A numerical study of the large-period limit of a Zakharov–Shabat eigenvalue problem with periodic potentials

Deconinck and Kutz (2006 J. Comput. Phys. 219 296?321) developed an efficient algorithm for solving the Zakharov?Shabat eigenvalue problem with periodic potentials numerically. It is natural to use the same algorithm for solving the problem for non-periodic potential (decaying potentials defined over the whole real line) using large periods. In this paper, we propose the use of a specific value of the Floquet exponent. Our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of Gardner (1997 J. Reine Angew. Math. 491 149?81) to be valid. We also illustrate the rather complicated path to convergence of some nonlinear Schr?dinger potentials.

[1]  Mark J. Ablowitz,et al.  Nonlinear differential−difference equations , 1975 .

[2]  J. K. Shaw,et al.  Purely imaginary eigenvalues of Zakharov-Shabat systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .

[4]  Bernard Deconinck,et al.  On the convergence of Hill's method , 2010, Math. Comput..

[5]  Kevin Zumbrun,et al.  Convergence of Hill's Method for Nonselfadjoint Operators , 2010, SIAM J. Numer. Anal..

[6]  D. McLaughlin,et al.  Coherence and chaos in the driven damped sine-Gordon equation: measurement of the soliton spectrum , 1986 .

[7]  V. Matveev,et al.  Schrödinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation , 1975 .

[8]  S. Lafortune,et al.  Spectral stability analysis for periodic traveling wave solutions of NLS and CGL perturbations , 2008 .

[9]  J. Satsuma,et al.  B Initial Value Problems of One-Dimensional self-Modulation of Nonlinear Waves in Dispersive Media (Part V. Initial Value Problems) , 1975 .

[10]  Mark J. Ablowitz,et al.  Method for Solving the Sine-Gordon Equation , 1973 .

[11]  M. G. Forest,et al.  Numerical inverse spectral transform for the periodic sine-Gordon equation: theta function solutions and their linearized stability , 1991 .

[12]  Bernard Deconinck,et al.  Computing spectra of linear operators using the Floquet-Fourier-Hill method , 2006, J. Comput. Phys..

[13]  R. Gardner,et al.  Spectral analysis of long wavelength periodic waves and applications. , 1997 .

[14]  M. Ablowitz,et al.  On the solution of a class of nonlinear partial di erence equations , 1977 .

[15]  Peter D. Lax,et al.  Periodic solutions of the KdV equation , 2010 .

[16]  M. Wadati,et al.  The Exact Solution of the Modified Korteweg-de Vries Equation , 1972 .

[17]  M. Ablowitz,et al.  The Periodic Cubic Schrõdinger Equation , 1981 .

[18]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[19]  Marcel Novaes,et al.  Resonances in open quantum maps , 2012, 1211.7248.

[20]  A. Osborne,et al.  Computation of the direct scattering transform for the nonlinear Schroedinger equation , 1992 .

[21]  B. Herbst,et al.  The nonlinear Schro¨dinger equation: asymmetric perturbations traveling waves and chaotic structures , 1997 .

[22]  Thiab R. Taha,et al.  Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical , 1984 .

[23]  Annalisa Calini,et al.  Finite-Gap Solutions of the Vortex Filament Equation: Genus One Solutions and Symmetric Solutions , 2005, J. Nonlinear Sci..

[24]  B. Herbst,et al.  Numerical homoclinic instabilities and the complex modified Korteweg-de Vries equation , 1991 .

[25]  M. Gregory Forest,et al.  Spectral theory for the periodic sine‐Gordon equation: A concrete viewpoint , 1982 .

[26]  Guo-Wei Wei,et al.  Discrete singular convolution for the sine-Gordon equation , 2000 .