Ginzburg-Landau system of complex modulation equations for distributed nonlinear dissipative transmission lines

The envelope modulation of a monoinductance transmission line is reduced to generalized coupled Ginzburg-Landau equations, from which a single cubic-quintic Ginzburg-Landau equation containing derivatives with respect to the space variable in the cubic terms is deduced. We investigate the modulational instability of the space wave solutions of both the system and the single equation. For the generalized coupled Ginzburg-Landau system, we consider only the zero wave numbers of the perturbations whose modulational instability conditions depend only on the coefficients of the system and the wave numbers of the carriers. In this case, a modulational instability criterion is established, which depends on both the perturbation wave numbers and the carrier. We also study the coherent structures of the generalized coupled Ginzburg-Landau system and present some numerical results. Published in Neliniini Kolyvannya, Vol. 9, No. 4, pp. 451–489, October–December, 2006.     

Exponential attractors for a generalized ginzburg-landau equation

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Asymptotic behavior of 2D generalized stochastic Ginzburg-Landau equation with additive noise

The 2D generalized stochastic Ginzburg-Landau equation with additive noise is considered. The compactness of the random dynamical system is established with a priori estimate method, showing that the random dynamical system possesses a random attractor in H^1 0.     

Bifurcations of traveling wave solutions and exact solutions to generalized Zakharov equation and Ginzburg-Landau equation

This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.     

A non isothermal Ginzburg-Landau model for phase transitions in shape memory alloys

A thermodynamical model for martensitic phase transitions in shape memory alloys is formulated in this paper in the framework of the Ginzburg-Landau approach to phase transitions. A single order parameter is chosen to represent the austenite parent phase and two mirror related martensite variants. A free energy previously proposed in the literature (Levitas et al. in Phys. Rev. B 66:134206, 2002; Phys. Rev. B 66:134207, 2002; Phys. Rev. B 68:134201, 2003) is employed, in its simplest form, as the main constitutive content of the model. In this paper we treat time-dependent Ginzburg-Landau equation as a balance law on the structure order and we couple it to a energy balance equation, thus allowing to account of heat transfer processes. We obtain a coupled thermo-mechanical problem whose consistency with the Second Law is verified.     

On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation

We present a formal asymptotic analysis which suggests a model for three-phase boundary motion as a singular limit of a vector-valued Ginzburg-Landau equation. We prove short-time existence and uniqueness of solutions for this model, that is, for a system of three-phase boundaries undergoing curvature motion with assigned angle conditions at the meeting point. Such models pertain to grain-boundary motion in alloys. The method we use, based on linearization about the initial conditions, applies to a wide class of parabolic systems. We illustrate this further by its application to an eutectic solidification problem.     

Localization and approximation of attractors for the Ginzburg-Landau equation

This paper studies the long-term behavior of solutions to the Ginzburg-Landau partial differential equation. For each positive integerm we explicitly produce a sequence of approximate inertial manifolds _m,j ,j = 1, 2,..., of dimensionm and associate with each manifold a thin neighborhood into which the orbits enter with an exponential speed and in a finite time. Of course this neighborhood contains the universal attractor which embodies the large time dynamics of the equations. The thickness of these neighborhoods decreases with increasingm for a fixed orderj; however, for a fixedm no conclusion can be made about the thickness of the neighborhoods associated to two differentj's. The neighborhoods associated to the manifolds localize the universal attractor and provide computabie large time approximations to solutions of the Ginzburg-Landau equation.     

On the equations of motion for accelerated frames

In this communication we present the equations of Euler generalized for the motion of a body in an accelerated reference frame using the generalized work-energy principle. The equivalence among the generalized Euler equation, the generalized Lagrange equation, and the generalized Kane equation are shown when applied to the motion of a body of a holonomic system that depend onn generalized coordinates. Therefore when the generalized coordinates can be reduced to two sets of independent coordinates, the generalized Euler equation can be split into two uncoupled equations that are not independent of each other.Universidade da Beira Interior, Covilhã, Portugal. Published in Prikladnaya Mekhanika, Vol. 31, No. 9, pp. 79–89, September, 1995.     

