Spectral analysis of periodic $b$-KP equation under transverse perturbation

Abstract

The $b$-family-Kadomtsev-Petviashvili equation ($b$-KP) is a two dimensional generalization of the $b$-family equation. In this paper, we study the spectral stability of the one-dimensional small-amplitude periodic traveling waves with respect to two-dimensional perturbations which are either co-periodic in the direction of propagation, or nonperiodic (localized or bounded). We perform a detailed spectral analysis of the linearized problem associated to the above mentioned perturbations, and derive various stability and instability criteria which depends in a delicate way on the parameter value of $b$, the transverse dispersion parameter $σ$, and the wave number $k$ of the longitudinal waves.

Figures (3)

• Figure 1: A schematic plot of the stability region for the periodic perturbation of the $b$-KP flow. The two shaded regions correspond to $\{(b, k^2) : b < -1, k^2 > \frac{b+1}{2b-7}\}$ and $\{(b, k^2) : b > \frac{7}{2}, k^2 > \frac{b+1}{2b-7}\}$.
• Figure 2: [$b$-KP-II equation] (a) Graph of the map $n \mapsto \kappa \frac{k^2(n^2-1)}{1+k^2}- \frac{ \ell^2}{n ^ { 2 }}$ for $\kappa =2, k=1$ and $\ell=0.8$. The eigenvalues of $\mathcal{K}_0(\ell)$ are found by taking $n=q, q \in \mathbb{Z}^*$. (b) Graph of the dispersion relation $n \mapsto \nu(n) = n\left(\frac{\kappa}{1+k^2}-\frac{\kappa}{1+k^2 n^2}- \frac{ \ell^2}{n ^ { 2 } ( 1 + k ^ { 2 } n ^ { 2 })}\right)$ for $\kappa =2, k=1$ and $\ell=0.8$. The imaginary parts of the eigenvalues of $\mathcal{A}_0(\ell)$ are found by taking $k=n, n \in \mathbb{Z}^*$. Notice that the zeros of the two maps are the same.
• Figure 3: [$b$-KP-I equation] (a) Graph of the map $n \mapsto \frac{ \ell^2}{n ^ { 2 }} - \frac{ \kappa k^2(1 - n^2)}{1+k^2}$ for $\kappa =2, k=1$ and $\ell=0.8$ and $\ell=0.3$ (from left to right). The eigenvalues of $\mathcal{K}_0(\ell)$ are found by taking $n=p+\xi, p \in \mathbb{Z}$. (b) Graph of the dispersion relation $n \mapsto n\left(\frac{\kappa}{1+k^2}-\frac{\kappa}{1+k^2 n^2}+ \frac{ \ell^2}{n ^ { 2 } ( 1 + k ^ { 2 } n ^ { 2 })}\right)$ for the same values of $\ell, \kappa$ and $k$. The imaginary parts of the eigenvalues of $\mathcal{A}_0(\ell)$ are found by taking $n=p+\xi, p \in \mathbb{Z}$. Notice that the zeros of the two maps are the same.

Theorems & Definitions (31)

• Theorem 1.1: Informal statement of transverse stability for periodic perturbation
• Lemma 2.1
• Definition 3.1: Transverse spectral stability
• Proposition 4.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Theorem 4.1
• proof
• Lemma 4.4