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IMA Journal of Applied Mathematics Advance Access originally published online on January 27, 2007
IMA Journal of Applied Mathematics 2007 72(2):206-222; doi:10.1093/imamat/hxl034
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Solitary wave interaction for a higher-order nonlinear Schrödinger equation

S. M. Hoseini** and T. R. Marchant***

School of Mathematics and Applied Statistics, The University of Wollongong, Wollongong, New South Wales 2522, Australia

** Email: smh33{at}uow.edu.au

*** Corresponding author. Email: tim_marchant{at}uow.edu.au


  Abstract

Solitary wave interaction for a higher-order version of the nonlinear Schrödinger (NLS) equation is examined. An asymptotic transformation is used to transform a higher-order NLS equation to a higher-order member of the NLS integrable hierarchy, if an algebraic relationship between the higher-order coefficients is satisfied. The transformation is used to derive the higher-order one- and two-soliton solutions; in general, the N-soliton solution can be derived. It is shown that the higher-order collision is asymptotically elastic and analytical expressions are found for the higher-order phase and coordinate shifts. Numerical simulations of the interaction of two higher-order solitary waves are also performed. Two examples are considered, one satisfies the algebraic relationship derived from asymptotic theory, and the other does not. For the example which satisfies the algebraic relationship, the numerical results confirm that the collision is elastic. The numerical and theoretical predictions for the higher-order phase and coordinate shifts are also in strong agreement. For the example which does not satisfy the algebraic relationship, the numerical results show that the collision is inelastic and radiation is shed by the solitary wave collision. As the bed of radiation shed by the waves decays very slowly (like t–1/2), it is computationally infeasible to calculate the final phase and coordinate shifts for the inelastic example. An asymptotic conservation law is derived and used to test the finite-difference scheme for the numerical solutions.

Keywords: NLS equation; solitary waves; asymptotic transformation; elastic and inelastic collisions; higher-order phase and coordinate shifts.

Received on 9 July 2006. accepted on 8 December 2006.


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