Viscous falling film instability around a vertical moving cylinder

The present work discusses both the linear and nonlinear stability conditions of a viscous falling film down the outer surface of a solid vertical cylinder which moves in the direction of its axis with a constant velocity.After studying the linear conditions,a generalized nonlinear kinematic model is then derived to present the physical system.Applying the boundary conditions,analytical solutions are obtained using the long-wave perturbation method.In the first step,the normal mode method is used to characterize the linear behaviors.In the second step,the nonlinear film flow model is solved by using the method of multiple scales,to obtain Ginzburg-Landau equation.The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem,and the results are displayed in many plots showing the stability criteria in various parameter planes.     

Nonlinear dynamics between two differentially heated vertical plates in the presence of stratification

We consider the numerical simulation of the flow between infinite, differentially heated vertical plates with positive stratification. We use a two-dimensional Boussinesq approximation, with periodic boundary conditions in the vertical direction. The relative stratification parameter ${\gamma=(\frac{1}{4}Ra S)^{1/4}}$ , where Ra is the Rayleigh number and S the adimensional stratification, is kept constant and equal to 8. The Prandtl number is 0.71. We derive a complex Ginzburg-Landau equation from the equations of motion. Coefficients are computed analytically, but we find that the domain of validity of these coefficients is small and rely on the numerical simulation to adjust the coefficients over a wider range of Rayleigh numbers. We show that the Ginzburg-Landau equation is able to accurately predict the characteristics of the periodic solution at moderate Rayleigh numbers. Above the primary bifurcation at Ra = 1.63 × 10⁵, the Ginzburg-Landau model is found to be Benjamin-Feir unstable and to be characterized by modulated traveling waves and phase-defect chaos, which is supported by evidence from the DNS. As the Rayleigh number is increased beyond Ra = 2.7 × 10⁵, nonlinearities become strong and the flow is characterized by cnoidal waves.     

Multi-symplectic method for the generalized (2+1)-dimensional KdV-mKdV equation

In the present paper,a general solution involving three arbitrary functions for the generalized(2+1)dimensional KdV-mKdV equation,which is derived fromthe generalized(1+1)-dimensional KdV-mKdV equation,is first introduced by means of the Wiess,Tabor,Carnevale(WTC) truncation method.And then multisymplectic formulations with several conservation lawstaken into account are presented for the generalized(2+1)dimensional KdV-mKdV equation based on the multisymplectic theory of Bridges.Subsequently,in order tosimulate the periodic wave solutions in terms of rationalfunctions of the Jacobi elliptic functions derived from thegeneral solution,a semi-implicit multi-symplectic schemeis constructed that is equivalent to the Preissmann scheme.From the results of the numerical experiments,we can conclude that the multi-symplectic schemes can accurately simulate the periodic wave solutions of the generalized(2+1)dimensional KdV-mKdV equation while preserve approximately the conservation laws.     

Generalized Euler–Lagrange equation for nonsmooth calculus of variations

In this paper, we first introduce a novel generalized derivative and obtain the generalized first-order Taylor expansion of the nonsmooth functions. Then we derive the generalized Euler–Lagrange equation for the nonsmooth calculus of variations and solve this equation by using Chebyshev pseudospectral method, approximately. Finally, the optimal solutions of some problems in the nonsmooth calculus of variations are approximated.     

On a generalized Broer-Kaup-Kupershmidt system for the long waves in shallow water

Describing the long waves in shallow water, a generalized Broer-Kaup-Kupershmidt system is investigated in this paper. With respect to the horizontal velocity of the water wave and the height of the water surface, we use symbolic computation to build up (A) a scaling transformation, (B) a set of the hetero-Bäcklund transformations, from that generalized system to a known linear partial differential equation, as well as (C) two sets of the similarity reductions, each of which from that generalized system to a known ordinary differential equation. Our results depend on all the shallow-water coefficients for that generalized system.

