Effect of a Localized Random Forcing Term on the Korteweg-De Vries Equation

Arnaud Debussche1, Jacques Printems2
1Laboratorie d'Analyse Numérique et EDP d'Orsay, Université Paris-Sud et CNRS, Orsay Cedex, France
2Laboratorie d'Analyse, Géométrie et Applications, Institut Galilée, Université Paris-Nord, Villetaneuse, France

Tóm tắt

In this work, we numerically investigate the influence of a white noise-type forcing on the phenomenon of forced generation of solitons by a localized moving disturbance. Our numerical method is based on finite elements and least-squares. We present numerical experiments for different values of noise amplitude and Froude number, which describe some damping effects on the emission of solitons.

Tài liệu tham khảo

T. R. Akylas, On the excitation of long nonlinear water waves by a moving pressuredistribution, J. Fluid. Mech. 141, 455–466 (1984). J. L. Bona and B.-Y. Zhang, The initial valueproblem for the forced Korteweg–de Vries equation, Proc. Roy. Soc. Edinburgh 126A, 571–598 (1996). G. F. Carey and Y. Shen, Approximations of the KdV equation by least squares finite elements, Comput.Methods Appl. Mech. Engrg. 93, 1–11 (1991). H. Y. Chang, Ch. Lien, S. Sukarto, S. Raychaudhury, J. Hill, E. K. Tsikis, and K. E. Lonngren, Propagation of ion-acoustic solitons in a non-quiescent plasma, Plasma Phys. Controlled Fusion 28, 675–681 (1986). S. L. Cole, Transcient waves produced by flow pasta bump, Wave motion 7, 579–587 (1985). A. Debussche and A. de Bouard, On the stochasticKorteweg–de Vries equation, J. Funct. Anal. 154, 215–251 (1998). A. Debussche, A. de Bouard, and Y. Tsutsumi, White noise driven stochastic Korteweg–de Vries equation, J. Funct. Anal. 169, 532–558 (1999). A. Debussche and J. Printems, Numerical simulation of the stochastic Korteweg–de Vries equation,Physica D 134(2), 200–226 (1999). R. Grimshaw, Resonant flow of a rotating fluid past anobstacle: the general case, Stud. Appl. Math. 83, 249–269 (1990). R. Grimshaw, Resonantforcing of barotropic coastally trapped waves, J. Phys. Ocean. 17, 53–65 (1987). R. Grimshaw, E. Pelinovsky, and X. Tian, Interaction of a solitary wave with an external force, Physica D 77, 405–433 (1994). R. Grimshaw and N. Smyth, Resonant flow of a stratified fluid over topography, J. Fluid Mech.169, 429–464 (1986). R. Grimshaw and Z. Yi, Resonant generation of finite-amplitude waves byflow past topography on a beta-plane, Stud. Appl. Math. 88, 89–112 (1993). R. Herman, Thestochastic, damped Korteweg–de Vries equation, J. Phys. A: Math. Gen. 23, 1063–1084 (1990). V.V. Konotop and L. Vasquez, Nonlinear Random Waves, World Scientific, Singapore, 1994. S.-J. Lee, G. T. Yates, and T. Y. Wu, Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances, J. Fluid. Mech. 199, 569–593 (1989). W. K. Melville and K. R. Helfrich,Transcritical two-layer flow over topography, J. Fluid Mech. 178, 31–52 (1987). H. Mitsuderaand R. Grimshaw, Generation of mesoscale variability by resonant interaction between a baroclinic current and localized topography, J. Phys. Ocean. 21, 737–765 (1991). A. Patoine and T. Warn, Theinteraction of long, quasi-stationary baroclinic waves with topography, J. Atmos. Sci. 39, 1018–1025 (1982). J. Printems, The stochastic Korteweg–de Vries equation in L 2(ℝ), J. Diff. Equations 153,338–373 (1999). M. Scalerandi, A. Romano, and C. A. Condat, Korteweg–de Vries solitons underadditive stochastic perturbations, Phys. Rev. E 58, 4166–4173 (1998). M. Wadati, StochasticKorteweg–de Vries equation, J. Phys. Soc. Jpn 52, 2642–2648 (1983). T. Warn and B. Brasnett,The amplification and capture of atmospheric solitons by topography: a theory of the onset of regional blocking, J. Atmos. Sci. 40, 28–38 (1983). G. B. Whitham, Linear and Nonlinear Waves (Pure and AppliedMathematics), Wiley Interscience, New York, 1974. T. Y. Wu, Generation of upstream advancingsolitons by moving disturbances, J. Fluid Mech. 184, 75–99 (1987).