On the Stability of Blowup Solutions to the Complex Ginzburg-Landau Equation in $$\mathbb{R}^d$$

Dissipative optical solitons in asymmetric Rosen-Morse potential

We present analytical methods whereby weak and strong turbulence are predicted in the D dimensional complex Ginzburg Landau (CGL) equation \(A_{t}=RA+(1+i\nu)\Delta A-(1+i\mu)A\vert A\vert^{2q}\) on a periodic domain [0,1]. Strong (hard) turbulence is characterised by large fluctuations away from space & time averages while no such fluctuations occur in weak (soft) turbulence. In the Δ-ν plane, there are different areas where weak & strong behaviour can occur. In the strong case (Δ, ν → ±∞, ‡∞), the corresponding areas go out to the inviscid limit where the CGL equation becomes the NLS equation in which a finite time singularity occurs when qD ≥ 2. A new infinite set of differential inequalities for the “lattice” of functionals \(F_{n,m}=\int\left[{\vert\nabla^{n-1}A\vert}^{2m}+\alpha_{n,m}\vert A\vert^{2m\left[q(n-1)+1\right]}\right]d\underline{x}\) enables us to construct large time upper bounds on the F n,m . The occurrence of strong spiky turbulence is predicted for qD = 2 by showing that exponents of R in the upper bounds of F n,m & ‖A‖∞ in the strong regions are dependent on the quantity |ν| which gets large in the inviscid limit. The critical value qD = 2 plays an important role: when qD > 2 the CGL equation has some similarities with the 3D Navier equations. A comparison is made between the two & the possibility of having a 1D system which mimics some limited features of the Navier Stokes equations is discussed.

It is shown that the energy of a thermoelastic bar and plate decays exponentially fast. The energy method, combined with a multiplier technique and compactness property, is used.

In this paper, complex Ginzburg-Landau (CGL) equations governed by p-Laplacian are studied. We discuss the global existence of solutions for the initial-boundary value problem of the equation in general domains. The global solvability of the initialboundary value problem for the case when p = 2 is already examined by several authors provided that parameters appearing in CGL equations satisfy a suitable condition. Our approach to CGL equations is based on the theory of parabolic equations with nonmonotone perturbations. By using this method together with some approximate procedure and a diagonal argument, the global solvability is shown without assuming any growth conditions on the nonlinear terms.

The complex Ginzburg-Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of different symmetries. Solutions of the CGL equation may or may not be fixed by the action of these symmetries. We investigate the stability of chaotic solutions with some reflectional symmetry to perturbations which break that symmetry. This can be achieved by considering the isotypic decomposition of the space and finding the dominant Lyapunov exponent associated with each isotypic component. Our numerical results indicate that for most parameter values, chaotic solutions that have been restricted to lie in invariant subspaces are unstable to perturbations out of these subspaces, leading us to conclude that for these parameter values arbitrary initial conditions will generically evolve to a solution with the minimum amount of symmetry allowable. We have also found a small region of parameter space in which chaotic solutions that are even are stable with respect to odd perturbations.

On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation

The existence and stability of exact continuous-wave and dark-soliton solutions to a system consisting of the cubic complex Ginzburg-Landau (CGL) equation linearly coupled with a linear dissipative equation is studied. We demonstrate the existence of vast regions in the system's parameter space associated with stable dark-soliton solutions, having the form of the Nozaki-Bekki envelope holes, in contrast to the case of the conventional CGL equation, where they are unstable. In the case when the dark soliton is unstable, two different types of instability are identified. The proposed stabilized model may be realized in terms of a dual-core nonlinear optical fiber, with one core active and one passive.

In this work we consider a quite general class of two-species hyperbolic reaction-advection-diffusion system with the main aim of elucidating the role played by inertial effects in the dynamics of oscillatory periodic patterns. To this aim, first, we use linear stability analysis techniques to deduce the conditions under which wave (or oscillatory Turing) instability takes place. Then, we apply multiple-scale weakly nonlinear analysis to determine the equation which rules the spatiotemporal evolution of pattern amplitude close to criticality. This investigation leads to a cubic complex Ginzburg-Landau (CCGL) equation which, owing to the functional dependence of the coefficients here involved on the inertial times, reveals some intriguing consequences. To show in detail the richness of such a scenario, we present, as an illustrative example, the pattern dynamics occurring in the hyperbolic generalization of the extended Klausmeier model. This is a simple two-species model used to describe the migration of vegetation stripes along the hillslope of semiarid environments. By means of a thorough comparison between analytical predictions and numerical simulations, we show that inertia, apart from enlarging the region of the parameter plane where wave instability occurs, may also modulate the key features of the coherent structures, solution of the CCGL equation. In particular, it is proven that inertial effects play a role, not only during transient regime from the spatially-homogeneous steady state toward the patterned state, but also in altering the amplitude, the wavelength, the angular frequency, and even the stability of the phase-winding solutions.

Scaling of turbulent spike amplitudes in the complex Ginzburg-Landau equation

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.

