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

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 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𝜕_𝑡𝑢−6𝑢𝜕_𝑥𝑢+𝜕³_𝑥𝑢=0 (1.1) can be transferred with the help of the Miura transformation𝑢=𝑣²+𝜕_𝑥𝑣 (1.2) via direct substitution 2𝑣𝜕_𝑡𝑣+𝜕_𝑥𝜕_𝑡𝑣−6𝑣²+𝜕_𝑥𝑣2𝑣𝜕_𝑥𝑣+𝜕²_𝑥𝑣+6𝜕_𝑥𝑣𝜕²_𝑥𝑣+2𝑣𝜕³_𝑥𝑣+𝜕⁴_𝑥𝑣=2𝑣+𝜕_𝑥𝜕_𝑡𝑣−6𝑣²𝜕_𝑥𝑣+𝜕³_𝑥𝑣=0 (1.3) into the mKdV equation𝜕_𝑡𝑣−6𝑣²𝜕_𝑥𝑣+𝜕³_𝑥𝑣=0. (1.4) In order to derive the NLS equation we make an ansatz𝜀𝜓_𝑣(𝑥,𝑡)=𝜀𝐴𝜀(𝑥−𝑐𝑡),𝜀²𝑡𝑒^{𝑖(𝑘𝑥−𝜔𝑡)}+𝑐.𝑐. (1.5) for the solutions 𝑣=𝑣(𝑥,𝑡) of (1.4), where 0<𝜀≪1 is a small perturbation paramater. Equating the coefficient at 𝜀𝑒^{𝑖(𝑘𝑥−𝜔𝑡)} to zero yields the linear dispersion relation 𝜔=−𝑘³. At 𝜀²𝑒^{𝑖(𝑘𝑥−𝜔𝑡)} we find the linear group velocity 𝑐=−3𝑘² and at 𝜀³𝑒^{𝑖(𝑘𝑥−𝜔𝑡)} we find that the complex-valued amplitude 𝐴 satisfies the NLS equation𝜕₂𝐴=−3𝑖𝑘𝜕²₁𝐴−6𝑖𝑘𝐴|𝐴|². (1.6)

2. Approximation of the mKdV Equation via the NLS Equation

Our first approximation result is as follows.

Theorem 2.1. Fix 𝑠≥2 and let 𝐴∈𝐶([0,𝑇₀],𝐻^𝑠+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 sup_{𝑡∈[0,𝑇0/𝜀2]}𝑣(⋅,𝑡)−𝜀𝜓_𝑣(⋅,𝑡)_𝐻𝑠≤𝐶𝜀^3/2. (2.1)

Proof. The error function 𝑅 defined by 𝑣(𝑥,𝑡)=𝜀𝜓_𝑣(𝑥,𝑡)+𝜀^3/2𝑅(𝑥,𝑡) satisfies 𝜕_𝑡𝑅+𝜕³_𝑥𝑅−6𝜀²𝜕_𝑥𝜓²_𝑣𝑅−6𝜀^5/2𝜕_𝑥𝜓_𝑣𝑅²−2𝜀³𝜕_𝑥𝑅³+𝜀^−3/2Res𝜀𝜓_𝑣=0, (2.2) with Res𝜀𝜓_𝑣=E𝜀⁴𝜕³₁𝐴−6𝜀⁴𝜕₁𝐴|𝐴|²−E³2𝜀³𝑖𝑘+𝜀𝜕₁𝐴³+𝑐.𝑐., (2.3) where E=𝑒^{𝑖(𝑘𝑥−𝜔𝑡)}. In order to eliminate the 𝒪(𝜀³) terms we modify the previous ansatz (1.5) by adding 𝜀³6𝑖𝑘3𝑖𝜔−(3𝑖𝑘)³𝐴³E³+𝑐.𝑐.. (2.4) After this modification the residual Res(𝜀𝜓_𝑣) is of formal order 𝒪(𝜀⁴). When evaluated in 𝐻^𝑠 there is a loss of 𝜀^−1/2 due to the scaling properties of the 𝐿²-norm. Hence there exist 𝜀₀>0 and 𝐶_res>0 such that for all 𝜀∈(0,𝜀₀) we have sup_{𝑡∈[0,𝑇0/𝜀2]}𝜀^−3/2Res𝜀𝜓_𝑣(⋅,𝑡)_𝐻𝑠≤𝐶_res𝜀². (2.5) By partial integration we find for 𝑠≥2 and all 𝑚∈{0,…,𝑠} that 𝜕_𝑡𝜕^𝑚_𝑥𝑅(𝑥,𝑡)²𝑑𝑥≤𝐶₁𝜀²𝜓_𝑣(⋅,𝑡)²_{𝐶𝑏𝑠+1}‖𝑅(⋅,𝑡)‖²_𝐻^𝑠+𝐶₂𝜀^5/2𝜓_𝑣(⋅,𝑡)_{𝐶𝑏𝑠+1}‖𝑅(⋅,𝑡)‖³_𝐻^𝑠+𝐶₃𝜀³‖𝑅(⋅,𝑡)‖⁴_𝐻^𝑠+𝐶₄𝜀²‖𝑅(⋅,𝑡)‖_𝐻^𝑠, (2.6) with 𝜀-independent constants 𝐶_𝑗. Hence using 𝑎≤1+𝑎² shows that the energy 𝑦(𝑡)=‖𝑅(⋅,𝑡)‖²_𝐻^𝑠 satisfies 𝜕_𝑡𝑦(𝑡)=𝐶₅𝜀²𝑦(𝑡)+𝐶₆𝜀^5/2𝑦(𝑡)^3/2+𝐶₇𝜀³𝑦(𝑡)²+𝐶₈𝜀². (2.7) 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 𝜀𝜓_𝑣 defines an approximation𝜀𝜓_𝑢=𝜀²𝜓²_𝑣+𝜀𝜕_𝑥𝜓_𝑣=𝜀𝑖𝑘𝐴𝜀(𝑥−𝑐𝑡),𝜀²𝑡𝑒^{𝑖(𝑘𝑥−𝜔𝑡)}+𝑐.𝑐.+𝒪𝜀² (3.1) of the solution 𝑢 of the KdV equation (1.1). Since 𝑢−𝜀𝜓_{𝑢𝐻𝑠−1}=𝑣²+𝜕_𝑥𝑣−𝜀²𝜓²_𝑣−𝜀𝜕_𝑥𝜓_{𝑣𝐻𝑠−1}≤𝑣−𝜀𝜓_𝑣𝐻𝑠+𝒪𝜀²=𝒪𝜀^3/2 (3.2) the approximation theorem in the original variables follows.

Theorem 3.1. Fix 𝑠≥1 and let 𝐴∈𝐶([0,𝑇₀],𝐻^𝑠+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 sup_{𝑡∈[0,𝑇0/𝜀2]}𝑢(⋅,𝑡)−𝜀𝜓_𝑢(⋅,𝑡)_𝐻𝑠≤𝐶𝜀^3/2. (3.3)

Acknowledgment

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