Localized structures on librational and rotational travelling waves in the sine-Gordon equation

We derive exact solutions to the sine-Gordon equation describing localized structures on the background of librational and rotational travelling waves. In the case of librational waves, the exact solution represents a localized spike in space-time coordinates (a rogue wave) that decays to the periodic background algebraically fast. In the case of rotational waves, the exact solution represents a kink propagating on the periodic background and decaying algebraically in the transverse direction to its propagation. These solutions model the universal patterns in the dynamics of fluxon condensates in the semi-classical limit. The different dynamics are related to modulational instability of the librational waves and modulational stability of the rotational waves.

Modulational Stability of Envelope Soliton in a Quantum Electron-Ion Plasma in Three Dimension - A generalised Nonlinear Schr ̈odinger Equ ation in 3D

Modulational stability of envelope soliton is studied in a quantum dusty plasma in three dimension.<br>The Krylov-Bogoliubov-Mitropolsky method is applied to the three dimension plasma governing<br>equations. A generalised form of Nonlinear Schr¨odinger equation is obtained whose dispersive term<br>has a tensorial character. Stability condition is deduced abintio and the stability zones are plotted<br>as a function of plasma parameters

Modulational Stability of Envelope Soliton in a Quantum Electron-Ion Plasma in Three Dimension - A generalised Nonlinear Schr ̈odinger Equ ation in 3D

Modulational stability of envelope soliton is studied in a quantum dusty plasma in three dimension.<br>The Krylov-Bogoliubov-Mitropolsky method is applied to the three dimension plasma governing<br>equations. A generalised form of Nonlinear Schr¨odinger equation is obtained whose dispersive term<br>has a tensorial character. Stability condition is deduced abintio and the stability zones are plotted<br>as a function of plasma parameters

Generalized modulation theory for strongly nonlinear gravity waves in a compressible atmosphere

This study investigates strongly nonlinear gravity waves in the compressible atmosphere from the Earth’s surface to the deep atmosphere. These waves are effectively described by Grimshaw’s dissipative modulation equations which provide the basis for finding stationary solutions such as mountain lee waves and testing their stability in an analytic fashion. Assuming energetically consistent boundary and far-field conditions, that is no energy flux through the surface, free-slip boundary, and finite total energy, general wave solutions are derived and illustrated in terms of realistic background fields. These assumptions also imply that the wave-Reynolds number must become less than unity above a certain height. The modulational stability of admissible, both non-hydrostatic and hydrostatic, waves is examined. It turns out that, when accounting for the self-induced mean flow, the wave-Froude number has a resonance condition. If it becomes 1/ 1 / 2 1/\sqrt 2 , then the wave destabilizes due to perturbations from the essential spectrum of the linearized modulation equations. However, if the horizontal wavelength is large enough, waves overturn before they can reach the modulational stability condition.

Experimental Observation of Modulational Instability in Crossing Surface Gravity Wavetrains

The coupled nonlinear Schrödinger equation (CNLSE) is a wave envelope evolution equation applicable to two crossing, narrow-banded wave systems. Modulational instability (MI), a feature of the nonlinear Schrödinger wave equation, is characterized (to first order) by an exponential growth of sideband components and the formation of distinct wave pulses, often containing extreme waves. Linear stability analysis of the CNLSE shows the effect of crossing angle, θ , on MI, and reveals instabilities between 0 ∘ < θ < 35 ∘ , 46 ∘ < θ < 143 ∘ , and 145 ∘ < θ < 180 ∘ . Herein, the modulational stability of crossing wavetrains seeded with symmetrical sidebands is determined experimentally from tests in a circular wave basin. Experiments were carried out at 12 crossing angles between 0 ∘ ≤ θ ≤ 88 ∘ , and strong unidirectional sideband growth was observed. This growth reduced significantly at angles beyond θ ≈ 20 ∘ , reaching complete stability at θ = 30–40 ∘ . We find satisfactory agreement between numerical predictions (using a time-marching CNLSE solver) and experimental measurements for all crossing angles.

L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves

Assuming spectral stability conditions of periodic reaction-diffusion waves u¯(x), we consider L1(R)-nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initial modulations. Lp(R)-nonlinear stability of such waves has been studied in Johnson et al. (2013) for p≥2 by using Hausdorff-Young inequality. In this note, by using the pointwise estimates obtained in Jung, (2012) and Jung and Zumbrun (2016), we extend Lp(R)-nonlinear stability (p≥2) in Johnson et al. (2013) to L1(R)-nonlinear stability. More precisely, we obtain L1(R)-estimates of modulated perturbations u~(x-ψ(x,t),t)-u¯(x) of u¯ with a phase function ψ(x,t) under small initial perturbations consisting of localized initial perturbations u~(x-h0(x),0)-u¯(x) and nonlocalized initial modulations h0(x)=ψ(x,0).