Low regularity justification results for envelope approximations of nonlinear wave packets in periodic media

1. Introduction

Amplitude equations are an ubiquitous tool to describe complicated physical systems modeled by partial differential equations. A phenomenon of particular interest is the propagation of wave packets, e.g. light pulses in dispersive media. This has been addressed in various physical settings, i.e. in linear and nonlinear media with or without oscillating coefficients. A non-exhaustive list of work on amplitude equations for wave packet propagation is [7,10–12,15–21]. In this amplitude formalism (AF), first by a multiple scaling ansatz the amplitude equation is derived, which typically takes the form of a (nonlinear) Schrödinger equation with constant so called effective coefficients. Then, in a second step, the amplitude equation is justified by estimates for the error between solutions of the full system and the approximation by the AF. This is also related to certain WKB methods (in geometric and diffractive optics, see, e.g., [2,8,13]), but since we restrict to one space dimension and to almost monochromatic waves (no explicit leading order phase modulation in the initial conditions) we prefer the more specific name amplitude formalism.

Our purpose here is to improve the AF to regularity assumptions on the data that are lower than in previous results. We consider the cubic Klein–Gordon equation (1) ε 2 ∂ t 2 u ε − ∂ x 2 u ε + ε − 2 u ε + c ε u ε 3 = 0 , where u ε = u ε ( t , x ) ∈ R , t ⩾ 0 , x ∈ R , and where c ε is highly oscillating with period ε, i.e., c ε ( x ) : = c ( x / ε ) with small ε and c ∈ L 2 ( ( 0 , 1 ) ) extended via c ( y + 1 ) = c ( y ) for y ∈ R . This can be generalized in various ways, for instance to oscillating coefficients for the linear terms, but for simplicity we stick to (1). On the other hand, the oscillating nonlinear coefficient c ε gives some averaging effects which go beyond the case of constant c; see Remark 1.1 for further comments.

The conserved energy for (1) is (2) E ( t ) = E ( u ε ( t ) ) : = ∫ 1 2 [ ( ε 2 ∂ t u ε ( t , x ) ) 2 + ( ε ∂ x u ε ( t , x ) ) 2 + u ε 2 ( t , x ) ] + ε 2 4 c ε u ε 4 ( t , x ) d x . If for simplicity we restrict to the case c ( y ) ⩾ 0 for all y ∈ [ 0 , 1 ) , then we immediately obtain the uniform a priori estimate (3) ‖ u ε ( t ) ‖ E 2 : = ε 4 ‖ ∂ t u ε ( t ) ‖ 2 + ε 2 ‖ ∂ x u ε ( t ) ‖ 2 + ‖ u ε ( t ) ‖ 2 ⩽ 2 E ( t ) = 2 E ( 0 ) for all t in the energy norm ‖ u ε ‖ E , where here and in the following always ‖ · ‖ = ‖ · ‖ L 2 ( R ) , i.e., ‖ u ε ( t ) ‖ 2 = ∫ u ε 2 ( t , x ) d x . From (3) we obtain global existence of solutions of (1). However, without the sign condition on c ( y ) , which we do not assume in this paper, (3) is wrong, and good estimates for ‖ u ε ( t ) ‖ E depend on the class of initial conditions (IC).

We consider IC in the form of wave-packets (4) u ε ( 0 , x ) = v 0 ( x ) e 1 ( 0 , x / ε ) + ε g 1 ( x ) + v ¯ 0 ( x ) e − 1 ( 0 , x / ε ) + ε g ¯ 1 ( x ) , ∂ t u ε ( 0 , x ) = ε − 2 v 1 ( x ) e 1 ( 0 , x / ε ) + ε − 1 g 2 ( x , x / ε ) + ε − 2 v ¯ 1 ( x ) e − 1 ( 0 , x / ε ) + ε − 1 g ¯ 2 ( x , x / ε ) , with e ± 1 ( τ , y ) = exp ( ± i ( k 0 y − ω 0 τ ) ) , y = x / ε , τ = t / ε 2 , v 0 , g 1 ∈ H 1 ( R ) , v 1 , g 2 ∈ L 2 ( R ) , and where v 1 is related to v 0 in a certain way, specified in Theorems 2.3 and 2.5 below. Here 0 ¯ denotes the complex conjugate, from now on we will use the shorthand notation c . c . to denote the complex conjugate of the preceding terms. The spatial wave number k 0 ∈ R and the temporal wavenumber ω 0 are related by the dispersion relation, which for (1) takes the simple form (5) ω 2 = k 2 + 1 . On a the level of formal asymptotic expansions it is well known, e.g., [9], that (1) has approximate solutions of the form (6) u ε ( t , x ) = ψ v ( t , x ) = v ( t , x − ν t / ε ) e 1 ( t / ε 2 , x / ε ) + c . c . , where ν = ∂ k ω ( k 0 ) (the group velocity) and v ( t , x ) fulfills the Nonlinear Schrödinger equation (NLS) (7) ∂ t v = i 2 ∂ k 2 ω 0 ∂ x 2 v − 3 i 2 ω 0 c ∗ | v | 2 v , c ∗ = ∫ 0 1 c ( y ) d y .

