Nonlinear Dynamics

, Volume 73, Issue 3, pp 1587–1599 | Cite as

The use of Volterra series in the analysis of the nonlinear Schrödinger equation

  • L. Z. Guo
  • Y. Z. Guo
  • S. A. Billings
  • D. Coca
  • Z. Q. Lang
Original Paper


A Volterra series analysis is used to analyse the dispersive behaviour in the frequency domain for the non-linear Schrödinger equation (NLS). It is shown that the solution of the initial value problem for the nonlinear Schrödinger equation admits a local multi-input Volterra series representation. Higher order spatial frequency responses of the nonlinear Schrödinger equation can therefore be defined in a similar manner as for lumped parameter non-linear systems. A systematic procedure is presented to calculate these higher order spatial frequency response functions analytically. The frequency domain behaviour of the equation, subject to Gaussian initial waves, is then investigated to reveal a variety of non-linear phenomena such as self-phase modulation (SPM), cross-phase modulation (CPM), and Raman effects modelled using the NLS.


Volterra series Nonlinear systems Nonlinear frequency response Partial differential equations 



The authors gratefully acknowledge support from the UK Engineering and Physical Sciences Research Council (EPSRC) and the European Research Council (ERC).


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • L. Z. Guo
    • 1
  • Y. Z. Guo
    • 1
  • S. A. Billings
    • 1
  • D. Coca
    • 1
  • Z. Q. Lang
    • 1
  1. 1.Department of Automatic Control and Systems EngineeringThe University of SheffieldSheffieldUK

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