Stability Of Plane Wave Solutions Research Articles

Plane wave stability analysis of Hartree and quantum dissipative systems

We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment. The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential; such a coupling lead to challenging issues compared to the usual nonlinear Schrödinger equations. The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals. We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime. In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter.

Long time stability of plane wave solutions to Schrödinger equation on Torus

ABSTRACT We prove the long time orbital stability of the plane wave solutions to the nonlinear Schrödinger equation (NLS) in the defocusing () or focusing () case, More precisely, in a Gevrey space for some positive constant σ, we show that solution with the initial datum in the -neighborhood of the plane wave solution still stays in the -neighborhood ( C>4 ) of the plane wave solution for a subexponential long time where and .

Quantum breathers and intrinsic localized excitation associated with the modulational instability in 1D Bose–Hubbard chain

In this work, the energy spectrum and intrinsic localized modes associated with the modulational instability of boson chains is described by a Bose–Hubbard model. By using of number states method combined with the numerical diagonalization, we show that the energy spectrum of the system exhibits bound states signature. With the help of multiple scales method in addition to a quasidiscreteness approximation, we obtain bright and dark type intrinsic localized modes at the center and at the edges of the Brillouin zone respectively and their appearance conditions. We found that the inclusion of interaction of two bosons at the same site can influence the appearance conditions of bright and dark soliton solutions. Considering the fact that nowadays, the forthright way to predict the formation of intrinsic localized modes in nonlinear systems is the modulational instability. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in the Bose–Hubbard chain. It is found that the shape of the region of modulational instability and the instability growth rates of the plane wave can be more affected dramatically when the interaction of two bosons at the same site is involved in the system. Furthermore, we calculate the instability growth rates of plane wave at the center and at the edge of the Brillouin zone for different values of the interaction of two bosons to analyze their formation conditions which are in full agreement with the method of multiple scale in addition to a quasi-discreteness approximation analysis.

Numerical study of blow-up to the purely elliptic generalized Davey–Stewartson system

Blow-up solutions for the purely elliptic generalized Davey–Stewartson system are studied by using a relaxation numerical method. The numerical method is based on an implicit finite-difference scheme with a second-order accuracy in both time and space. The stability of the numerical method is analyzed by investigating the linear stability of plane wave solutions. To evaluate the ability of the relaxation method to detect blow-up, numerical simulations are conducted for several test problems. A particular attention is paid to the gap interval neither a global existence nor a blow-up result is established. The monotonicity properties of blow-up time on the coupling parameter are also investigated numerically.

Sobolev Stability of Plane Wave Solutions to the Nonlinear Schrödinger Equation

This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.

Stability of Plane Wave Solutions in Complex Ginzburg--Landau Equation with Delayed Feedback

We perform bifurcation analysis of plane wave solutions in a one-dimensional complex cubic-quintic Ginzburg--Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delays, using an asymptotic approach we classify plane wave solutions into strongly unstable, weakly unstable, and stable. The results of analytical bifurcation analysis are in agreement with those obtained by direct numerical integration of the model equation.

Sobolev Stability of Plane Wave Solutions to the Cubic Nonlinear Schrödinger Equation on a Torus

It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.

Wave patterns in α-helix proteins with interspine coupling

Modulational instability is a direct way by which localized structures emerge in nonlinear systems. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in α-helix proteins with three spines. Through numerical simulations, trains of pulses are found and confirm our analytical predictions. The presence of higher-order interactions between adjacent spines tends to suppress the formed localized structures for erratic ones to emerge. These erratic structures are highly localized and rather reinforce the idea that the energy to be used in metabolic processes is rather confined to specific regions for its efficiency.

Stability of plane waves in two-phase porous media flow

We examine the Saffman–Taylor instability for oil displaced by water in a porous medium. The model equations are based on Darcy's law for two-phase flow, with dependent variables pressure and saturation. Stability of plane wave solutions is governed by the hyperbolic/elliptic system obtained by ignoring capillary pressure, which adds diffusion to the hyperbolic equation. Interestingly, the growth rate of perturbations of unstable waves is linear in the wave number to leading order, whereas a naive analysis would indicate quadratic dependence. This gives a sharp boundary in the state space of upstream and downstream saturations separating stable from unstable waves. The role of this boundary, derived from the linearized hyperbolic/elliptic system, is verified by numerical simulations of the full nonlinear parabolic/elliptic equations.

Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation

In this paper, a three dimensional Ginburg-Landau type equation isconsidered. Firstly, two families of new traveling wave solutions interm of explicit functions are presented by using the homogeneousbalance method, in which one consists of variable-amplitude solutions and the other constant-amplitude solutions (namely, plane wave solutions). Moreover,the stability of plane wave solutions is analyzed by using theregular phase plane techniques.