Hard turbulence in a finite dimensional dynamical system?

Cubic-quartic optical soliton perturbation in polarization-preserving fibers with complex Ginzburg-Landau equation having five nonlinear refractive index structures

We study the effect of nonlinear gradient terms on breathing localized solutions in the complex Ginzburg-Landau equation. It is found that even small nonlinear gradient terms — which appear at the same order as the quintic term — can cause dramatic changes in the behavior of the solution, such as causing opposite sides of an otherwise monoperiodic symmetrically breathing solution to breathe at different frequencies, thus causing the solution to breathe periodically or chaotically on only one side or the solution to rapidly spread. [S0031-9007(98)07488-2] PACS numbers: 47.20.Ky, 03.40.Gc, 03.40.Kf, 05.70.Ln For over thirty years now it has been known that stable localized solutions can exist for certain nonlinear partial differential equations. The best-known example of such solutions is the soliton [1,2], a localized solution which occurs in purely dispersive systems such as the nonlinear Schrodinger equation. More recently stable localized solutions have been found to occur in quintic complex Ginzburg-Landau (CGL) equations [3‐ 7] —generic equations with both dissipation and dispersion and which describe systems near a subcritical bifurcation to traveling waves. These dissipative-dispersive localized (DDL) solutions can be considered to be the analog of the solitons that occur in purely dispersive systems. Although these DDL solutions share some properties with solitons, such as a fixed shape for the modulus and interaction behavior in which shape and size are preserved during collisions [4,5], there are fundamental differences. For example, these DDL solutions also exhibit mutual annihilation during collisions [4,5], a property which does not occur for solitons. Also, in contrast to solitons which require no energy input for their existence, the DDL solutions depend on a constant influx of energy in order to overcome the dissipation. Stable DDL solutions have also been studied in a two-dimensional (2D) quintic CGL equation [3,8], in a 2D equation for systems with broken rotational symmetry [9], and in equations describing systems in nonlinear optics — a dye laser with saturable absorber [10] and a system exhibiting optical bistability [11]. Experimentally, stable DDL solutions have been found in binary fluid convection [12,13] and in a dye laser with saturable absorber [14]. Until recently the behavior of localized solutions of prototype equations has been limited to solutions with fixed modulus such as the solitons and DDL solutions discussed above, or to solutions which oscillate periodically about zero (for real equations) such as the “breathers” of the sine Gordon equation. Therefore, an interesting discovery was that of stable localized solutions for which the modulus breathes periodically, quasiperiodically, or even chaotically [15]. By stable is meant that the solution lies on an attractor. These breathing DDL solutions, which were found for the quintic CGL equation, bare no relation to the “breathers” of the sine-Gordon equation. Also they are very different from the slowly spreading chaotic localized solutions of the quintic CGL equation [16,17]. The breathing DDL solutions exhibit interesting interaction behavior such as dependence on initial conditions for the outcome of collisions and even sensitive dependence on initial conditions for the outcome of collisions involving chaotic breathing DDL solutions [18]. For the breathing DDL solutions of the quintic CGL equation, nonlinear gradient terms have thus far been neglected for simplicity. However, for an actual physical system nonlinear gradient terms will always be present since they occur at the same order as the quintic term [19]. Therefore, an important question is whether there are any qualitative changes in the behavior as a result of the nonlinear gradient terms. In this Letter we study the effect of nonlinear gradient terms on the breathing DDL solutions of the quintic CGL equation. We find that even small nonlinear gradient terms can dramatically alter the behavior of solutions, such as causing opposite sides of an otherwise monoperiodic symmetrically breathing solution to breathe at different frequencies, causing the solution to breathe periodically or chaotically on only one side, or causing the solution to rapidly spread. We find that it is also possible for nonlinear gradient terms to cause an otherwise fixed-shape solution to breathe periodically or even chaotically. The quintic CGL equation with nonlinear gradient terms reads

We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time T T only at one blow-up point. We also give a sharp description of its profile. In the first part, we formally construct a blow-up solution. In the second part we give the rigorous proof. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution. We would like to mention that the asymptotic profile of our solution is different from previously known profiles for CGL or for the semilinear heat equation.

This paper is the second of a two-stage exposition, in which we study the nonequilibrium dynamics of the complex Ginzburg-Landau (CGL) equation. We use spiral defects to characterize the system evolution and morphologies. In the first paper of this exposition [S.K. Das, S. Puri, and M.C. Cross, Phys. Rev E 64, 046206 (2001)], we presented analytical results for the correlation function of a single spiral defect, and its short-distance singular behavior. We had also examined the utility of the Gaussian auxiliary field ansatz for characterizing multispiral morphologies. In this paper, we present results from an extensive numerical study of nonequilibrium dynamics in the CGL equation with dimensionality d=2,3. We discuss the behavior of domain growth laws; real-space correlation functions; and momentum-space structure factors. We also compare numerical results for the correlation functions and structure factors with analytical results presented in our first paper.