Stability Of Planar Waves Research Articles

Nonlinear Schrödinger equation for a two-dimensional plasma: Solitons, breathers, and plane wave stability

Nonlinear Schrödinger equation for a two-dimensional plasma: Solitons, breathers, and plane wave stability

Lyapunov Stability of Planar Waves to the Reaction-Diffusion Equation with a Non-Lipschitzian Reaction Term

Extending (Drábek and Takáč 2017), we investigate the Lyapunov stability of planar waves for the reaction-diffusion equation on ℝ n , n ≥ 2 , with a α -H o ¨ lder continuous ( 0 < α < 1 ), but not necessarily smooth reaction term. We first consider an initial value problem for the equation and then construct sub- and supersolutions to the problem by a subtle modification of the planar wave. Our main result states that a bounded classical solution to the problem stays near the planar wave for all time whenever an initial data is close enough to the planar wave.

On the characterization and stability of plane waves under hyperbolic two-temperature generalized thermoelasticity

The propagation and stability (Whitham’s criteria) of harmonic plane waves are described in the context of the hyperbolic two-temperature generalized thermoelasticity in which heat conduction in deformable bodies depends upon the difference between the double derivative of conductive and dynamic temperature. The exact dispersion relation solutions for the longitudinal plane wave are derived analytically. Several characterizations of the wave field, like phase velocity, specific loss, penetration depth, amplitude coefficient factor, and phase shift are examined for the low as well as high frequency asymptotic expansions. For the validity of analytical findings and to study the effect of varying hyperbolic two-temperature parameter on different characterizations, the numerical computation of a particular example is illustrated and displayed graphically. The results of some earlier works have been deduced and discussed from the present investigation as speciallimiting cases.

MULTIDIMENSIONAL STABILITY OF PLANAR WAVES FOR DELAYED REACTION-DIFFUSION EQUATION WITH NONLOCAL DIFFUSION

In this paper, we consider the multidimensional stability of planar waves for a class of nonlocal dispersal equation in $n$-dimensional space with time delay. We prove that all noncritical planar waves are exponentially stable in $L^{\infty}(\mathbb{R}^n )$ in the form of ${\rm{e}}^{-\mu_{\tau} t}$ for some constant $\mu_{\tau} =\mu(\tau)>0$($\tau >0$ is the time delay) by using comparison principle and Fourier transform. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the critical planar waves, we prove that they are asymptotically stable by establishing some estimates in weighted $L^1(\mathbb{R}^n)$ space and $H^k(\mathbb{R}^n) (k \geq [\frac{n+1}{2}])$ space.

Stability of planar waves in a Lotka–Volterra system

This paper is concerned with the large time behavior of disturbed planar fronts in the Lotka–Volterra system in Rn(n⩾2). We first show that the large time behavior of the disturbed fronts can be approximated by that of the mean curvature flow with a drift term for all large time up to t=+∞. And then we prove that the planar front is asymptotically stable in L∞(Rn) under ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases.

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.

Helicity and Wave Switching in a Nonlinear Model of DNA Dynamics

We have introduced helicity in the revised Hamiltonian in which the dipole-dipole interaction and dipoleinduced- dipole interaction have taken into account. The system has been shown to be govern by a set of coupled discrete sine- Gordon equations, which describe the bending and oscillations of the hydrogen bonds. The multiple scale expansion has been used and the stability of planar waves, solutions of the system, has been investigated. We have brought out the effect of discreteness and have shown that helicity brings about wave switching in the two-component DNA model under study. PACS number(s): 87.14.gk, 05.45.Yv, 87.15.H.

Multidimensional Stability of Subsonic Phase Transitions in a Non-Isothermal Van Der Waals Fluid

We show the multidimensional stability of subsonic phase transitions in a non-isothermal van der Waals fluid. Based on the existence result of planar waves in our previous work [1], a jump condition is posed on non-isothermal phase boundaries which makes the argument possible. Stability of planar waves both in one dimensional and multidi-mensional spaces are proved.

Stability of planar waves in reaction-diffusion system

This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on ℝ n , where n ⩾ 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t → ∞. The convergence is uniform in ℝ n . Moreover, the stability of planar waves in reaction-diffusion equations with nonlocal delays is also established by transforming the delayed equations into a non-delayed reaction-diffusion system.

Stability of planar waves in mono-stable reaction-diffusion equations

Stability of planar waves in mono-stable reaction-diffusion equations

Stability of plane waves on deep water with dissipation

The Benjamin–Feir modulational instability effects the evolution of perturbed plane-wave solutions of the cubic nonlinear Schrödinger equation (NLS), the modified NLS, and the band-modified NLS. Recent work demonstrates that the Benjamin–Feir instability in NLS is “stabilized” when a linear term representing dissipation is added. In this paper, we add a linear term representing dissipation to the modified NLS and band-modified NLS equations and establish that the plane-wave solutions of these equations are linearly stable. Although the plane-wave solutions are stable, some perturbations grow for a finite period of time. We analytically bound this growth and present approximate time-dependent regions of wave-number space that correspond to perturbations that have increasing amplitudes.

On the propagation of plane waves in type–III thermoelastic media

The propagation of plane waves in infinite, three–dimensional, type–III thermoelastic media is investigated. Exact dispersion relation solutions are determined and several characterizations of the wavefield are examined. Low– and high–frequency asymptotic expressions are given, small–coupling limit results are derived, and special/limiting cases, including those corresponding to thermoelastic media of types II and I, are noted. Computational tools are used to illustrate the analytical findings and to study the effects of varying the physical parameters. Of the findings presented, the following are most significant: (i) the determination of critical values of the physical parameters and their impact on the wavefields; (ii) ascertaining that type–III media behave, essentially, like type–II (respectively, type–I) media with respect to low– (respectively, high–) frequency plane waves; (iii) establishing the Whitham stability of plane waves in type–III media; and (iv) the determination of the dispersion characteristics of type–III media.

Existence and linear stability of planar waves in the Oregonator model

In this paper, various waves in the Oregonator model are investigated in detail. The existence of planar waves (in particular, spiral waves) is rigorously proven and their linear stability is analysed. In addition, some non-planar waves in the model are also presented.