Advances in Mathematical Physics

Volume 2011 (2011), Article ID 854719, 4 pages

http://dx.doi.org/10.1155/2011/854719

## Justification of the NLS Approximation for the KdV Equation Using the Miura Transformation

IADM, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Received 4 March 2011; Accepted 16 March 2011

Academic Editor: Pavel Exner

Copyright © 2011 Guido Schneider. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is the purpose of this paper to give a simple proof of the fact that solutions of the KdV equation can be approximated via solutions of the NLS equation. The proof is based on an elimination of the quadratic terms of the KdV equation via the Miura transformation.

#### 1. Introduction

The NLS equation describes slow modulations in time and space of an oscillating and advancing spatially localized wave packet. There exist various approximation results, cf. [1–4] showing that the NLS equation makes correct predictions of the behavior of the original system. Systems with quadratic nonlinearities and zero eigenvalues at the wave number turn out to be rather difficult for the proof of such approximation results, cf. [5, 6]. The water wave problem falls into this class. Very recently, this long outstanding problem [7] has been solved [8] for the water wave problem in case of no surface tension and infinite depth by using special properties of this problem. Another equation which falls into this class is the KdV equation. The connection between the KdV and the NLS equation has been investigated already for a long time, cf. [9]. In [10, 11] the NLS equation has been derived as a modulation equation for the KdV equation, and its inverse scattering scheme has been related to the one of the KdV equation. It is the purpose of this paper to give a simple proof of the fact that solutions of the KdV equation can be approximated via solutions of the NLS equation. Beyond things this has been shown by numerical experiments in [12]. An analytical approximation result has been given by a rather complicated proof in [5] with a small correction explained in [6]. The much simpler proof of this fact presented here is based on an elimination of the quadratic terms of the KdV equation via the Miura transformation.

Following [13] the KdV equation can be transferred with the help of the Miura transformation via direct substitution into the mKdV equation In order to derive the NLS equation we make an ansatz for the solutions of (1.4), where is a small perturbation paramater. Equating the coefficient at to zero yields the linear dispersion relation . At we find the linear group velocity and at we find that the complex-valued amplitude satisfies the NLS equation

#### 2. Approximation of the mKdV Equation via the NLS Equation

Our first approximation result is as follows.

Theorem 2.1. *Fix and let be a solution of the NLS equation (1.6). Then there exist and such that for all there are solutions of the mKdV equation (1.4) such that
*

*Proof. *The error function defined by satisfies
with
where . In order to eliminate the terms we modify the previous ansatz (1.5) by adding
After this modification the residual is of formal order . When evaluated in there is a loss of due to the scaling properties of the -norm. Hence there exist and such that for all we have
By partial integration we find for and all that
with -independent constants . Hence using shows that the energy satisfies
Rescaling time and using Gronwall’s inequality immediately shows the boundedness of for all , respectively all . Therefore, we are done.

#### 3. Transfer to the KdV Equation

Applying the Miura transformation (1.2) to the approximation defines an approximation of the solution of the KdV equation (1.1). Since the approximation theorem in the original variables follows.

Theorem 3.1. *Fix and let be a solution of the NLS equation (1.6). Then there exist and such that for all there are solutions of the KdV equation (1.1) such that
*

#### Acknowledgment

This paper is partially supported by the Deutsche Forschungsgemeinschaft DFG under Grant Schn520/8-1.

#### References

- L. A. Kalyakin, “Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium,”
*Matematicheskiĭ Sbornik. Novaya Seriya*, vol. 132(174), no. 4, pp. 470–495, 592, 1987. View at Google Scholar · View at Zentralblatt MATH - P. Kirrmann, G. Schneider, and A. Mielke, “The validity of modulation equations for extended systems with cubic nonlinearities,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 122, no. 1-2, pp. 85–91, 1992. View at Google Scholar · View at Zentralblatt MATH - G. Schneider, “Justification of modulation equations for hyperbolic systems via normal forms,”
*NoDEA. Nonlinear Differential Equations and Applications*, vol. 5, no. 1, pp. 69–82, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Schneider, “Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances,”
*Journal of Differential Equations*, vol. 216, no. 2, pp. 354–386, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Schneider, “Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation,”
*Journal of Differential Equations*, vol. 147, no. 2, pp. 333–354, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W.-P. Düll and G. Schneider, “Justification of the nonlinear Schrödinger equation for a resonant Boussinesq model,”
*Indiana University Mathematics Journal*, vol. 55, no. 6, pp. 1813–1834, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. E. Zakharov, “Stability of periodic waves of finite amplitude on the surface of a deep fluid,”
*Journal of Applied Mechanics and Technical Physics*, vol. 4, pp. 190–194, 1968. View at Google Scholar - N. Totz and S. Wu, “A rigorous justification of the modulation approximation to the 2d full water wave problem,” Preprint, 2011. View at Publisher · View at Google Scholar
- R. S. Johnson, “On the modulation of water waves on shear flows,”
*Proceedings of the Royal Society of London A*, vol. 347, pp. 537–546, 1976. View at Publisher · View at Google Scholar - V. E. Zakharov and E. A. Kuznetsov, “Multiscale expansions in the theory of systems integrable by the inverse scattering transform,”
*Physica D*, vol. 18, no. 1–3, pp. 455–463, 1986. View at Publisher · View at Google Scholar - E. R. Tracy, J. W. Larson, A. R. Osborne, and L. Bergamasco, “On the nonlinear Schrödinger limit of the Korteweg-de Vries equation,”
*Physica D*, vol. 32, no. 1, pp. 83–106, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. P. Boyd and G.-Y. Chen, “Weakly nonlinear wavepackets in the Korteweg-de Vries equation: the KdV/NLS connection,”
*Mathematics and Computers in Simulation*, vol. 55, no. 4–6, pp. 317–328, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. G. Drazin and R. S. Johnson,
*Solitons: An Introduction*, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 1989.