Transport and generation of macroscopically modulated waves in diatomic chains

We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of a single plane wave in a nonlinear diatomic chain with stabilizing on-site potentials including the case where a wave generates another wave via self-interaction. More precisely, we show that in typical chains acoustical waves can generate optical but not acoustical waves, while optical waves are always closed with respect to self-interaction.

The linearized modelü = Lu admits for non-trivial plane-wave solutions provided the frequency ω ∈ R and the wave number ϑ ∈ (−π, π] satisfy the dispersion relation where c i := 2v i,1 + w i,1 . This is equivalent to for all ϑ ∈ (−π, π] and the additional assumption c 1 = c 2 yields the strict separation of the optical and acoustical branches of the frequency, . All of the above assumptions are satisfied in the case w i,1 > 0, 4v i,1 + w i,1 > 0, v 1,1 v 2,1 > 0, 2v 2,1 + w 2,1 > 2v 1,1 + w 1,1 , which we assume in the following. The eigenvectors A to the eigenfrequencies ω = ω ± (θ) are given by The plan of the paper is as follows. In Section 2 we discuss whether a given plane wave solution E can generate via self-interaction another plane wave E 2 . Then, taking into account also this possibility, in Section 3 we derive formally the macroscopic equations for the first order amplitudes A 1,n of two waves n = 1, 2, and finally, in Section 4, we justify the derived equations.

Resonances
Since we are interested in the self-interaction of a plane wave E, which means that E 2 is also a plane wave, in a diatomic chain we are interested in resonance conditions like the ones on the left hand side below. Making in (6) the substitutions c := (cos ϑ + 1)/2 ∈ [0, 1], d 1 := (c 1 + c 2 ) 2 /f > 0, d 2 := (c 1 − c 2 ) 2 /f > 0 with f := 16v 1,1 v 2,1 > 0 and d 1 − d 2 > 1, the problem of finding a ϑ ∈ (−π, π] satisfying one of these resonance conditions is equivalent to finding a c ∈ [0, 1] for given d 1 > d 2 + 1 > 1 satisfying the corresponding equation on the right hand side: By the positivity of all appearing square roots we immediately see that a resonance 2ω + (ϑ) = ω − (2ϑ), i.e., an optical wave generating an acoustical one, is not possible. Moreover, since we see that an optical wave can not generate another optical one, i.e., 2ω + (ϑ) = ω + (2ϑ) ∀ ϑ ∈ (−π, π]. Thus, an optical wave is closed under self-interaction of order 2. However, an acoustical wave can generate an optical one by self-interaction, i.e., for appropriate choice of the harmonic parts of the interaction and on-site potentials there exist ϑ ∈ (−π, π] such that 2ω − (ϑ) = ω + (2ϑ). After taking squares on the left and right hand sides, the corresponding condition (9) reads and we want to prove the existence of a c ∈ [0, 1] that satisfies this condition for the d 1 , d 2 given above. We restrict ourselves to the case v 1,1 = a > 0, v 2,1 = γa, γ > 1, w 1,1 = w 2,1 = b > 0. This setting satisfies all conditions posed so far on the harmonic coefficients, and we obtain Inserting these values into (10), we get Hence, for every c ∈ [0, 1] such that 16c ≥ 9 − 8δ there exists a b a such that (10) is satisfied. Since δ > 0, we can always find such a c.
Furthermore, the resonance condition for the generation of an acoustical wave from an acoustical one, 2ω − (ϑ) = ω − (2ϑ), is equivalent to Concerning the case just considered, we observe that for d 2 = δ, the r.h.s. is nonnegative only for c ∈ [c e , 1] with c e := max{0, 5− √ 15δ+24 2 } (and hence for all c ∈ [0, 1] when δ ≥ 1/15). Restricting our analysis to the set [c e , 1] (non-empty for all δ > 0), we obtain by squaring and insertion of the values (11) as above although with a minus sign in front of the square root. Due to the existent on-site potential (where b > 0), in order to obtain resonances the r.h.s. g needs to be strictly positive for some c ∈ [c e , 1]. However, a careful analysis reveals that g(c) ≤ 0 for c ∈ [c e , 1], and we obtain that in the case v 2,1 = γa > a = v 1,1 , w 1,1 = w 2,1 = b > 0, an acoustical wave can not generate another acoustical one by self-interaction. Finally, we conclude by showing that ω − (ϑ) + ω + (2ϑ) = ω + (3ϑ) for all ϑ ∈ (−π, π]. Indeed, after squaring the left and right hand sides we see that the equality is equivalent to , the r.h.s. of this equation is always < 0, and it suffices to show that the l.h.s. is ≥ 0 even for . Comparing in this modified l.h.s. the square of the first two terms with the square of the third one, and adding a suitable term, this is equivalent to showing that . Since the r.h.s. is positive and strictly decreasing as a function of , which holds true (as an elementary analysis shows), with g(1) = 0.

