Numerical Study of Oscillatory Regimes in the Kadomtsev–Petviashvili Equation

Abstract The aim of this paper is the accurate numerical study of the Kadomtsev–Petviashvili (KP) equation. In particular, we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end, we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey–Stewartson system. In a second step, we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg–de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.

[1]  Mark J. Ablowitz,et al.  Wave collapse and instability of solitary waves of a generalized Kadomtsev-Petviashvili equation , 1994 .

[2]  B. Kadomtsev,et al.  On the Stability of Solitary Waves in Weakly Dispersing Media , 1970 .

[3]  Paul H. Roberts,et al.  Motions in a Bose condensate: IV. Axisymmetric solitary waves , 1982 .

[4]  K. Khusnutdinova,et al.  Double waves in multi-dimensional systems of hydrodynamic type: the necessary condition for integrability , 2004, nlin/0412064.

[5]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[6]  Christian Klein,et al.  Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation , 2006 .

[7]  Valeri A. Makarov,et al.  Two-dimensional solitons in Bose-Einstein condensates with a disk-shaped trap , 2003 .

[8]  Senatorski,et al.  Simulations of Two-Dimensional Kadomtsev-Petviashvili Soliton Dynamics in Three-Dimensional Space. , 1996, Physical review letters.

[9]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

[10]  Igor Krichever,et al.  Method of averaging for two-dimensional "integrable" equations , 1988 .

[11]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[12]  Fei-Ran Tian,et al.  Oscillations of the zero dispersion limit of the korteweg‐de vries equation , 1993 .

[13]  Li-Yeng Sung,et al.  On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations , 1992 .

[14]  Y. Kodama.  A method for solving the dispersionless KP equation and its exact solutions , 1988 .

[15]  R. H. Hardin.  Application of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations , 1973 .

[16]  Jean Bourgain,et al.  On the Cauchy problem for the Kadomstev-Petviashvili equation , 1993 .

[17]  John Strain,et al.  Computing the weak limit of KdV , 1994 .

[18]  M. Boiti,et al.  Properties of solutions of the Kadomtsev–Petviashvili I equation , 1994 .

[19]  Yvan Martel,et al.  Non existence of L2–compact solutions of the Kadomtsev–Petviashvili II equation , 2004 .

[20]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[21]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[22]  T. Hou,et al.  Removing the stiffness from interfacial flows with surface tension , 1994 .

[23]  J. D. Lawson.  Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants , 1967 .

[24]  正人 木村.  Max-Planck-Institute for Mathematics in the Sciences(海外,ラボラトリーズ) , 2001 .

[25]  T. Mitsui,et al.  A Conservative Spectral Method for Several Two-Dimensional Nonlinear Wave Equations , 1999 .

[26]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[27]  T. Driscoll.  A composite Runge-Kutta method for the spectral solution of semilinear PDEs , 2002 .

[28]  Senatorski,et al.  Numerical simulations of Kadomtsev-Petviashvili soliton interactions. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  N. Mauser,et al.  Numerical study of the Davey-Stewartson system , 2004 .

[30]  N. Tzvetkov.  Bilinear estimates related to the KP equations , 2000 .

[31]  Jean-Claude Saut,et al.  Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation , 2002 .

[32]  H. Takaoka,et al.  On the local regularity of the Kadomtsev-Petviashvili-II equation , 2001 .

[33]  Li-Yeng Sung,et al.  The Cauchy problem for the Kadomtsev–Petviashvili–I equation without the zero mass constraint , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[34]  Tamara Grava,et al.  Mathematik in den Naturwissenschaften Leipzig Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations , 2005 .

[35]  Jean-Claude Saut,et al.  Remarks on the Mass Constraint for KP-Type Equations , 2006, SIAM J. Math. Anal..

[36]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[37]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.

[38]  Abdul-Majid Wazwaz,et al.  A computational approach to soliton solutions of the Kadomtsev-Petviashvili equation , 2001, Appl. Math. Comput..

[39]  Hideo Takaoka,et al.  Well-posedness for the Kadomtsev-Petviashvili II equation , 2000, Advances in Differential Equations.

[40]  J. Colliander,et al.  Low regularity solutions for the Kadomtsev-Petviashvili I equation , 2002 .

[41]  Salah M. El-Sayed,et al.  Numerical soliton-like solutions of the potential Kadomtsev–Petviashvili equation by the decomposition method , 2003 .

[42]  Andrei V. Faminskii,et al.  The Cauchy problem for the Kadomtsev-Petviashvili equation , 1990 .

[43]  G. Schneider.  Approximation of the Korteweg-de Vries Equation by the Nonlinear Schrödinger Equation , 1998 .

[44]  B. Minchev,et al.  A review of exponential integrators for first order semi-linear problems , 2005 .

[45]  C. David Levermore,et al.  The Small Dispersion Limit of the Korteweg-deVries Equation. I , 1982 .

[46]  Christof Sparber,et al.  NUMERICAL SIMULATION OF GENERALIZED KP TYPE EQUATIONS WITH SMALL DISPERSION , 2006 .

[47]  Robert L. Pego,et al.  On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation , 1997 .

[48]  Tamara Grava,et al.  The generation, propagation, and extinction of multiphases in the KdV zero‐dispersion limit , 2002 .

[49]  Stephanos Venakides,et al.  The zero dispersion limit of the korteweg‐de vries equation for initial potentials with non‐trivial reflection coefficient , 1985 .

[50]  Catherine Sulem,et al.  The nonlinear Schrödinger equation , 2012 .

[51]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[52]  C. David Levermore,et al.  The hyperbolic nature of the zero dispersion Kdv limit , 1988 .

[53]  Anna Rozanova-Pierrat,et al.  Mathematical analysis of Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation , 2006 .

[54]  Y. Kodama,et al.  A method for solving the dispersionless KP hierarchy and its exact solutions. II , 1989 .

[55]  Stephanos Venakides,et al.  The Small Dispersion Limit of the Korteweg-De Vries Equation , 1987 .

[56]  Vladimir E. Zakharov,et al.  Multi-scale expansions in the theory of systems integrable by the inverse scattering transform , 1986 .

[57]  Razvan C. Fetecau,et al.  A Hamiltonian Regularization of the Burgers Equation , 2006, J. Nonlinear Sci..

[58]  Lloyd N. Trefethen,et al.  Fourth-Order Time-Stepping for Stiff PDEs , 2005, SIAM J. Sci. Comput..

[59]  George Rowlands,et al.  Instabilities and oscillations of one- and two-dimensional Kadomtsev-Petviashvili waves and solitons II. Linear to nonlinear analysis , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[60]  G. Rowlands,et al.  Instabilities and oscillations of one–and two–dimensional Kadomtsev—Petviashvili waves and solitons , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[61]  R. Johnson,et al.  The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations , 2003 .