The KdV approximation for a system with unstable resonances

The KdV equation can be derived via multiple scaling analysis for the approximate description of long waves in dispersive systems with a conservation law. In this paper, we justify this approximation for a system with unstable resonances by proving estimates between the KdV approximation and true solutions of the original system. By working in spaces of analytic functions, the approach will allow us to handle more complicated systems without a detailed discussion of the resonances and without finding a suitableenergy.

Remark 1. Such an approximation result is nontrivial since solutions of order ( 2 ) have to be controlled on an (1∕ 3 ) timescale. The estimate (6) and Sobolev's embedding theorem imply Remark 2. The linearized problem is solved by In Fourier space, the KdV equation describes the modes in the u equation, which are strongly concentrated around the wave number k = 0; cf Figure 1. Therefore, the expansion u (k) = k − 1 2 k 3 + (k 5 ) at k = 0 plays an important role for the dynamics. We have 1 = − 1 2 in (4).
Remark 3. Historically, the KdV equation has been derived for the so-called water wave problem first. Approximation results have been established in a number of papers. They are either based on energy estimates (cf Craig 1 and Schneider and Wayne, 2 Schneider and Wayne, 3 and Duell 4 ) or on the use of analytic functions (cf Kano and Nishida 5 and Schneider 6 ).
Remark 4. Although the BKG system looks less complicated than the water wave problem, for the KdV approximation of the BKG system, some features occur, which are not present for the water wave problem over a flat bottom, namely, the occurrence of quadratic resonances, like they occur for the water wave problem over a periodic bottom. The linearized water wave problem over a periodic bottom, which is solved by Bloch modes, has been analyzed in Craig et al. 7

FIGURE 1
The curves of eigenvalues ± u , ± v for the linearized BKG system plotted as a function over the Fourier wave numbers in case 2 = 1 (left) and 2 = 5 (right). The modes in the circles are described by the KdV approximation

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[Colour figure can be viewed at wileyonlinelibrary.com] Remark 5. For > 2, the curves u and v intersect at two wave numbers k 1 and k 2 ; cf the right panel of Figure 1. In Bauer et al, 8 it has been explained that there are 2 k 1 spatially periodic solutions of the form where = 2 t. By suitably choosing the coefficients a uu , a uv , b uu , and b uv , the matrix M has eigenvalues with nonvanishing real part. Hence, growth rates e = e 2 t = e T∕ with a > 0 occur. These allow us to bring n A 1 and n B 1 , which are both initially of order ( n ), to an order ( 2 ) at a time T = ((n − 2) | ln( )|) ≪ 1. Therefore, we have that v = ( 2 ) on a timescale much smaller than the natural timescale of the KdV equation. See Figure 2.  Remark 8. The BKG system is a prototype model for a whole class of systems. Elements of this class are the poly-atomic FPU problem and the water wave problem over a periodic bottom with (1) periodicity and bottom variations of (1). The transfer of the following analysis to these systems will be the subject of future research. The main strength of the approach of the present paper is that it will allow us to handle such more complicated systems without a detailed discussion of the resonances and without finding a suitable energy, which will be different for every system. Except of very few exceptions (cf Chong and Schneider, 9 Bauer et al, 10 Chirilus-Bruckner et al, 12 and Gaison et al 13 ), the KdV approximation so far has only been justified for systems with a single pair of curves of eigenvalues ±i u . Notation. The Fourier transform of a function u is denoted by  u orû. Possibly different constants that can be chosen independently of the small perturbation parameter 0 < 2 ≪ 1 are denoted with the same symbol C.

