Instability Of Plane Waves Research Articles

Nonlinear analysis of the Eckhaus instability: modulated amplitude waves and phase chaos with nonzero average phase gradient

We analyze the Eckhaus instability of plane waves in the one-dimensional complex Ginzburg–Landau equation (CGLE) and describe the nonlinear effects arising in the Eckhaus unstable regime. Modulated amplitude waves (MAWs) are quasi-periodic solutions of the CGLE that emerge near the Eckhaus instability of plane waves and cease to exist due to saddle-node (SN) bifurcations. These MAWs can be characterized by their average phase gradient ν and by the spatial period P of the periodic amplitude modulation. A numerical bifurcation analysis reveals the existence and stability properties of MAWs with arbitrary ν and P. MAWs are found to be stable for large enough ν and intermediate values of P. For different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude. Defects form from perturbed plane waves for parameter values above the SN of the corresponding MAWs. The break-down of phase chaos with average phase gradient ν≠0 (“wound-up phase chaos”) is thus related to these SNs. A lower bound for the break-down of wound-up phase chaos is given by the necessary presence of SNs and an upper bound by the absence of the splitting instability of MAWs.

Modulational instability in periodic quadratic nonlinear materials.

We investigate the modulational instability of plane waves in quadratic nonlinear materials with linear and nonlinear quasi-phase-matching gratings. Exact Floquet calculations, confirmed by numerical simulations, show that the periodicity can drastically alter the gain spectrum but never completely removes the instability. The low-frequency part of the gain spectrum is accurately predicted by an averaged theory and disappears for certain gratings. The high-frequency part is related to the inherent gain of the homogeneous non-phase-matched material and is a consistent spectral feature.

Nonlinear wavefields in optical fibers with finite time response and amplification effects

A higher order nonlinear integro-differential equation of the Schrödinger type modeling the evolution of finite amplitude wavepackets in the presence of optical amplification and finite material- and polarization time responses is analyzed with respect to the dynamical evolution of localized solutions. Nonexistence of localized travelling and stationary wave solutions is pointed out for certain parameter regimes. Finally the linear stage of the modulational instability of plane waves is studied and it is shown that the finite time responses produce finite bandwidth instabilities.

Nonfocusing Instabilities in Coupled, Integrable Nonlinear Schrödinger pdes

instabilities that exist independently of the well-known modulational instability of the focusing NLS equation. The focusing versus defocusing behavior of scalar NLS fields is a well-known model for the corresponding behavior of pulse transmission in optical fibers in the anomalous (focusing) versus normal (defocusing) dispersion regime [19], [20]. For fibers with birefringence (induced by an asymmetry in the cross section), the scalar NLS fields for two orthogonal polarization modes couple nonlinearly [26]. Experiments by Rothenberg [32], [33] have demonstrated a new type of modulational instability in a birefringent normal dispersion fiber, and he proposes this cross-phase coupling instability as a mechanism for the generation of ultrafast, terahertz optical oscillations. In this paper the nonfocusing plane wave instability in an integrable coupled nonlinear Schrodinger (CNLS) partial differential equation system is contrasted with the focusing instability from two perspectives: traditional linearized stability analysis and integrable methods based on periodic inverse spectral theory. The latter approach is a crucial first step toward a nonlinear , nonlocal understanding of this new optical instability analogous to that developed for the focusing modulational instability of the sine-Gordon equations by Ercolani, Forest, and McLaughlin [13], [14], [15], [17] and the scalar NLS equation by Tracy, Chen, and Lee [36], [37], Forest and Lee [18], and McLaughlin, Li, and Overman [23], [24].

Stabilization of dark and vortex parametric spatial solitons

We demonstrate that a weak defocusing Kerr effect in an optical medium with predominantly quadratic [or chi((2))] nonlinear response can eliminate the parametric modulational instability of plane waves, leading to the existence of stable two-wave dark and vortex spatial solitons.

On the asymptotic evolution of a localized perturbation of the one-dimensional Landau-Lifshits equation with uniaxial anisotropy

The witham method is applied to analyze the modulation instability of plane waves in a one-dimensional ferromagnet described by the Landau-Lifshits equation with uniaxial anisotropy. It is shown that the instability results in the formation of a domain structure in the system.

Modulational instability of plane waves in a two-dimensional jet and wake

The modulational instability problem is reformulated by the amplitude expansion method. A generalized version of the Stewartson-Stuart equation is derived, which is applicable to any supercritical state as long as the amplitude of the disturbance is small. The modulational instability of plane waves in a two-dimensional jet and wake is investigated on the basis of this equation, and it is found that the plane waves are stable to side-band disturbances in supercritical states except in the close neighbourhood of the critical state.

Nonlinear modulation of Love waves in a compressible hyperelastic layered half space

The derivative expansion method is used to examine the nonlinear modulation of Love waves in a half space covered by a layer of uniform thickness having different mechanical properties. Both half space and layer are assumed to be homogeneous, isotropic and compressible hyperelastic materials. It is shown that the nonlinear modulation of Love waves is governed by the nonlinear Schrödinger (NLS) equation. By taking into account the known properties of the solutions of NLS equation, the possibilities of soliton solutions and the modulational instability of plane waves for various nonlinear material properties of the layer and half space are investigated. It is remarked that the surface envelope solitary waves may exist depending on the nonlinear constitution of the layered half space.

Long Wave/Short Wave Resonance in Equatorial Waves

Abstract It is shown that resonant coupling between ultra long equatorial Rossby waves and packets of either short Rossby or short westward-traveling gravity waves is possible. Simple analytic formulas give the discrete value of the packet wave number k, for which the group velocity of the packet of meridional mode number n matches the group velocity of a nondispersive long Rossby wave of odd mode number m. The equations that describe the coupling are derived via the method of multiple scale and tables of the interaction coefficients are numerically calculated. For realistic parameter values, it appears this coupling could be important in the tropical ocean. The principal physics of the coupled equations is threefold: 1) modulational or “side band” instability of plane waves, 2) instability of a short wave packet with respect to growing long waves if no long waves are initially present, and 3) solitary waves which consist of an envelope soliton of short waves of Nonlinear Schrodinger type traveling in con...