A direct method of solution for the Fokas–Lenells derivative nonlinear Schrödinger equation: I. Bright soliton solutions

We develop a direct method of solution for finding the bright N-soliton solution of the Fokas–Lenells derivative nonlinear Schrodinger equation. The construction of the solution is performed by means of a purely algebraic procedure using an elementary theory of determinants and does not rely on the inverse scattering transform method. We present two different expressions of the solution both of which are expressed as a ratio of determinants. We then investigate the properties of the solutions and find several new features. Specifically, we derive the formula for the phase shift caused by the collisions of bright solitons.

[1]  Yuri S. Kivshar,et al.  Optical Solitons: From Fibers to Photonic Crystals , 2003 .

[2]  Bo Tian,et al.  Dark and antidark solitons in the modified nonlinear Schrödinger equation accounting for the self-steepening effect. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  V.,et al.  On the theory of two-dimensional stationary self-focusing of electromagnetic waves , 2011 .

[4]  Yoshimasa Matsuno,et al.  Bilinear Transformation Method , 1984 .

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

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

[7]  A. Fordy.  Derivative nonlinear Schrodinger equations and Hermitian symmetric spaces , 1984 .

[8]  Susumu Takeda,et al.  Modified Nonlinear Schrödinger Equation for Alfvén Waves Propagating along the Magnetic Field in Cold Plasmas , 1975 .

[9]  Mark J. Ablowitz,et al.  On discretizations of the vector nonlinear Schrödinger equation , 1998, solv-int/9810014.

[10]  M. Wadati,et al.  Integrable Multi-Component Hybrid Nonlinear Schrödinger Equations , 1995 .

[11]  Bo Tian,et al.  Novel behavior and properties for the nonlinear pulse propagation in optical fibers , 2012 .

[12]  Jonatan Lenells,et al.  Dressing for a Novel Integrable Generalization of the Nonlinear Schrödinger Equation , 2009, J. Nonlinear Sci..

[13]  E. Mjølhus,et al.  On the modulational instability of hydromagnetic waves parallel to the magnetic field , 1976, Journal of Plasma Physics.

[14]  N-soliton solution for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions , 2006, nlin/0602044.

[15]  THE DIRECT METHOD , 1917 .

[16]  V. E. Vekslerchik,et al.  Lattice representation and dark solitons of the Fokas–Lenells equation , 2011, 1103.4701.

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

[18]  Hiroshi Inoue,et al.  Soliton Solutions of the Derivative Nonlinear Schrödinger Equation , 1979 .

[19]  O. C. Wright.  Some homoclinic connections of a novel integrable generalized nonlinear Schrödinger equation , 2009 .

[20]  V. M. Lashkin,et al.  N-soliton solutions and perturbation theory for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions , 2007, nlin/0701039.

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

[22]  Hiroshi Inoue,et al.  Exact Solutions of the Derivative Nonlinear Schrödinger Equation under the Nonvanishing Conditions , 1978 .

[23]  A. Kundu.  Two-fold integrable hierarchy of nonholonomic deformation of the derivative nonlinear Schrödinger and the Lenells–Fokas equation , 2009, 0910.0383.

[24]  Kimiaki Konno,et al.  Spiky Soliton in Circular Polarized Alfvén Wave , 1979 .

[25]  広田 良吾,et al.  The direct method in soliton theory , 2004 .

[26]  Y. Matsuno.  The bright N-soliton solution of a multi-component modified nonlinear Schrödinger equation , 2011, 1110.5421.

[27]  Jingsong He,et al.  The Darboux transformation of the derivative nonlinear Schrödinger equation , 2011, 1109.0674.

[28]  M. Wadati,et al.  Coupled Hybrid Nonlinear Schrödinger Equation and Optical Solitons , 1994 .

[29]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[30]  Robert Vein,et al.  Determinants and their applications in mathematical physics , 1998 .

[31]  H. Steudel,et al.  The hierarchy of multi-soliton solutions of the derivative nonlinear Schrödinger equation , 2003 .

[32]  E. Mjølhus.  A note on the modulational instability of long Alfvén waves parallel to the magnetic field , 1978, Journal of Plasma Physics.