DERIVATION OF THE KDV APPROXIMATION
into the BKG system gives The residuals Res u and Res v contain the terms, which do not cancel after inserting the approximation into the BKG system. For our subsequent error estimates, we need Res u = ( 8 ) and Res v = ( 8 ). In order to achieve this goal, we have to extend our approximation of v by higher order terms. Therefore, our final approximation is given by For this improved approximation, we find if we choose B 1 and B 2 to satisfy 1∕2 , we lose a factor −1/2 if we compute the magnitude of the residual w.r.t. powers of in L 2 -based spaces. Therefore, we have the following.
Proof. Counting the powers of is straightforward. Hence, it remains to discuss the assumption s A − s ≥ 8. The term that loses most regularity is 2 T B 2 , which can be expressed in terms of 2 T (AB 1 ) and 2 We have that T A can be expressed via the right-hand side of the KdV equation in terms of A, … , 3 X A. Differentiating the KdV equation w.r.t. T shows that 2 T (A 2 ) can be expressed in terms of A, … , 6 X A. Therefore, 2 Then, writing the equations for the error, obtained from the BKG systems (1) and (2), as a first order system, a term −1 x Res u occurs.

Lemma 2. Under the assumption of Lemma 1, we have the estimate
Proof. The loss of −1 comes from −1 x = −1 −1 X . We need to show that −1 x Res u is again in L 2 . This is obvious for all terms, which have a derivative X in front, ie, all terms except 2 Therefore, we are done.

THE EQUATIONS FOR THE ERROR
The error functions where 3 2 This system is written as first-order system, with u , v from (7), x Res u in 3 i u f u is of order ( 3 ) due to Lemma 2. After diagonalization, of the linear part, we obtain and similar for  −1 and  −2 . Since 2 u is strongly concentrated at k = 0, we separate 2 u into a part concentrated close to k = 0 and into the rest. For > 0, we define the mode projection E viaÊ u =Êû, whereÊ (k) = 1 for |k| ≤ andÊ (k) = 0 elsewhere. Corollary 3 and Remark 11 in Appendix A), we write the equations for the error as and similar for  −1 and  −2 , where ).

THE FUNCTIONAL ANALYTIC SET-UP
In order to control the unstable resonances, we introduce a number of function spaces. By (·, ·), we denote the Euclidean inner product, and by | · |, the associated Euclidean norm in R d . The Fourier transform is denoted by For m ≥ 0, we define the Sobolev spaces equipped with the inner product For any m ∈ N, the induced norm is equivalent to the usual H m -norm. Finally, for m ≥ 0, we introduce By Sobolev's embedding theorem, the space H m+ (R) is continuously embedded into W m for each > 1∕2. Moreover, every u ∈ W m is ⌊m⌋ times continuously differentiable with finite C ⌊m⌋ b (R)-norm. In order to control the positive growth rates, possibly occurring at the resonances (cf Remark 5), we work in the space where ≥ 0 and m ≥ 0. Functions u ∈ H ∞ ,0 can be extended to functions that are analytic on the strip {z ∈ C ∶ |Im(z)| < }; cf Reed and Simon. 11, Theorem IX. 13 Similarly, we define the spaces W ∞ ,m . In our notations of the spaces and norms, we do not distinguish between scalar and vector-valued functions. The spaces H ∞ ,m are closed under point-wise multiplication for every ≥ 0 and m > 1∕2, and the spaces W ∞ ,m for every ≥ 0 and m ≥ 0. For details, see Lemma 6, Corollary 1, and Corollary 2.

SOME FIRST ESTIMATES
In this section, we collect various estimates, which are necessary for the proof of Theorem 1. We start by rewriting Lemma 1 and Lemma 2 into H ∞ ,s spaces.
be a solution of the KdV Equation (9), and let 2 u and 4 v be defined as above. For this approximation, then there exist 0 > 0 and C res > 0 such that for all ∈ (0, 0 ), we have Proof. Using Lemma 6 from the appendix the proof goes line for line as the proofs of Lemma 1 and Lemma 2.
Since u is a bounded operator in H ∞ ,s and since ( v ) −1 is a bounded operator from H ∞ ,s to H ∞ ,s+1 we have, using Lemma 3, that With these estimates and Corollary 3, we find

FROM ANALYTIC TO SOBOLEV FUNCTIONS
In order to control the unstable resonances, we solve the equations for the error in H ∞ ,s spaces with s ≥ 1 and = (t) decreasing in time. In detail, we choose for 0 ≤ t ≤ T 1 ∕ 3 with T 1 = A ∕ . In order to satisfy the subsequent condition (17), we have to choose > 0 sufficiently large, which gives the restriction on the approximation time.
In order to work in usual Sobolev, we introduce with S (t) a multiplication operator defined in Fourier space bŷ As a direct consequence of the definitions, we have the following.

