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

Authors: Guido Schneider
Journal: Advances in Mathematical Physics
 (2011)

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## 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.


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## Body

## 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 k=0 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(1.1)∂tu-6u∂xu+∂x3u=0
can be transferred with the help of the Miura transformation(1.2)u=v2+∂xv
via direct substitution (1.3)2v∂tv+∂x∂tv-6(v2+∂xv)(2v∂xv+∂x2v)+6(∂xv)∂x2v+2v∂x3v+∂x4v=(2v+∂x)(∂tv-6v2∂xv+∂x3v)=0
into the mKdV equation(1.4)∂tv-6v2∂xv+∂x3v=0.
In order to derive the NLS equation we make an ansatz(1.5)εψv(x,t)=εA(ε(x-ct),ε2t)ei(kx-ωt)+c.c.
for the solutions v=v(x,t) of (1.4), where 0<ε≪1 is a small perturbation paramater. Equating the coefficient at εei(kx-ωt) to zero yields the linear dispersion relation ω=-k3. At ε2ei(kx-ωt) we find the linear group velocity c=-3k2 and at ε3ei(kx-ωt) we find that the complex-valued amplitude A satisfies the NLS equation(1.6)∂2A=-3ik∂12A-6ikA|A|2.


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

Our first approximation result is as follows.Theorem 2.1.
Fixs≥2 and let A∈C([0,T0],Hs+3) be a solution of the NLS equation (1.6). Then there exist ε0>0 and C>0 such that for all ε∈(0,ε0) there are solutions of the mKdV equation (1.4) such that
(2.1)sup⁡t∈[0,T0/ε2]‖v(⋅,t)-εψv(⋅,t)‖Hs≤Cε3/2.Proof.
The error functionR defined by v(x,t)=εψv(x,t)+ε3/2R(x,t) satisfies
(2.2)∂tR+∂x3R-6ε2∂x(ψv2R)-6ε5/2∂x(ψvR2)-2ε3∂x(R3)+ε-3/2Res(εψv)=0,
with
(2.3)Res(εψv)=E(ε4∂13A-6ε4∂1(A|A|2))-E3(2ε3(ik+ε∂1)(A3))+c.c.,
where E=ei(kx-ωt). In order to eliminate the 𝒪(ε3) terms we modify the previous ansatz (1.5) by adding
(2.4)ε36ik3iω-(3ik)3(A3)E3+c.c..
After this modification the residual Res(εψv) is of formal order 𝒪(ε4). When evaluated in Hs there is a loss of ε-1/2 due to the scaling properties of the L2-norm. Hence there exist ε0>0 and Cres>0 such that for all ε∈(0,ε0) we have
(2.5)sup⁡t∈[0,T0/ε2]‖ε-3/2Res(εψv(⋅,t))‖Hs≤Cresε2.
By partial integration we find for s≥2 and all m∈{0,…,s} that
(2.6)∂t∫(∂xmR(x,t))2dx≤C1ε2‖ψv(⋅,t)‖Cbs+12‖R(⋅,t)‖Hs2+C2ε5/2‖ψv(⋅,t)‖Cbs+1‖R(⋅,t)‖Hs3+C3ε3‖R(⋅,t)‖Hs4+C4ε2‖R(⋅,t)‖Hs,
with ε-independent constants Cj. Hence using a≤1+a2 shows that the energy y(t)=∥R(·,t)∥Hs2 satisfies
(2.7)∂ty(t)=C5ε2y(t)+C6ε5/2y(t)3/2+C7ε3y(t)2+C8ε2.
Rescaling time T=ε2t and using Gronwall’s inequality immediately shows the 𝒪(1) boundedness of y for all T∈[0,T0], respectively all t∈[0,T0/ε2]. Therefore, we are done.


## 3. Transfer to the KdV Equation

Applying the Miura transformation (1.2) to the approximation εψv defines an approximation(3.1)εψu=ε2ψv2+ε∂xψv=εikA(ε(x-ct),ε2t)ei(kx-ωt)+c.c.+O(ε2)
of the solution u of the KdV equation (1.1). Since (3.2)‖u-εψu‖Hs-1=‖v2+∂xv-ε2ψv2-ε∂xψv‖Hs-1≤‖v-εψv‖Hs+O(ε2)=O(ε3/2)
the approximation theorem in the original variables follows.Theorem 3.1.
Fixs≥1 and let A∈C([0,T0],Hs+4) be a solution of the NLS equation (1.6). Then there exist ε0>0 and C>0 such that for all ε∈(0,ε0) there are solutions of the KdV equation (1.1) such that
(3.3)sup⁡t∈[0,T0/ε2]‖u(⋅,t)-εψu(⋅,t)‖Hs≤Cε3/2.



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*Source: 854719-2011-05-17.xml*