Effective Slow Dynamics Models for a Class of Dispersive Systems

Abstract

We consider dispersive systems of the form

$$\begin{aligned} \partial _t U = \Lambda _U U + B_U(U,V) , \qquad \varepsilon \partial _t V = \Lambda _V V + B_V(U,U) \end{aligned}$$

in the singular limit $\varepsilon \rightarrow 0$, where $\Lambda _U,\Lambda _V$ 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

$$\begin{aligned} \partial _t \psi _U = \Lambda _U \psi _U - B_U(\psi _U, \Lambda _V^{-1} B_V(\psi _U,\psi _U) ) \end{aligned}$$

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.

Local Existence and Uniqueness of the Limit Systems

For completeness we add the following local existence and uniqueness result for (16).

Theorem A.1

For all $C_0 > 0$ there exists a $T_0 > 0$ such that for all $U_0 \in X_U$ with $\Vert U_0 \Vert _{X_U} \le C_0$ there exists a unique mild solution $\psi _U \in C([0,T_0], X_U)$ of (16) with $\psi _U|_{t = 0} = U_0$.

Proof

We consider the variation of constant formula

$$\begin{aligned} \psi _U(t) = e^{\Lambda _U t} U_0 - \int _0^t e^{\Lambda _U (t-\tau )} B_U(\psi _U, \Lambda _V^{-1} B_V(\psi _U,\psi _U) )(\tau )d\tau \end{aligned}$$

(60)

associated to (16). Due to the assumptions (S1), (B1), and (I), for all fixed $C_1 > 0$ the right-hand side of (60) is a contraction in a ball

$$\begin{aligned} \{ \psi _U \in C([0,T_0], X_U) : \Vert \psi _U - e^{\Lambda _U t} U_0 \Vert _{X_U} \le C_1 \} \end{aligned}$$

for a $T_0 > 0$ sufficiently small. By the contraction mapping principle there is a unique fixed point of the right-hand side of (60) which by definition is the unique mild solution $\psi _U \in C([0,T_0], X_U)$ of (16) with $\psi _U|_{t = 0} = U_0$. $\square$

Remark A.2

In one of the previous applications, for estimating the residual terms, we used solutions $\psi _U \in C([0,T_0], X_{\psi })$ with $X_{\psi } \subset X_U$ another suitably chosen Banach space. The local existence and uniqueness proof in $X_{\psi }$ will work exactly the same since $X_U$ and $X_{\psi }$ have been chosen as Sobolev spaces $H^{s_U}$ and $H^{s_{\psi }}$ with $s_U < s_{\psi }$.