Lemma 4. For t ∈ [0, A ∕( 3 )], the linear mappings S (t) ∶ H ∞ (t),s → H s and S (t) ∶ W ∞ (t),s → W s , with (t) = ( A − 3 t)∕ , are bijective and bounded with bounded inverse.
The new variables satisfy and similar for R −1 and R −2 , where The operator |k| op is defined via its operation in Fourier space|k| op R(k) = |k|R(k). In order to estimate R 1 and R 2 on the long (1∕ 3 ) timescale, the terms of order ( 3 ) in (13) and (6) are no problem. In the end, they can be controlled easily with Gronwall's inequality. Using Lemma 4, Lemma 6, and the previous estimates, we find Hence, the major difficulty to come to the long (1∕ 3 ) timescale is the control of the terms of order ( 2 ) in (13) and (6). Our strategy to handle the terms of order ( 2 ) is as follows. The modes to wave numbers outside a neighborhood of zero are controlled with the sectorial operator − 2 |k| op . For terms not vanishing at k = 0, this operator is of no use. For wave numbers in a 0 neighborhood of the origin with 0 > 0 small, but fixed, the error equations are simplified by a number of normal form transformations, ie, by a number of near identity change of variables. These normal form transformations are provided in Section 7. In Section 8, the solutions of the transformed system are then estimated by energy estimates.
Remark 9. Since the BKG systems (1) and (2) are a semilinear system local existence and uniqueness in H ∞ ,s spaces follow from a simple application of the contraction mapping principle to the variation of constant formula. The subsequent error estimates serve as a priori estimates to guarantee existence and uniqueness on the required time interval.

THE NORMAL FORM TRANSFORMATION
In this section, we provide a number of near identity change of variables to eliminate terms close to the wave number k = 0 of formal order ( 2 ), which cannot be controlled by the sectorial operator − 2 |k| op or which finally turn out to be of order ( 3 ). Before we do so, we recall the basics of normal form transformations.
Remark 10. For the abstract evolutionary system we seek a near identity change of coordinates v = u − 2 K(u) to eliminate the terms 2 N Q (u) and to transfer them into higher order terms, like 3 N c (u). We find In order to eliminate 2 N Q , we choose 2 K to satisfy such that after the transformation . Hence, after the transform, the terms of order ( 2 ) are eliminated, and only terms of order ( 3 ) remain.
In the following, we provide such normal form transformations with the goal, which have explained above.
• The term 2 E 0 i u S (t)(a uv (E u )S −1 (t)(R 2 + R −2 )) in the R 1 equation can be written in Fourier space as and similarly for q 1,−2 (k, k − l, l). Following the existing literature (cf Sanders et al 14 and Schneider and Uecker 15 ), it is obvious that this term can be eliminated by a near identity change of variableŝ .

Lemma 5.
There exist 0 > 0 and C > 0 such that for all ∈ (0, 0 ], the transformation (R 1 , R 2 )  → (R 3 , R 4 ) is bijective with After the transformation, our system is of the form and similar for R −3 and R −4 . The terms with g 3 and g 4 come from g 1 and g 2 and from higher order terms obtained via the normal form transformation. Therefore, using s A − s ≥ 2, the terms with g 3 and g 4 obey the estimates where C 1 is a constant independent of 0 < ≪ 1, solely depending on ||A|| W ∞ A ,s A .