As our main results we give justifications of (7) with low regularity requirements on v 0 . For v 0 ∈ H 2 ( R ) , letting u ε ( x , t ) = ψ v ( x , t ) + ε r ( x , t ) we prove, on an O ( 1 ) time scale, the error estimate (8) ‖ r ( t ) ‖ E ⩽ C . For v 0 ∈ H s ( R ) , 1 < s < 2 , letting u ε ( x , t ) = ψ v ( x , t ) + ε s / 2 r ( x , t ) we show that (9) ‖ r ( t ) ‖ H ε 1 ⩽ C with the scaled H 1 norm ‖ u ‖ H ε 1 ( R ) : = ∫ R u 2 ( x ) + ε 2 ( ∂ x u ( x ) ) 2 d x . See Theorems 2.3 and 2.5 for the precise results. In particular, these also yield long time existence for (1) for the specific IC without assumptions on the sign of c. The reduced regularity assumptions are achieved by smooth approximations of the initial conditions via cut-off of Fourier modes of wave number k with | k | > n , and balancing of errors: the approximation error is small for large n, while the error estimates of the amplitude formalism will grow in n. We balance the growth in n with some power of ε, track the errors and choose an appropriate n ( ε ) to achieve the desired estimates.

Remark 1.1.

a) As already mentioned, many related results are known in the literature, see, e.g., [7,11,19,21] and the references therein. Our model (1), (4) is a simple extension of the constant coefficient cubic Klein–Gordon equation, and the use of c ε instead of a constant c ∈ R allows to illustrate some averaging effects in the effective coefficients. Moreover, it allows to illustrate a slight technical improvement (see Remark 2.1) which avoids the diagonalization of linear operators for instance used in [7] for the definition of improved approximations. On the other hand, contrary to, e.g., [3,4,7], we do not consider periodic coefficients in the linear part because we want to avoid the Bloch wave machinery needed in this case; the results given here can be extended to this case, but this requires some technical effort.

b) A more complicated version of (1) contains quadratic terms, e.g., (10) ε 2 ∂ t 2 u ε − ∂ x 2 u ε + ε − 2 u ε + ε − 1 b ε u ε 2 + c ε u ε 3 = 0 , where b ε ( x ) : = b ( x / ε ) , b sufficiently smooth with b ( y + 1 ) = b ( y ) . A priori estimates and justification results for this quadratic case are typically obtained via normal form transforms [7,17,18,22], and require certain non-resonance conditions and a careful handling of the regularity loss in the normal form transforms. We believe that the methods from this paper can be transferred to the quadratic case, but for now restrict to the simpler cubic case.

c) Other standard versions of (1), (4) are obtained from setting (11) u ˜ ε ( τ , ξ ) = ε u ε ( ε 2 τ , ε ξ ) , which yields (12) ∂ τ 2 u ˜ ε = ∂ ξ 2 u ˜ ε − u ˜ ε − b ( ξ ) u ˜ ε 2 − c ( ξ ) u ˜ ε 3 , (13) u ˜ ε ( 0 , ξ ) = ε v 0 ( ε ξ ) e ˜ 1 ( 0 , ξ ) + c . c . , ∂ τ u ˜ ε ( 0 , ξ ) = v 1 ( ε ξ ) e ˜ 1 ( 0 , ξ ) + c . c . with e ˜ 1 ( τ , ξ ) = e i ( k 0 ξ − ω 0 τ ) . Note that (12) does not explicitly depend on ε, which makes this scaling somewhat more natural, but the ε-dependence of the initial conditions justifies the subscript ε in u ˜ ε . The advantage of (1), (4) is that it is somewhat closer to the underlying physics: x, t and x / ε , t / ε 2 are called the macroscopic and microscopic scales, respectively, and consequently (7) is called the macroscopic equation.

d) A related class of problem, see, e.g., [14,15] and the references therein, is given by the case of spatially non-oscillatory initial data, corresponding to k 0 = 0 in (4), for (typically) constant coefficient and gauge invariant nonlinear Klein–Gordon equations with x ∈ R d . In this case, the solutions formally decompose into fast oscillations in time modulated by slow envelopes in time and space, which again can be described via solutions of NLS equations, and [14,15] give low regularity approximation results for this situation, which would essentially correspond to v 0 ∈ H 1 ( R ) (and k 0 = 0 ) in (4). While our proofs below use pointwise in time a priori estimates on the difference between the formal approximation and the solution u ε in some energy norms, the proofs in [14,15] are based on Strichartz estimates, which are not easily available in our case of highly oscillatory initial data ( k 0 ≠ 0 ) and non gauge invariant nonlinearity.

Remark 1.2.

Another approach to derive and justify effective equations for problems with rapidly varying data is the method of two-scale convergence (2SC). The 2SC method for wave equations is based on the same formal calculation as the amplitude formalism, and on an a priori estimate like (3). On the linear level, terms involving the solution u ε then have good convergence properties within the 2SC, and this allows to derive and justify a limiting equation in one step, for initial data v 0 ∈ H 1 ( R ) . See, e.g., [3], where this has been carried out for a linear wave equation with x ∈ R d , which moreover on the linear level is much more general than (1).

