598.
5. G.F. Carey, Y. Shen, Approximations of the KdV equation by least squares finite elements, Comput. Methods Appl. Mech. Eng. 93 (1991) 1-11.
6. H.Y. Chang, Ch. Lien, S. Sukarto, S. Raychaudhury, J. Hill, E.K. Tsikis, K.E. Lonngren, Propagation of ion-acoustic solitons in a non-quiescent plasma, Plasma Phys. Controlled Fusion 28 (1986) 675-681.
7. S. L. Cole, Transcient waves produced by flow past a bump, Wave Motion 7 (1985) 579-587.
8. P.G. Drazin, Solitons, in: London Mathematical Society Lecture Note Series, vol. 85, Cambridge University Press, Cambridge, 1983.
9. A. Debussche, A. de Bouard, On the stochastic Korteweg-de Vries equation, J. Funct. Anal. 154 (1998) 215-251.
10. A. Debussche, J. Printems, Effect of a localized random forcing on the Korteweg-de Vries equation, Prépublication d'Orsay 99-02, in press.
11. R. Grimshaw, Resonant flow of a rotating fluid past an obstacle: the general case, Stud. Appl. Math. 83 (1990) 249-269.
12. R. Grimshaw, Resonant forcing of barotropic coastally trapped waves, J. Phys. Ocean 17 (1987) 53-65.
13. R. Grimshaw, Z. Yi, Resonant generation of finite-amplitude waves by flow past topography on a beta-plane, Stud. Appl. Math. 88 (1993) 89-112.
14. I. Gyöngy, D. Nualart, Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise, Potential Anal. 7 (1997) 725-757.
15. R. Herman, The stochastic, damped Korteweg-de Vries equation, J. Phys. A: Math. Gen. 23 (1990) 1063-1084.
16. T. Kawahara, Effect of random inhomogeneities on nonlinear propagation of water waves, J. Phys. Soc. Jpn 41 (1976) 1402-1409.
17. V.V. Konotop, L. Vasquez, Nonlinear Random Waves, World Scientific, Singapore, 1994.
18. D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39 (1895) 422-443.
19. P.E. Kloeden, E. Platten, Numerical solution of stochastic differential equations, in: Applications of Mathematics, vol. 23, Springer, Berlin, 1992.
20. S.-J. Lee, G.T. Yates, T.Y. Wu, Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances, J. Fluid. Mech. 199 (1989) 569-593.
21. Milstein, Approximate integration of stochastic differential equations, Theor. Prob. Appl. 19 (1974) 557-562.
22. Milstein, Weak approximation of solutions of systems of stochastic differential equations, Theor. Prob. Appl. 30 (1985) 750-766.
23. H. Mitsudera, R. Grimshaw, Generation of mesoscale variability by resonant interaction between a baroclinic current and localized topography, J. Phys. Ocean 21 (1991) 737-765.
24. A. Nachbin, G.C. Papanicolaou, Water waves in shallow channels of rapidly varying depth, J. Fluid Mech. 241 (1992), 311-332.
25. A. Patoine, T. Warn, The interaction of long, quasi-stationary baroclinic waves with topography, J. Atmos. Sci. 39 (1982) 1018-1025.
26. 2(ℝ), J. Differential Equations 153 (1999) 338-373.
27. M.I. Rabinovich, T.D. Trubetskov, Introduction to Theory of Oscillations and Waves, Nauka, Moscow, 1984.
28. J.M. Sanz-Serna, J.G. Verwer, Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. Num. Anal. 6 (1986) 25-42.
29. M. Scalerandi, A. Romano, C.A. Condat, Korteweg-de Vries solitons under additive stochastic perturbations, Phys. Rev. E 58 (1998) 4166-4173.
30. B.K. Shivamoggi, Introduction to Nonlinear Fluid-Plasma Waves, Kluwer Academic Publishers, Dordrecht, 1988.
31. T.R. Taha, M.J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations III. Numerical, the Korteweg-de Vries equation, J. Comp. Phys. 55 (1984) 231-253.
32. D. Talay, Efficient numerical schemes for the approximation of expectation of functionals of the solutions of an stochastic differential equation and applications, Lecture Notes in Control and Inform. Sc. 61 (1984), 294-313
33. 2AN 20 (1986) 141-179.
34. M. Wadati, Stochastic Korteweg-de Vries equation, J. Phys. Soc. Jpn 52 (1983) 2642-2648.
35. M. Wadati, Y. Akutsu, Stochastic Korteweg-de Vries equation with and without damping, J. Phys. Soc. Jpn 53 (1984) 3342.
36. T. Warn, B. Brasnett, The amplification and capture of atmospheric solitons by topography: a theory of the onset of regional blocking, J. Atmos. Sci. 40 (1983) 28-38.
37. G.B. Whitham, Linear and nonlinear waves, in: Pure and Applied Mathematics, Wiley/Interscience, New York, 1974.
38. T.Y. Wu, Generation of upstream advancing solitons by moving disturbances, J. Fluid. Mech. 184 (1987) 75-99 .