Formal derivation
We are interested in solutions of (3) which in first order in ε are a sum of two macroscopically modulated plane-wave solutions with small amplitudes where A 1,n = A 1,n , A (2) 1,n T : R × R → C 2 and E n (t, j) := e i(ωnt+jϑn) with (ω n , ϑ n ) satisfying (4). However, due to the scaling of A 1,n by ε and the macroscopic nature of its time and space variables, its dynamics will include terms of second order in ε. Hence, taking into account the nonlinearity of our original system (3) and the fact that we consider two different plane waves, we insert into (3) the improved approximation 2,ι , A 2,ι (τ, y ± ξ 2 ε) with ξ 1 , ξ 2 ∈ (0, 1), assuming A 1,n (τ, ·) ∈ C 2 (R; C 2 ), A 2,ι (τ, ·) ∈ C 1 (R; C 2 ).
Carrying out the usual (lengthy but straightforward) formal expansion in terms of ε and E n , we obtain thatÜ A,2 with the explicit expressions for K ι , ι ∈ I, and res U A,2 ε = O(ε 3 ) given in the Appendix. Hence, in order for our ansatz (13) to satisfy (3) up to order ε, taking into account that E 1 = E 2 , the systems H(ω n , ϑ n )A 1,n = 0 have to be satisfied. As we have already seen, since det H(ω n , ϑ n ) = 0, this gives the relation between first and second component of A 1,n (7), (8) with A, ρ, ω, ϑ replaced by A 1,n , ρ n , ω n , ϑ n .
Next, we assume that (Note here that det H(0, 0) = 0 is always satisfied due to our stability assumption c 1 c 2 > 4v 1,1 v 2,1 .) In this case and for ϑ n = ±π, we obtain from the equations for ε 2 E n ρ n A 1,n + v 2,1 e −iϑn ∂ y A 1,n .
Inserting A 1,n = −ρ n A 1,n , and noting that (5) gives we obtain from the equality of the right hand sides of (15) 1,n = 0 for ω n = ω ± (ϑ n ). Analogously, in the case ϑ n = ±π we get from (8) Thus, we conclude that if the non-resonance conditions (14) hold, which means in particular that neither wave generates a new one via self-interaction, the dynamics of the amplitudes A 1,n are given by uncoupled transport equations where the velocity is the group velocity of the corresponding carrier wave. Hence, setting in particular A 1,2 (0, ·) = 0 we obtain that the dynamics of A 1,1 are given, unsurprisingly, by a homogeneous transport equation. Moreover, since the K ι , ι ∈ I, are known, as they depend only on the first order amplitudes A 1,n (see Appendix), and since (15), (17) 2 , (18) 2 determine the relation between the components of A 2,n , we obtain by (14) all A 2,ι except for one component of A 2,n , which can be assumed to be equivalently vanishing.

Justification
The equations obtained by the formal derivation constitute only necessary conditions on the amplitudes A 1,n of the ansatz (12). The purpose of the justification is to show that indeed solutions u of such a form exist.