Exact solutions to mKdV equation with variable coefficients

In this paper a special mKdV with variable coefficients is considered. A transformation of variables is first applied in order to obtain a mKdV equation with constant coefficients. Some its one-, two- and three-soliton as well as breather-type soliton solutions are derived by using Hirorta’s bilinear approach.     

Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions

A variable coefficient viscoelastic wave equation with acoustic boundary conditions and nonlinear source term is considered. Under suitable conditions on the initial data and the relaxation function g, we show the polynomial decay of the energy solution and the blow up of solutions by energy methods. The estimates for the lifespan of solutions are also given.     

Exact solutions of KdV equation with time-dependent coefficients

In this paper, we study the Korteweg-de Vries (KdV) equation having time dependent coefficients from the Lie symmetry point of view. We obtain Lie point symmetries admitted by the equation for various forms for the time-dependent coefficients. We use the symmetries to construct the group-invariant solutions for each of the cases of the arbitrary coefficients. Subsequently, the 1-soliton solution is obtained by the aid of solitary wave ansatz method. It is observed that the soliton solution will exist provided that these time-dependent coefficients are all Riemann integrable.     

Exact controllability of the wave equation with time-dependent and variable coefficients

This paper deals with boundary exact controllability for the dynamics governed by the wave equation with variable coefficients in time and space, subject to Dirichlet or Neumann boundary controls. The observability inequalities are established by the Riemannian geometry method under some geometric conditions.     

Existence of weak solutions to the time-dependent MHD equations coupled to the heat equation with nonlocal radiation boundary conditions

We study the coupling of the system of magnetohydrodynamics to the heat equation in the context of an application to crystal growth. The heat sources are given by the dissipation of current that occurs in the fixed conductors of the system. According to Boussinesq’s model, the dissipative heating is neglected in the fluid. We take into account the natural interface conditions for the magnetic field, and nonlocal radiation boundary conditions for the heat flux. We prove the existence of a weak solution with a defect measure, concentrated in a singular set.     

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

In this paper, we consider a viscoelastic wave equation of variable coefficients in the presence of past history with nonlinear damping and delay in the internal feedback and dynamic boundary conditions. Under suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumption on the behavior of the relaxation function at infinity by Riemannian geometry method and Lyapunov functional method.     

Exact solutions to nonlinear Schrödinger equation with variable coefficients

According to Ma-Fuchsseiter’s idea, a trial equation method was proposed to find the exact envelop traveling wave solutions to some nonlinear differential equations with variable coefficients. As an application, combining with the complete discrimination system for polynomial, some exact envelop traveling wave solutions to Schrödinger equation with variable coefficients were obtained. At the same time, the physical meanings of the obtained solutions are discussed, and the problem needed to further study is pointed out.     

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

Periodic solutions of the wave equation with nonconstant coefficients and with homogeneous Dirichlet and Neumann boundary conditions

We prove theorems on the existence and regularization of periodic solutions of the wave equation with variable coefficients on an interval with homogeneous Dirichlet and Neumann boundary conditions. The nonlinear term has a power-law growth or satisfies the nonresonance condition at infinity.     

Convex solutions of the polynomial-like iterative equation with variable coefficients

In this paper, by applying the Schauder''s fixed point theorem we prove the existence of increasing and decreasing solutions of the polynomial-like iterative equation with variable coefficients and further completely investigate increasing convex (or concave) solutions and decreasing convex (or concave) solutions of this equation. The uniqueness and continuous dependence of those solutions are also discussed     

Exact solutions of the combined KdV-Burgers equation with variable coefficients

In this Letter, the exp-function method is used to obtain generalized solitary solutions and periodic solutions of the general types of combined KdV-Burgers equation with variable coefficients. It is shown that the exp-function method via symbolic computation provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations with variable coefficients in mathematical physics.     

Some exact solutions of KdV equation with variable coefficients

In this paper, the extended mapping transformation method is used to obtain some new exact solutions of a variable-coefficient KdV equation arising in arterial mechanics. The obtained solutions include soliton solutions, periodic solutions and rational solutions.     

Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients

This paper studies the modified Korteweg–de Vries equation with time variable coefficients of the damping and dispersion using Lie symmetry methods. We carry out Lie group classification with respect to the time-dependent coefficients. Lie point symmetries admitted by the mKdV equation for various forms for the time variable coefficients are obtained. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are determined. These are then used to determine exact group-invariant solutions, including soliton solutions, and symmetry reductions for some special forms of the equations.     

Lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficients

In this paper, lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficients, which can provide some more realistic models than their constant-coefficient counterparts, is derived through selecting equilibrium distribution function and adding the compensate function, appropriately. Effects and approximate value range of the free parameters, which are introduced to adjust the single relaxation time and equilibrium distribution function, are discussed in detail, as well as the impact of the lattice space step and velocity. Numerical simulations in different situations of this equation are conducted, including the propagation and interaction of the solitons, the evolution of the non-propagating soliton and the propagation of the double-pole solutions. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current lattice Boltzmann model is a satisfactory and efficient algorithm.     

Dynamic problems for the sine-gordon equation with variable coefficients. Exact solutions

The sine-Gordon equation with dissipation and a variable coefficient on the non-linear term is considered. This equation describes waves in an energetically open system with an external field acting on it which varies monotonically with time. Scale transformations, matched with the external field and the dissipation, are introduced which reduce the generalized equation to the standard equation. It is shown that processes for controlling the oscillations and waves exist for which the equation is transformed to a form with constant coefficients and an effective dissipation which vanishes or is either positive (damping of the oscillations) or negative (their amplification). Waves, which propagate with a constant, decaying or increasing amplitude and variable frequency and velocity correspond to them.     

On the lindblad equation with unbounded time-dependent coefficients

We prove newa priori estimates for the resolvent of a minimal quantum dynamical semigroup. These estimates simplify well-known conditions sufficient for conservativity and impose continuity conditions on the time-dependent operator coefficients ensuring the existence of conservative solutions of the Markov evolution equations. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 125–140, January, 1997. Translated by A. M. Chebotarev