Effective Slow Dynamics Models for a Class of Dispersive Systems

Abstract

We consider dispersive systems of the form ∂ t U = Λ U U + B U (U, V), ε∂ t V = Λ V V + B V (U, U) in the singular limit ε → 0, where Λ U , Λ V are linear and B U , B V bilinear mappings. We are interested in deriving error estimates for the approximation obtained through the regular limit system ∂ t ψ U = Λ U ψ U − B U (ψ U , Λ −1 V B V (ψ U , ψ U)) from a more general point of view. Our abstract approximation theorem applies to a number of semilinear systems, such as the Dirac-Klein-Gordon system, the Klein-Gordon-Zakharov system, and a mean field polaron model. It extracts the common features of scattered results in the literature, but also gains an approximation result for the Dirac-Klein-Gordon system which has not been documented in the literature before. We explain that our abstract approximation theorem is sharp in the sense that there exists a quasilinear system of the same structure where the regular limit system makes wrong predictions.

The Dirac-Klein-Gordon system in the strong coupling limit

“…So, while in general (1.6) does not hold even approximately, Ψ is still well described by (1.7) in the limit. Similar singular limits have been studied in the literature [22,1,8,13,2], and we will comment in Section 1.3 below on the particularity of our situation compared to some of these works.…”

“…with respect to the H s+1 -norm would require an H s+1 -bound on S. Hence, the chain of estimates does not close due to a loss of derivatives. In the works [8,13,2], it turns out that this loss of derivatives does not occur due to an additional regularizing effect, and in these cases one can use such an argument. This loss of derivatives can be dealt with by considering the differences, or reduced variables,…”

The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability

“…We assume that the other eigenvalues (except for the three eigenvalues stated in Hypothesis A) have negative real part and are uniformly bounded away from the imaginary axis near the bifurcation point. A consequence of Hypothesis A is that in leading order the dynamics is in the direction of the corresponding eigenvectors, see (2). Other scenarios which are possible in the general case m ≥ 3 have been excluded by assuming Hypothesis A.…”

The monoatomic FPU system as a limit of a diatomic FPU system

“…A diatomic FPU system with heavy and light particles. 1 The dynamics of diatomic lattices, e.g. propagation of traveling solitary waves, has always been important in physical applications and has been studied in numerous works, e.g.…”

“…The approximation result of Theorem 1 is nontrivial since the right hand side of the associated first order system to (2), and (3) Remark 3. Approximation results for systems with a small perturbation parameter in front of one time derivative, similar to system (2)-(3) have been considered in [1]. However, the abstract theorem from [1] does not apply since the nonlinear interaction appearing here is different as the one considered in Eq.…”