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.     

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.     

The first integral method for modified Benjamin–Bona–Mahony equation

In this paper, we use the first integral method for analytic treatment of the modified Benjamin–Bona–Mahony equation. Some exact new solutions are formally derived.     

Exact traveling wave solutions for the Benjamin–Bona–Mahony equation by improved Fan sub-equation method

This paper combines the bifurcation theory of dynamical systems and the Fan sub-equation method to improve the Fan sub-equation method for solving the BBM equation. Periodic solutions, kink solutions and solitary solutions are formally derived in a general form. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

Convergence rate to traveling waves for generalized Benjamin–Bona–Mahony–Burgers equations

This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations

Convergence rate to traveling waves for generalized Benjamin–Bona–Mahony–Burgers equations

This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations

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.     

Traveling wave solutions of the nonlinear Drinfel’d–Sokolov–Wilson equation and modified Benjamin–Bona–Mahony equations

In this paper, the modified simple equation (MSE) method is implemented to find the exact solutions for the nonlinear Drinfel’d–Sokolov–Wilson (DSW) equation and the modified Benjamin–Bona–Mahony (mBBM) equations. The efficiency of this method for constructing these exact solutions has been demonstrated. It is shown that the MSE method is direct, effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this technique reduces the large volume of calculations.     

Error estimates of the spectral collocation method for the Ginzburg–Landau equation coupled with the Benjamin–Bona–Mahony equation

In this paper, we consider the spectral collocation method for the Ginzburg–Landau equation coupled with the Benjamin–Bona–Mahony equation. Semidiscrete and fully discrete spectral collocation schemes are given. In the fully discrete case, a three-level spectral collocation scheme is considered. An energy estimation method is used to obtain error estimates for the approximate solutions. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

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.     

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.     

Optimal decay rate of solutions for Cahn–Hilliard equation with inertial term in multi-dimensions

The Cauchy problem of the Cahn–Hilliard equation with inertial term is considered. Based on Green?s function method together with energy estimates, we get the global-in-time existence and optimal decay rate of solutions. Furthermore, the viscous case is investigated in the last section.     

Symmetry reduction,exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.