Exponential decay estimates for time-delayed Benjamin–Bona–Mahony equations

In this article, we consider Benjamin–Bona–Mahony equation with a time delay. By using the Liapunov function method, we show that the time-delayed Benjamin–Bona–Mahony equation is exponentially decay if the delay parameter is sufficiently small.     

Optimal decay rates of solutions for a multidimensional generalized Benjamin–Bona–Mahony equation

In this paper, we study the optimal decay rates of solutions for the generalized Benjamin–Bona–Mahony equation in multi-dimensional space (n≥3$n\ge 3$). By using Fourier transform and the energy method, we obtain the L^q(2≤q≤∞)$L^q(2\le q\le \infty )$ convergence rates of the solutions under the condition that the initial data is small. The optimal decay rates obtained in this paper are found to be the same as the decay rate for the Heat equation.     

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

In the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin–Bona–Mahony–Burgers (BBMB) equation. For the construction of the two-level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(τ² + h²) . For the three-level linearized scheme, the extrapolation technique is utilized to linearize the nonlinear term based on ψ function. We obtain the conservation, boundedness, unique solvability and convergence of numerical solutions with the convergence order O(τ² + h²) at length. Furthermore, extending our work to the BBMB equation with the nonlinear source term is considered and a Newton linearized method is inserted to deal with it. The applicability and accuracy of both schemes are demonstrated by numerical experiments.     

Large-time behavior for the solution to the generalized Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space

In this paper, we consider the global existence as well as the optimal decay estimates of the Cauchy problem for the multi-dimensional Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space. And these results are obtained by Green?s function method, Fourier analysis method, energy estimates method combined with the time-frequency decomposition method.     

Existence of standing waves for the complex Ginzburg–Landau equation

We study the existence of standing wave solutions of the complex Ginzburg–Landau equation

equation(GL)

_φt−e^iθ(ρI−Δ)φ−e^iγ|φ|^αφ=0${\phi }_t-e^{i\theta }(\rho I-\mathrm{\Delta })\phi -e^{i\gamma }{|\phi |}^{\alpha }\phi =0$

in R^N${\mathbb{R}}^N$, where α>0$\alpha >0$, (N−2)α<4$(N-2)\alpha <4$, ρ>0$\rho >0$ and θ,γ∈R$\theta ,\gamma \in \mathbb{R}$. We show that for any θ∈(−π/2,π/2)$\theta \in (-\pi /2,\pi /2)$ there exists ε>0$\epsilon >0$ such that (GL) has a non-trivial standing wave solution if |γ−θ|<ε$|\gamma -\theta |<\epsilon$. Analogous result is obtained in a ball Ω∈R^N$\Omega \in {\mathbb{R}}^N$ for ρ>−_λ1$\rho >-{\lambda }_1$, where _λ1${\lambda }_1$ is the first eigenvalue of the Laplace operator with Dirichlet boundary conditions.     

Conservation laws and exact solutions of a Generalized Benjamin–Bona–Mahony–Burgers equation

The concept of nonlinear self-adjointness given by Ibragimov is applied to a Generalized Benjamin–Bona–Mahony–Burgers equation. Then, a nonlinear self-adjoint classification has been achieved. Moreover, some nontrivial conservation laws are constructed by using the multipliers method which does not require the use of a variational principle. Finally, by applying the modified simplest equation method we derive new travelling wave solutions.     

Global attractor for the damped Benjamin–Bona–Mahony equations on R1

We utilize a new necessary and sufficient condition to verity the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood–Paley projection operators. We then use this condition to prove the existence of an attractor for the damped Benjamin–Bona–Mahony equation in the phase space H ¹(R ¹) by showing the solutions are point dissipative and asymptotically compact. Moreover the attractor is in fact smoother and it belongs to H ^3/2?? for every ?>0.     

Nehari method for the generalized Ginzburg–Landau system

The Dirichlet problem for the generalized Ginzburg–Landau system is considered. The existence of positive vector solutions is proved in the following three cases: (1) the cross term has weak growth; (2) the interaction constant is large enough; and (3) the cross term has strong growth and the interaction constant is positive and close to zero.     

Null controllability of the complex Ginzburg–Landau equation

The paper investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Zero-controllability results are derived from a new Carleman estimate and an analysis based upon the theory of sectorial operators.     

Recurrent solutions of the derivative Ginzburg–Landau equation with boundary forces

In this paper, we will establish the existence of the bounded solutions, periodic solutions, quasi-periodic solutions and almost periodic solutions for the derivative Ginzburg–Landau equation with time-dependent boundary external forces. A smoothing effect for the semigroup associated with linear Ginzburg–Landau operator is the crucial tool in establishing the main results.     

Random attractors for a Ginzburg–Landau equation with additive noise

The existence of a compact random attractor for the random dynamical system generated by the complex Ginzburg–Landau equation with additive white noise has been proved. And a precise estimate of the upper bound of the Hausdorff dimension of the random attractor is obtained.     

Finite-time blowup for a complex Ginzburg–Landau equation with linear driving

In this paper, we consider the complex Ginzburg–Landau equation ${u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u}$ on ${\mathbb{R}^N}$ , where ${\alpha > 0,\,\gamma \in \mathbb{R}}$ and ${-\pi /2 < \theta < \pi /2}$ . By convexity arguments, we prove that, under certain conditions on ${\alpha,\theta,\gamma}$ , a class of solutions with negative initial energy blows up in finite time.     

Breather type of chirped soliton solutions for the 2D Ginzburg–Landau equation

New exact solutions including bright soliton solutions, breather and periodic types of chirped soliton solutions, kink-wave and homoclinic wave solutions for the 2D Ginzburg–Landau equation are obtained using the special envelope transform and the auxiliary function method. It is shown that the specially envelope transform and the auxiliary function method provide a powerful mathematical tool for solving nonlinear equations arising in mathematical physics.