THE FINAL ENERGY ESTIMATES
We have now all ingredients to perform the final energy estimates. We define an operator via the multiplierΩ(k) = min( v (k), 4) in Fourier space. This operator is used in the estimates for the subsequent term Res 6 . It leads there to a cancelation, which shows that Res 6 is of order ( 3 ).
We start now to estimate the time derivative of We compute The terms s 1 , … s 8 are either of order ( 3 ) or can be estimated by the negative terms of order ( 2 ) collected in s good . The terms collected in s 1 and s 2 can easily be estimated to be ( 3 ). The terms s 3 , s 4 , s 5 , s 7 , and s 8 have either u or E c in front and thus vanish at the wave number k = 0. They can be estimated by the good terms collected in s good . The term s 6 can be shown to be of order ( 3 ) using the long wave character of the KdV approximation. for s good , s 1 , and s 2 (a) We start with the bound on the linear terms collected in s good . Using the Fourier representation of |k| op gives

Estimates
and similar for (Ω 1∕2 R 4 , − 2 |k| op Ω 1∕2 R 4 ) H s such that finally, (b) Using the skew symmetry of i u and i v yields (c) Using the Cauchy-Schwarz inequality and (7) yields for s 3 , s 4 , s 5 , s 7 , and s 8 The good terms collected in s good do not allow us to estimate terms at the wave number k = 0. We have to use the fact, that the terms s 3 , s 4 , s 5 , s 7 , and s 8 vanish at the wave number k = 0, too.

Estimates
(a) The terms s 3 and s 4 can be estimated by the "good" terms using the fact that |̂u(k)| ≤ C|k| for k → 0. The last estimate implies that the symbol of = |k| −1∕2 op u is bounded at the wave number k = 0. We find where we used that ||S −1 (t) 1∕2 u || W s = ( 1∕2 ) due to Corollary 3 applied to |̂1 ∕2 (k)| ≤ C|k| 1∕2 . In the last line, we used 5/2 ab ≤ 2 a 2 + 3 b 2 . The term s 4 can be estimated in exactly the same as the term s 3 . The last lines have to be modified into (b) The remaining terms s 5 , s 7 , and s 8 can be estimated by the "good" terms in exactly the same way as s 3 and s 4 using the fact that s 5 , s 7 , and s 8 have an E c in front which vanishes at the wave number k = 0, too. We finally obtain

Estimates for s 6
For the Fourier transform of Res 6 in case s = 0, we obtain where we usedû(l − k) =û(k − l) which holds due to the fact that u is real-valued. By definition we have b uvÊ (k − l) = b uvÊ (l − k) ∈ R, and so q 0 (k, 0, k) = 0 for all k ∈ R. Since we have a compact set of wave numbers involved here, this implies |q 0 (k, k−l, l)| ≤ C|k − l|. As a consequence, we can apply Corollary 3 and obtain ∫ ∫ q 0 (k, k − l, l)R −4 (k)̂u(k − l)R 4 (l)dldk = ( ), respectively, |s 6 | ≤ C 3 E s . The case s > 0 works very similarly. In order to estimate ( s x ΩR 4 , s we use the fact that after the comma, whenever a derivative falls on u , we gain an additional power of . Hence, there is only one term of order ( 2 ); namely, then, all s derivatives after the comma fall on R 4 . But this term can be estimated line for line as the case s = 0.

Putting all estimates together
Using the previous estimates yields with C a constant, which is independent of 0 < ≪ 1. Subsequently, we choose > 0 so large but independently of 0 < ≪ 1 that will be satisfied. Under this assumption, we have In the following, we choose > 0 so small that 1∕2 E 1∕2 s will be satisfied. Under this assumption, we then have and so Gronwall's inequality implies The constant M is independent of , respectively, T 1 and 0 < ≪ 1. We are done, if we choose 0 > 0 so small that 1∕2 0 M 1∕2 ≤ 1, which guarantees the validity of (18), and then > 0 so large that which guarantees the validity of (17).