On energy estimates for second order hyperbolic equations with Levi conditions for higher order regularity

We consider energy estimates for second order homogeneous hyperbolic equations with time dependent coefficients. The property of energy conservation, which holds in the case of constant coefficients, does not hold in general for variable coefficients; in fact, the energy can be unbounded as t → ∞ in this case. The conditions to the coefficients for the generalized energy conservation (GEC), which is an equivalence of the energy uniformly with respect to time, has been studied precisely for wave type equations, that is, only the propagation speed is variable. However, it is not true that the same conditions to the coefficients conclude (GEC) for general homogeneous hyperbolic equations. The main purpose of this paper is to give additional conditions to the coefficients which provide (GEC); they will be called as C ^k -type Levi conditions due to the essentially same meaning of usual Levi condition for the well-posedness of weakly hyperbolic equations.     

New periodic wave solutions for nonlinear evolution equations with variable coefficients via mapping method

An extended mapping method with a computerized symbolic computation is used for constructing a new exact travelling wave solutions for nonlinear evolution equations arising in physics, namely, generalized Zakharov Kuznetsov equation with variable coefficients. As a result, many exact travelling wave solutions are obtained which include new periodic wave solution, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations with variable coefficients arising in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemmannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua are generalized to the variable coefficient case by some computational techniques in Riemmannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalitities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.     

Internal and Boundary Observability Estimates for the Heterogeneous Maxwell's System

Observability estimates for Maxwell's system with variable coefficients are established using the differential geometry method recently developed for scalar wave equations. The main tool is that Maxwell's system is reducible to a perturbed vectorial wave equation with a decoupled principal part.     

变系数波方程振动传递的边界镇定

本文讨论了变系数波方程振动传递的边界镇定,应用黎曼几何方法和迹的正则性得到了所讨论问题的能量一致衰减率.     

Regularization of a Solution to the Cauchy Problem with Data on a Timelike Plane

A regularizing algorithm is constructed for solving the problem of extension of a wave field from the boundary to the interior of a half-plane. The cases are considered of the wave equation and hyperbolic equations with variable lower order coefficients.     

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.     

On the Integrability of a Generalized Variable‐Coefficient Forced Korteweg‐de Vries Equation in Fluids

With the inhomogeneities of media taken into account, under investigation is hereby a generalized variable‐coefficient forced Korteweg‐de Vries (vc‐fKdV) equation, which describes shallow‐water waves, internal gravity waves, etc. Under an integrable constraint condition on the variable coefficients, in this paper, the complete integrability of the generalized vc‐fKdV equation is proposed. By virtue of a generalization of Bells polynomials, we systematically present its bilinear representations, Bäcklund transformations, Lax pairs and Darboux covariant Lax pairs, which can be reduced to the ones of some integrable models, such as vcKdV model, cylindrical KdV equation, and an analytical model of tsunami generation. It is very interesting that its bilinear formulism is free for the integrable constraint condition. Besides, researching the Lax equations yield its infinitely conservation laws, all conserved densities and fluxes of them are obtained by explicit recursion formulas. Furthermore, by considering its bilinear formulism with an extra auxiliary variable, we present the soliton solutions and Riemann theta function periodic wave solutions of the equation. According to the constraint among the nonlinear, dispersive, and line‐damping coefficients, we further discuss the solitonic structures and interaction properties by some graphic analysis. Finally, the relationships between the periodic wave solutions and soliton solutions are presented in detail by a limiting procedure.     

On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback

The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in [A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24-52] where the system is claimed to be nondissipative.     

On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback

The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in [A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24–52] where the system is claimed to be nondissipative.     

On Stationary Nonequilibrium Measures for Wave Equations

Doklady Mathematics - In the paper, the Cauchy problem for wave equations with constant and variable coefficients is considered. We assume that the initial data are a random function with finite...     

(3+1)-维耦合高阶非线性Schrödinger方程的光学多畸形波

本文研究带有高阶项、时间色散项和非线性系数项的复杂（3+1）-维高阶耦合非线性Schrödinger（3DHCNLSE）方程的精确解. 首先,利用相似变换将非自治的方程转化为自治的耦合Hirota 方程; 其次,采用Darboux 变换方法得到耦合Hirota 方程带有任意常数的有理解; 最后,给出变系数3DHCNLSE方程带有任意常数的1 阶和2 阶多畸形波解. 本文获得的（3+1）-维（3D）多畸形波解可以用来描述深海动力学波和非线性光学纤维中出现的一些物理现象.     

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.     

Amplitude Modulation of Nonlinear Waves in Fluid-Filled Tapered Tubes

We study the modulation of nonlinear waves in fluid-filled prestressed tapered tubes. For this, we obtain the nonlinear dynamical equations of motion of a prestressed tapered tube filled with an incompressible inviscid fluid. Assuming that the tapering angle is small and using the reductive perturbation method, we study the amplitude modulation of nonlinear waves and obtain the nonlinear Schrödinger equation with variable coefficients as the evolution equation. A traveling-wave type of solution of such a nonlinear equation with variable coefficients is obtained, and we observe that in contrast to the case of a constant tube radius, the speed of the wave is variable. Namely, the wave speed increases with distance for narrowing tubes and decreases for expanding tubes.     

Auxiliary equation method for the mKdV equation with variable coefficients

By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of nonlinear evolution equations with variable coefficients. Being concise and straightforward, this method is applied to the mKdV equation with variable coefficients. As a result, new explicit solutions including solitary wave solutions and trigonometric function solutions are obtained with the aid of symbolic computation.     

Nonlinear Internal Damping of Wave Equations with Variable Coefficients

For wave equations with variable coefficients on regions which are not necessarily smooth, we study the energy decay rate when a nonlinear damping is applied on a suitable subrigion.     

Homogenization and corrector for the wave equation with discontinuous coefficients in time

In this paper we analyze the homogenization of the wave equation with bounded variation coefficients in time, generalizing the classical result, which assumes Lipschitz-continuity. We start showing a general existence and uniqueness result for a general sort of hyperbolic equations. Then, we obtain our homogenization result comparing the solution of a sequence of wave equations to the solution of a sequence of elliptic ones. We conclude the paper making an analysis of the corrector. Firstly, we obtain a corrector result assuming that the derivative of the coefficients in the time variable is equicontinuous. This result was known for non-time dependent coefficients. After, we show, with a counterexample, that the regularity hypothesis for the corrector theorem is optimal in the sense that it does not hold if the time derivative of the coefficients is just bounded.     

具时间依赖系数和耗散边界的非线性波方程的能量指数衰减性

利用能量法证明了具耗散边界条件和时间依赖系数的非线性波方程的能量指数衰减性.     

A modified Lyapunov method and its applications to ODE

Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some nonlinear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equations admitting periodic solutions, an averaging over the coefficients gives global solutions. We show that, under some restrictive condition, the method is equivalent to the Picard-Lindelöf method. After some numerical experiments showing the efficiency of the method, we apply it to equations of interest in physics, in which we show that our method possesses an excellent precision even with low iterations.