Integrability revisited: a necessary condition

Abstract In this Letter we revisit the subject of integrability in systems with infinite degrees of freedom. We will show via combined analytical and numerical results that such systems cannot be integrable if the solitons possess internal modes. The potential sufficiency of the condition is also discussed.

[1]  Aubry,et al.  Growth and decay of discrete nonlinear Schrodinger breathers interacting with internal modes or standing-wave phonons , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[3]  P. Kevrekidis,et al.  Stability of waves in discrete systems , 2001 .

[4]  P. Kevrekidis,et al.  Bifurcation of internal solitary wave modes from the essential spectrum , 2000 .

[5]  M. Ablowitz,et al.  Nonlinear differential–difference equations and Fourier analysis , 1976 .

[6]  R. Matzner,et al.  Fractal structure in the scalar λ ( φ 2 − 1 ) 2 theory , 1991 .

[7]  T. Sugiyama.  Kink-Antikink Collisions in the Two-Dimensional φ4 Model , 1979 .

[8]  J. Gibbon,et al.  Solitons and Nonlinear Wave Equations , 1982 .

[9]  J. M. Speight.  A discrete $\phi^4$ system without a Peierls - Nabarro barrier , 1997 .

[10]  Yuri S. Kivshar,et al.  Dynamics of Solitons in Nearly Integrable Systems , 1989 .

[11]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

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

[13]  R. S. Ward,et al.  Kink dynamics in a novel discrete sine-Gordon system , 1994, patt-sol/9911008.

[14]  Todd Kapitula,et al.  Exponentially small splitting of heteroclinc orbits: from the rapidly forced pendulum to discrete solitons , 2000 .

[15]  Yuri S. Kivshar,et al.  Internal Modes of Solitary Waves , 1998 .

[16]  Panayotis G. Kevrekidis,et al.  Breathers on a background: periodic and quasiperiodic solutions of extended discrete nonlinear wave systems , 2002, Math. Comput. Simul..

[17]  M. Peyrard,et al.  Solitonlike excitations in a one-dimensional atomic chain with a nonlinear deformable substrate potential , 1982 .