Dressing for a Novel Integrable Generalization of the Nonlinear Schrödinger Equation

We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the N-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen.

[1]  W. Strauss,et al.  Stability of peakons , 2000 .

[2]  A. Fokas On a class of physically important integrable equations , 1994 .

[3]  T. Tsuchida New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems , 2007, 0712.4373.

[4]  J. Lenells Traveling wave solutions of the Camassa-Holm equation , 2005 .

[5]  Athanassios S. Fokas,et al.  Symplectic structures, their B?acklund transformation and hereditary symmetries , 1981 .

[6]  J. Lenells Exactly Solvable Model for Nonlinear Pulse Propagation in Optical Fibers , 2008, 0810.5289.

[7]  A. Fokas,et al.  Explicit soliton asymptotics for the Korteweg–de Vries equation on the half-line , 2008, 0812.1579.

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

[9]  Sergei Sakovich,et al.  On Transformations of the Rabelo Equations , 2007, 0705.2889.

[10]  Vladimir E. Zakharov,et al.  The Inverse Scattering Method , 1980 .

[11]  Mauro Luiz Rabelo,et al.  On Equations Which Describe Pseudospherical Surfaces , 1989 .

[12]  V. Gerdjikov,et al.  Inverse scattering transform for the Camassa–Holm equation , 2006, Inverse Problems.

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

[14]  Richard Beals,et al.  Bäcklund Transformations and Inverse Scattering Solutions for Some Pseudospherical Surface Equations , 1989 .

[15]  A. Fokas,et al.  The linearization of the initial-boundary value problem of the nonlinear Schro¨dinger equation , 1996 .

[16]  David J. Kaup,et al.  An exact solution for a derivative nonlinear Schrödinger equation , 1978 .

[17]  R. L. Anderson,et al.  On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems , 1982 .

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

[19]  Derivation of Bäcklund Transformation for the AKNS-Integrable System by Means of the Riemann-Hilbert Problem , 1987 .

[20]  Jonatan Lenells,et al.  Stability of periodic peakons , 2004 .

[21]  Zhiwu Lin,et al.  Stability of peakons for the Degasperis‐Procesi equation , 2007, 0712.2007.

[22]  A. S. Fokas,et al.  On a novel integrable generalization of the nonlinear Schrödinger equation , 2008, 0812.1510.

[23]  A. S. Fokas,et al.  An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons , 2008, 0812.1335.

[24]  J. Lenells The Scattering Approach for the Camassa–Holm equation , 2002, nlin/0306021.

[25]  Yi Xiao Note on the Darboux transformation for the derivative nonlinear Schrodinger equation , 1991 .

[26]  A. S. Fokas,et al.  A unified transform method for solving linear and certain nonlinear PDEs , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.