Modulational instability of two pairs of counter-propagating waves and energy exchange in a two-component system

The modulational instability of two pairs of plane counter-propagating waves in a two-component system is considered within the framework of two coupled Sine-Gordon equations. The emphasis is on the generic case when the system is not integrable, and the group velocities of each pair of waves are arbitrary and usually different from each other. We consider the dynamics of weakly nonlinear wave packets, and using an asymptotic multiple-scales expansion we obtain several sets of evolution equations describing the two components of the system. Depending on the wave packet length scale vis-a-vis the wave amplitude scale, these evolution equations are either four non-dispersive and nonlinearly coupled envelope equations, or four non-locally coupled nonlinear Schrödinger equations. We also consider a set of fully coupled nonlinear Schrödinger equations, even though in general this system contains small dispersive terms which are strictly beyond the leading order of the asymptotic multiple-scales expansion procedure. Although each set of amplitude equations can be used to describe a range of dynamical phenomena, we focus here on stability of plane-wave solutions, and show that they may be modulationally unstable. We study these instabilities in the context of solutions exhibiting an energy exchange between the two physical components of the system. We also perform numerical simulations of the original unapproximated coupled Sine-Gordon equations to assess the accuracy of the instability predictions derived from the various asymptotic models, and consider the range of validity of each asymptotic model. We show that the instabilities can lead to the formation of localized structures, and to a modification of the linear energy exchange, which then continues for some time into the nonlinear regime as an energy exchange between these localized structures. When the system is close to being integrable, the time evolution is distinguished by a remarkable almost periodic sequence of energy exchange scenarios, with spatial patterns alternating between approximately uniform wavetrains and localized structures, which invites further theoretical study of this special case.

Analysis of a corner layer problem in anisotropic interfaces

We investigate a model of anisotropic diffuse interfaces in ordered FCC crystals introduced recently by Braun et al and Tanoglu et al [3, 18, 19], focusing on parametric conditions which give extreme anisotropy. For a reduced model, we prove existence and stability of plane wave solutions connecting the disordered FCC state with the ordered $Cu_3Au$ state described by solutions to a system of three equations. These plane wave solutions correspond to planar interfaces. Different orientations of the planes in relation to the crystal axes give rise to different surface energies. Guided by previous work based on numerics and formal asymptotics, we reduce this problem in the six dimensional phase space of the system to a two dimensional phase space by taking advantage of the symmetries of the crystal and restricting attention to solutions with corresponding symmetries. For this reduced problem a standing wave solution is constructed that corresponds to a transition that, in the extreme anisotropy limit, is continuous but not differentiable. We also investigate the stability of the constructed solution by studying the eigenvalue problem for the linearized equation. We find that although the transition is stable, there is a growing number $0(\frac{1}{\epsilon})$, of critical eigenvalues, where $\frac{1}{\epsilon}$ » $1$ is a measure of the anisotropy. Specifically we obtain a discrete spectrum with eigenvalues $\lambda_n = \e^{2/3}\mu_n$ with $\mu_n$ ~ $Cn^{2/3}$, as $n \to + \infty$. The scaling characteristics of the critical spectrum suggest a previously unknown microstructural instability.

Stability of plane wave solutions for the Ginzburg-Landau-BBM equations (II)

In part (I) of this paper, the stability of plane waves for the Ginzburg-Landau-BBM equations has been proved. Here, the perturbation decays are obtained by using the Fourier transformation and semigroup estimates.

Wave solutions for the Ginzburg-Landau-BBM equations (I)

In this paper, the stability of plane wave solutions for the Ginzburg-Landau-BBM equations are considered. When the coefficients are suitably constrained, these waves are stable.

On the nonlinear stability of plane waves for the ginzburg-landau equation

I consider the nonlinear stability of plane wave solutions to a Ginzburg-Landau equation with additional fifth-order terms and cubic terms containing spatial derivatives. I show that, under the constraint that the diffusion coefficient be real, these waves are stable. Furthermore, it is shown that the radial component of the perturbation decays at a faster rate than the phase component of the perturbation as t → ∞. The result is also applicable to the classical Ginzburg-Landau equation. © 1994 John Wiley & Sons, Inc.

Stability of plane wave solutions of complex Ginzburg-Landau equations

We consider the stability of plane wave solutions of both single and coupled complex Ginzburg-Landau equations and determine stability domains in the space of coefficients of the equations.

Stability of plane wave solutions of nonlinear systems

The linear stability of plane wave solutions of various systems of nonlinear partial differential equations is treated. They include an equation for the envelope of a surface wave train on deep water, Zakharov's system for Langmuir waves in plasmas, coupled Schrödinger and Klein-Gordon equations for nucleon and meson fields, and a pair coupled Schrödinger equations. The method of analysis, which was presented previously by the authors, is explained and its scope is broadened.

Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation

The stability of plane, periodic solutions of the two-dimensional nonlinear Schrödinger equation to infinitesimal, two-dimensional perturbation has been calculated and verified numerically. For standing wave disturbances, instability is found for both odd and even modes; as the period of the unperturbed solution increases, the instability associated with the odd modes remains but that associated with the even mode disappears, which is consistent with the results of Zakharov and Rubenchik [8], Saffman and Yuen [4] and Ablowitz and Segur [1] on the stability of solitons. In addition, we have identified travelling wave instabilities for the even mode perturbations which are absent in the long-wave limit. Extrapolation to the case of an unperturbed solution with infinite period suggests that these instabilities]may also be present for the soliton. In other words, the soliton is unstable to odd, standing-wave perturbations, and very likely also to even, travelling-wave perturbations.