Discrete conservation laws and the convergence of long time simulations of the mkdv equation

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. The standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions. In this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order. Pseudospectral spatial discretization appear as the most robust of the numerical methods, but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence.

[1]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[2]  Kazuaki Nakayama,et al.  Motion of Curves in the Plane , 1993 .

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

[4]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[5]  E. Hairer.  Symmetric Projection Methods for Differential Equations on Manifolds , 2000 .

[6]  Kenji Ohkuma,et al.  Multiple-Pole Solutions of the Modified Korteweg-de Vries Equation , 1982 .

[7]  Terence Tao,et al.  Why are solitons stable , 2008, 0802.2408.

[8]  Avinash Khare,et al.  On some classes of mKdV periodic solutions , 2004, nlin/0410047.

[9]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[10]  J. M. Keiser,et al.  A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs , 1998 .

[11]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[12]  Francisco de la Hoz,et al.  Numerical Study of a Flow of Regular Planar Curves That Develop Singularities at Finite Time , 2008, SIAM J. Appl. Math..

[13]  C. Kenig,et al.  Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle , 1993 .

[14]  M. A. Helal,et al.  A Chebyshev spectral method for solving Korteweg-de Vries equation with hydrodynamical application , 2001 .

[15]  J. Nathan Kutz,et al.  Numerical simulation of the semi-classical limit of the focusing nonlinear Schrödinger equation , 1999 .

[16]  T. Chan,et al.  FOURIER METHODS WITH EXTENDED STABILITY INTERVALS FOR THE KORTEWEG-DE VRIES EQUATION. , 1985 .

[17]  M.W.Kalinowski.  Nonlinear Waves , 2016, 1611.10114.

[18]  Héctor D. Ceniceros,et al.  A numerical study of the semi-classical limit of the focusing nonlinear Schrödinger equation , 2002 .

[19]  Nakayama,et al.  Integrability and the motion of curves. , 1992, Physical review letters.

[20]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[21]  Jesús María Sanz-Serna,et al.  Convergence of method of lines approximations to partial differential equations , 1984, Computing.

[22]  D. Gottlieb,et al.  Numerical analysis of spectral methods , 1977 .

[24]  M. Helal,et al.  A comparison between two different methods for solving KdV–Burgers equation , 2006 .

[25]  Luis Vega,et al.  On the ill-posedness of some canonical dispersive equations , 2001 .

[26]  L. Gardner,et al.  Solitary wave solutions of the MKdV− equation , 1995 .

[27]  Qiang Du,et al.  Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications , 2005 .

[28]  Gulcin M. Muslu,et al.  A split-step Fourier method for the complex modified Korteweg-de Vries equation☆ , 2003 .

[29]  Gijs de Boer,et al.  with Numerical , 2000 .

[30]  G. Lamb.  Elements of soliton theory , 1980 .

[31]  M. A. Helal,et al.  Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics , 2002 .

[32]  Osamu Inoue,et al.  Fourier expansion solution of the Korteweg-de Vries equation , 1980 .

[33]  B. Guo,et al.  Spectral Methods and Their Applications , 1998 .

[34]  Bengt Fornberg,et al.  A numerical and theoretical study of certain nonlinear wave phenomena , 1978, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[35]  M. A. Helal,et al.  A comparative study between two different methods for solving the general Korteweg–de Vries equation (GKdV) , 2007 .

[36]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[37]  Petrich,et al.  The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane. , 1991, Physical review letters.

[38]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[39]  A. B. Shamardan,et al.  Central finite difference schemes for nonlinear dispersive waves , 1990 .

[40]  C. Wexler,et al.  Contour dynamics, waves, and solitons in the quantum Hall effect , 1999 .