Asymptotic expansion of wave scattering in a periodic 2d-plane

Abstract

We give a counter part of Sommerfeld outging radiation condition for waves propagating in a 2d periodic medium under generical assumptions and provide a uniqueness theorem for outgoing solutions.

Figures (6)

• Figure 1: Bloch variety (left) and Fermi level $F_{k^2}^0$ (right) with $k=1.2$ for $\alpha=1+0.8\cos(x)\cos(y)$ and $\beta=1$.
• Figure 2: The set $F_{k^2}^+(x)$ with $x=(1,1)$ and $\alpha,k$ as in Figure \ref{['fig1']}.
• Figure 3: A closed component of $F_{k^2}^0$ with the points $\underline\ell_j$ and $\ell_n(\theta)\in F_{k^2}^+(x)$
• Figure 4: Unfolding the Fermi level for $k=1.2$ (left) leads (right) to ${\mathscr C}_0(k^2)=c_0$, ${\mathscr C}_1(k^2)=c_1$ and ${\mathscr C}_2(k^2)=\cup_{2\leq j\leq 9}c_j$ using B-translations (red curve)
• Figure 5: Real Fermi level $F_{k^2}^0$ for $k=1.06$ (left) and $k=1.081$ (right, folded)

Theorems & Definitions (32)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Definition 2.4
• Theorem 2.7
• Lemma 3.1
• proof
• Theorem 3.2: KP
• proof
• Remark 3.3