A justification of (7) for (1) for v 0 ∈ H 1 ( R ) using methods from 2SC would be an important achievement, and an improvement of, or at least an alternative to, our results (8) and (9). However, 2SC is considerably harder for nonlinear equations, as so called strong 2SC is needed to obtain some information on nonlinear expressions involving u ε . In [23], 2SC methods have been combined with multiple scales expansions to prove justification results for the homogenization of nonlinear Schrödinger equations with a large rapidly varying potential and highly regular initial data. The linear Schrödinger case with v 0 ∈ H 1 is considered in [2,5], and in [1, Remark 7.5] it is claimed that the results of [1,5] for the linear case in some special cases generalize to the nonlinear case. However, it appears that sufficient information to treat nonlinear terms in the framework of 2SC is not easily available for solutions of (1), and we failed to make 2SC methods work for (1), (4) without directly using the results (8) or (9), which in particular means that again we cannot deal with v 0 ∈ H 1 ∖ H s , s > 1 .

2. The justification results

Our derivation and justification of (7) proceeds in two steps. First we plug the ansatz (14) u ε ( t , x ) = ψ v ( t , x ) + h . o . t . : = v ( t , ξ ) e 1 ( t / ε 2 , x / ε ) + c . c . + h . o . t . , ξ = x − ν t / ε , e j = e i j ( k 0 y − ω 0 τ ) , into (1), where y = x / ε , τ = t / ε 2 , and h . o . t . denotes at this point unspecified higher order (in ε) terms. Sorting with respect to ε then yields a hierarchy of equations, to be successively solved. All O ( ε − 2 ) terms vanish due to the dispersion relation ω 0 2 = k 0 2 + 1 , all O ( ε − 1 ) terms vanish by the choice ν = ∂ k ω ( k 0 ) , and at O ( ε 0 ) we obtain (15) [ − 2 i ω 0 ∂ t v + ( ν 2 − 1 ) ∂ ξ 2 v + 3 c ( x / ε ) | v | 2 v ] e 1 + c ( x / ε ) v 3 e 3 + c . c . = 0 . Clearly, the splitting into harmonics e 1 and e 3 in (15) is not self-evident, unless c is constant. However, based on the separation of scales between the arguments ξ = x − ν t / ε of v and y = x / ε of c, the standard procedure is to average the coefficients of e 1 in y over the periodicity cell ( 0 , 1 ) of c, i.e., to require (16) ∫ 0 1 ( − 2 i ω 0 ∂ t v + ( ν 2 − 1 ) ∂ ξ 2 v + 3 c ( y ) | v | 2 v ) d y = − 2 i ω 0 ∂ t v + ( ν 2 − 1 ) ∂ ξ 2 v + 3 c ∗ | v | 2 v = ! 0 , which is the NLS (7) for v. Thus, the O ( ε 0 e 1 ) terms vanish in an averaged sense, i.e., at O ( ε 0 ) the so called residual has the form (17) Res 0 ( ψ v ) : = ( c ( x / ε ) − c ∗ ) | v | 2 v e 1 + c ( x / ε ) v 3 e 3 . The complete residual is Res ( ψ v ) = ε 2 ∂ t 2 ψ v − ∂ x 2 ψ v + ε − 2 ψ v + c ε ψ v 3 = Res 0 + ε Res 1 , and it is easy to see that under mild conditions on v the terms Res 1 are O ( 1 ) bounded in natural norms (see below).

Given a solution v of the NLS, the question is whether the ansatz (14) gives an approximation of a solution of (1) on an O ( 1 ) time-scale, which is the natural time-scale for the NLS evolution. In the context of the AF this problem of justification is usually dealt with as follows. Defining the (scaled) error ρ ( x , t ) via u ε ( x , t ) = ψ v ( x , t ) + ε ρ ( x , t ) , we want to use a-priori estimates on the error equation (18) ε 2 ∂ t 2 ρ = ∂ x 2 ρ − ε − 2 ρ − f , f = f ( t , x ) = c ε ( 3 ψ v 2 ρ + 3 ε ψ v ρ 2 + ε 2 ρ 3 ) − ε − 1 Res ( ψ v ) , to show that ρ stays O ( 1 ) bounded in a suitable norm, on an O ( 1 ) time scale. In the energy-norm we obtain (19) d d t ‖ ρ ‖ E 2 = 2 ε 2 ∫ ( ∂ t ρ ) f d x ⩽ 2 ε 2 ‖ ∂ t ρ ‖ L 2 ‖ f ‖ L 2 ⩽ 2 ‖ ρ ‖ E ‖ f ‖ L 2 and using a Gronwall argument it would be sufficient to have (20) ‖ ε − 1 Res ( ψ v ) ‖ L 2 = O ( 1 ) to have ‖ ρ ‖ E bounded. However, already from (17) we see that (20) does not hold in general if v is O ( 1 ) in L 2 ∩ L ∞ , say.