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

## Abstract

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.

## Keywords

Volterra series Nonlinear systems Nonlinear frequency response Partial differential equations## Notes

### Acknowledgements

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

## References

- 1.Bedrosian, E., Rice, S.O.: The output properties of Volterra systems driven by harmonic and Gaussian inputs. Proc. IEEE
**59**, 1688–1707 (1971) MathSciNetCrossRefGoogle Scholar - 2.Billings, S.A., Peyton-Jones, J.C.: Mapping nonlinear integrodifferential equations into the frequency domain. Int. J. Control
**52**, 863–879 (1990) MATHCrossRefGoogle Scholar - 3.Billings, S.A., Tsang, K.M.: Spectral analysis for non-linear systems, part I: parametric non-linear spectral analysis. Mech. Syst. Signal Process.
**3**(4), 319–339 (1989) MATHCrossRefGoogle Scholar - 4.Billings, S.A., Tsang, K.M.: Spectral analysis for non-linear systems, part II: interpretation of non-linear frequency response functions. Mech. Syst. Signal Process.
**3**(4), 341–359 (1989) MATHCrossRefGoogle Scholar - 5.Guo, L.Z., Guo, Y.Z., Billings, S.A., Coca, D., Lang, Z.Q.: A Volterra series approach to the frequency domain analysis of nonlinear viscous Burgers’ equation. J. Nonlinear Dyn. (2012). doi: 10.1007/s11071-012-0571-3 MathSciNetGoogle Scholar
- 6.Helie, T., Hasler, M.: Volterra series for solving weakly non-linear partial differential equations: application to a dissipative Burger’s equation. Int. J. Control
**77**(12), 1071–1082 (2004) MathSciNetMATHCrossRefGoogle Scholar - 7.Khelifa, S., Cherruault, Y.: New results for the Adomian method. Kybernetes
**29**(3), 332–354 (1999) MathSciNetCrossRefGoogle Scholar - 8.Lang, Z.Q., Billings, S.A.: Output frequencies of nonlinear systems. Int. J. Control
**713**, 730 (1997) Google Scholar - 9.Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983) MATHCrossRefGoogle Scholar
- 10.Peddanarappagari, K.V., Brandt-Pearce, M.: Volterra series transfer function of single-mode fibers. J. Lightwave Technol.
**15**, 2232–2241 (1997) CrossRefGoogle Scholar - 11.Peddanarappagari, K.V., Brandt-Pearce, M.: Volterra series approach for optimizing fiber-optic communications systems designs. J. Lightwave Technol.
**15**, 2046–2055 (1998) CrossRefGoogle Scholar - 12.Peyton-Jones, J.C., Billings, S.A.: A recursive algorithm for computing the frequency response of a class of nonlinear difference equation model. Int. J. Control
**50**, 1925–1940 (1989) MathSciNetMATHCrossRefGoogle Scholar - 13.Rugh, W.J.: Nonlinear System Theory: The Volterra/Wiener Approach. Johns Hopkins University Press, Baltimore (1981) MATHGoogle Scholar
- 14.Sansen, W.: Distortion in elementary transistor circuits. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process.
**46**(3), 315–325 (1999) CrossRefGoogle Scholar - 15.Schetzen, M.: The Volterra and Wiener Theories of Nonlinear Systems. Wiley, Chichester (1980) MATHGoogle Scholar
- 16.Tao, T.: In: Nonlinear Dispersive Equations: Local and Global Analysis. CBMS, vol. 106. University of California, Los Angeles (2006) Google Scholar
- 17.Worden, K., Manson, G., Tomlinson, G.R.: A harmonic probing algorithm for the multi-input Volterra series. J. Sound Vib.
**201**(1), 67–84 (1997) MathSciNetMATHCrossRefGoogle Scholar - 18.Wu, X.F., Lang, Z.Q., Billings, S.A.: Analysis of output frequencies of nonlinear systems. IEEE Trans. Signal Process.
**55**(7), 3239–3246 (2007) MathSciNetCrossRefGoogle Scholar