Multiscale reduction of discrete nonlinear Schrödinger equations

We use a discrete multiscale analysis to study the asymptotic integrability of differential–difference equations. In particular, we show that multiscale perturbation techniques provide an analytic tool to derive necessary integrability conditions for two well-known discretizations of the nonlinear Schrödinger equation.

[1]  A. Fokas,et al.  Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman , 2011 .

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

[3]  A. B. Datseff.  On the nonlinear schrödinger equation , 2009 .

[4]  A. Degasperis.  Multiscale Expansion and Integrability of Dispersive Wave Equations , 2009 .

[5]  D. Levi,et al.  On the integrability of the discrete nonlinear Schrödinger equation , 2008, 0808.0837.

[6]  D. Levi,et al.  Multiscale expansion on the lattice and integrability of partial difference equations , 2007, 0710.5299.

[7]  D. Levi,et al.  Multiscale Expansion and Integrability Properties of the Lattice Potential KdV Equation , 2007, 0709.3704.

[8]  D. Levi,et al.  Multiscale expansion of the lattice potential KdV equation on functions of an infinite slow-varyness order , 2007, 0706.1046.

[9]  D. Levi,et al.  Discrete reductive perturbation technique , 2005, math-ph/0510084.

[10]  I. V. Barashenkov,et al.  Translationally invariant discrete kinks from one-dimensional maps. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  D. Levi.  Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV , 2005, nlin/0505061.

[12]  A. Khare,et al.  Exact solutions of the saturable discrete nonlinear Schrödinger equation , 2004, nlin/0409057.

[13]  J. Eilbeck,et al.  The Discrete Nonlinear Schr\"odinger equation - 20 Years on , 2002, nlin/0211049.

[14]  V. Konotop,et al.  Nonlinear excitations in arrays of Bose-Einstein condensates , 2001, cond-mat/0106042.

[15]  Alwyn C. Scott,et al.  Nonlinear Science: Emergence and Dynamics of Coherent Structures , 1999 .

[16]  J. Leon,et al.  Multiscale analysis of discrete nonlinear evolution equations , 1999, solv-int/9902005.

[17]  D. Hennig,et al.  Wave transmission in nonlinear lattices , 1999 .

[18]  Mark J. Ablowitz,et al.  On integrability and chaos in discrete systems , 1998, solv-int/9810020.

[19]  A. Degasperis,et al.  Multiple-scale perturbation beyond the nonlinear Schroedinger equation. I , 1997 .

[20]  I. Gelfand,et al.  Algebraic Aspects of Integrable Systems , 1996 .

[21]  Yuji Kodama,et al.  Solitons in optical communications , 1995 .

[22]  Vladimir E. Zakharov,et al.  What Is Integrability , 1991 .

[23]  W. Eckhaus,et al.  Nonlinear evolution equations, rescalings, model PDEs and their integrability: II , 1987 .

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

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

[26]  Mark J. Ablowitz,et al.  A Nonlinear Difference Scheme and Inverse Scattering , 1976 .

[27]  A. Davydov,et al.  Solitary excitons in one‐dimensional molecular chains , 1973 .

[28]  T. Holstein,et al.  Studies of polaron motion: Part II. The “small” polaron , 1959 .

[29]  Károly Jordán.  Calculus of finite differences , 1951 .