Normal form for transverse instability of the line soliton with a nearly critical speed of propagation

Abstract

In the context of the line solitons in the Zakharov-Kuznetsov (ZK) equation, there exists a critical speed of propagation such that small transversely periodic perturbations are unstable if the soliton speed is larger than the critical speed and orbitally stable if the soliton speed is smaller than the critical speed. The normal form for transverse instability of the line soliton with a nearly critical speed of propagation is derived by means of symplectic projections and near-identity transformations. Justification of this normal form is provided with the energy method. The normal form predicts a transformation of the unstable line solitons with larger-than-critical speeds to the orbitally stable transversely modulated solitary waves. Transverse instability of line solitons for the ZK equation (1.1)We are concerned here with the transverse instability of the line solitons under periodic transverse perturbations in the ZK equation (1.1). First, we review relevant properties of the line solitons of the KdV equation. Next, we obtain spectral transverse stability results for the line solitons with a nearly critical speed of propagation. Further, we study bifurcations of the transversely modulated solitary waves. Finally, we present the main result on the normal form for transverse instability of the line solitons with a nearly critical speed of propagation.

Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

“…For concreteness, we choose 𝑟 1 = 0 and 𝑟 2 = 𝑟 3 = 6𝑐 (with the specific parameterization chosen so as to simplify the calculations that follow, similarly to Ref. 42), and 𝑞 = 0. We then have 2𝐾 𝑚 𝑍 = √ 𝑐(𝑥 + 𝑞𝑦 − 4𝑐𝑡) = √ 𝑐𝜉, where 𝜉 = 𝑥 − 4𝑐𝑡, and, as per (4),…”

“…(However, these studies are concerned with solutions that vanish as $$|\mathbf {x}|\rightarrow \infty$$, and are therefore not directly applicable to the present work, which deals with solutions that are periodic with respect to each spatial dimension.) The stability of the solitary wave solutions of () was studied with various methods, 9,15,17,21,30,31,42,51 and that of its periodic solutions was studied in Ref. 26.…”

Numerical study of the transverse stability of line solitons of the Zakharov-Kuznetsov equations

“…This result was completed in [45] where Yamazaki was able to construct center stable manifolds around unstable line solitary waves to the Zakharov-Kuznetsov equation on R×T L . Pelinovsky [35] proved the asymptotic stability of the transversely modulated solitary waves of the Zakharov-Kuznetsov equation on R × T L in exponentially weighted spaces.…”

Center stable manifolds around line solitary waves of the Zakharov--Kuznetsov equation

“…By the Lyapunov-Schmidt reduction method, the existence and the stability of the transversely modulated solitary waves was showed in [44]. Using the normal form which describes the motion of the amplitude of the transversely modulated solitary waves, Pelinovsky proved the asymptotic behavior of solutions near by the transversely modulated solitary waves for the Zakharov-Kuznetsov equation in [36]. Moreover, Pelinovsky showed the asymptotic stability of the transversely modulated solitary waves in the sense by Pego and Winstein [35].…”