     

Characteristic fast marching method on triangular grids for the generalized eikonal equation in moving media

The governing equation of the first arrival time of a monotonically propagating front (wavefront or shock front) in an inhomogeneous moving medium is an anisotropic eikonal equation, called the generalized eikonal equation in moving media. When the ambient medium is at rest, this equation reduces to the well-known (isotropic) eikonal equation in which the characteristic direction coincides with the normal direction of the propagating front. The fast marching method is an efficient method for computing the first arrival time of a propagating front as the approximate solution of the isotropic eikonal equation. The fast marching method inherits the property that the characteristic direction coincides with the normal direction at every point on the propagating wavefront and therefore is well suited for the eikonal equation. Due to anisotropic nature, this property does not hold in the case of front propagation in a moving medium. Thus, the fast marching method cannot be directly used for the generalized eikonal equation and needs some suitable modifications. We recently proposed a characteristic fast marching method on a rectangular grid for the generalized eikonal equation (Dahiya et al., 2013) and shown numerically that this method is stable, accurate, and easy to update to second order approximations. In the present work, we generalize the method on structured triangular grids. We compare the numerical solution obtained using our method with the ray theory solution to show that the method captures accurately the viscosity solution of the generalized eikonal equation. We use the method to study some interesting geometrical features of an initially planar wavefront propagating in a medium with Taylor–Green type vortices.     

Stability of shock waves in the modified quintic complex Ginzburg-Landau equation

We consider a shock-type wave solution of the modified quintic complex Ginzburg-Landau equation and make a numerical study of its spatiotemporal stability. Discussions related to the behavior of this front wave are introduced and it is shown how the velocities of the wave can be used to collect information concerning the pattern formation in the system. Published in Neliniini Kolyvannya, Vol. 10, No. 2, pp. 270–276, April–June, 2007.     

Tempered Mittag-Leffler noise-induced resonant behaviors in the generalized Langevin system with random mass

Nonlinear Dynamics - In this study, we propose the fluctuating-mass generalized Langevin equation (GLE) with a tempered Mittag-Leffler (M-L) noise, and investigate the generalized stochastic...     

Evolution of a long-wave train in loss of stability of a phase interface in geothermal systems

The transition to instability of phase interfaces in geothermal systems when a water stratum overlies a steam stratum and the most unstable mode corresponds to zero wavenumber is considered. The nonlinear Kolmogorov-Petrovskii-Piskunov equation describing the evolution of a narrow strip of weakly unstable modes is obtained. This equation is an analog of the well-known Ginzburg-Landau equation corresponding to the case of destabilization of modes with finite wavenumbers. It is shown that in the neighborhood of the critical points there exist two locations of the plane phase interface which coincide at the instant at which the instability threshold is reached and then disappear.     

Travelling, non-periodic patterns in nonlinear stability problems

In this short note we present results on the existence of several classes of travelling, non-periodic solutions of the complex Ginzburg-Landau equation. First we give a very short introduction to the G-L equation and show its importance in nonlinear stability theory. We then study the G-L equation with complex coefficients and establish the existence of a 2-parameter family of quasi-periodic solutions and two different types of one-parameter families of heteroclinic orbits; all members of these families travel with a well-defined wave-speed. The heteroclinic solutions correspond to (travelling) soliton-like localized structures which connect different (stable) periodic patterns. Mathematically, these families of travelling solutions (quasi-periodic and heteroclinic) are continuations into the complex case of the stationary solutions of the real G-L equation.     

A reaction-diffusion equation and its traveling wave solutions

In the present paper, we study a non-linear reaction-diffusion equation, which can be considered as a generalized Fisher equation. An exact solution and traveling wave solutions to the generalized Fisher equation are obtained by means of the Cole-Hopf transformation and the Lie symmetry method.