Large time behavior of free boundary for a degenerate parabolic equation with absorption

This work studies the large time behavior of free boundary and continuous dependence on nonlinearity for the Cauchy problem of a degenerate parabolic partial differential equation with absorption. Our objective is to give an explicit expression of speed of propagation of the solution and to show that the solution depends on the nonlinearity of the equation continuously.     

The expanding behavior of positive set H u (t) of a degenerate parabolic equation

This work studies the expanding behavior of the positive set of solutions and the continuous dependence on the nonlinearity for a degenerate parabolic partial differential equation u_t=Df(u){u_{t}=\Delta\phi(u)} . Our objective is to give an explicit expression of speed of propagation of the solution and to show that the solution continuously depends on the nonlinearity of the equation.     

Properties of simple model problems for reacting gas flows

The compressible Navier–Stokes equations for reacting gases are extremely complex. Simpler models have been considered, and for these completely non-physical propagation speeds have been observed. These model problems are stiff, meaning that several different scales are present in the solution. Numerical solution of non-reacting flows almost always involves addition of extra dissipation. It will be shown that this action will render a totally wrong propagation speed for a simple model equation of reacting flows. This problem will be accentuated by increasing stiffness of the problem. Existence and uniqueness of a solution to this model equation is proved. The dependence of the propagation speed on the viscosity and a term governing the stiffness (comparable to the reaction rate for a more complete model) is investigated. A remedy for the wrong propagation speed for this simple model equation is proposed such that the speed is correct although the front is smeared out.     

Some properties of solutions to the weakly dissipative Degasperis-Procesi equation

In this paper, we consider the weakly dissipative Degasperis-Procesi equation. The present paper is concerned with some aspects of existence of global solutions, persistence properties and propagation speed. First we try to discuss the local well-posedness and blow-up scenario, then establish the sufficient conditions on global existence of the solution. Finally, persistence properties on strong solutions and the propagation speed for the weakly dissipative Degasperis-Procesi equation are also investigated.     

Solitary wave solutions for high dispersive cubic-quintic nonlinear Schrödinger equation

We consider the higher-order dispersive nonlinear Schrödinger equation including fourth-order dispersion effects and a quintic nonlinearity. This equation describes the propagation of femtosecond light pulses in a medium that exhibits a parabolic nonlinearity law. By adopting the ansatz solution of Li et al. [Zhonghao Li, Lu Li, Huiping Tian, Guosheng Zhou. New types of solitary wave solutions for the higher-order nonlinear Schrödinger equation. Phys Rev Lett 2000;84:4096], we find two different solitary wave solutions under certain parametric conditions. These solutions are in the form of bright and dark soliton solutions.     

Min–max formulæ for the speeds of pulsating travelling fronts in periodic excitable media

This paper is concerned with some nonlinear propagation phenomena for reaction–advection–diffusion equations in a periodic framework. It deals with travelling wave solutions of the equation $u_t =\nabla\cdot(A(z)\nabla u)\;+q(z)\cdot\nabla u+\,f(z,u),\qquad t\in\mathbb{R},\quad z\in\Omega,$ propagating with a speed c. In the case of a “combustion” nonlinearity, the speed c exists and it is unique, while the front u is unique up to a translation in t. We give a min–max and a max–min formula for this speed c. On the other hand, in the case of a “ZFK” or a “KPP” nonlinearity, there exists a minimal speed of propagation c*. In this situation, we give a min–max formula for c*. Finally, we apply this min–max formula to prove a variational formula involving eigenvalue problems for the minimal speed c* in the “KPP” case.     

Limit behavior of the global solutions to the Degasperis-Procesi-type equation

In this paper, we study the Degasperis-Procesi equation with a physically perturbation term—a linear dispersion. Based on the global existence result, we show that the solution of the Degasperis-Procesi equation with linear dispersion tends to the solution of the corresponding Degasperis-Procesi equation as the dispersive parameter goes to zero. Moreover, we prove that smooth solutions of the equation have finite propagation speed: they will have compact support if its initial data has this property.     

Infinite propagation speed of the Camassa-Holm equation

We use the inverse scattering transform to show that a solution of the Camassa-Holm equation is identically zero whenever it vanishes on two horizontal half-lines in the x-t space. In particular, a solution that has compact support at two different times vanishes everywhere, proving that the Camassa-Holm equation has infinite propagation speed.     

On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient

In this paper, we consider the dissipative Camassa–Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated.     

Asymptotic Speed of Wave Propagation for A Discrete Reaction-Diffusion Equation

We deal with asymptotic speed of wave propagation for a discrete reactlon-diffusion equation. We find the minimal wave speed c★ from the characteristic equation and show that c★ is just the asymptotic speed of wave propagation. The isotropic property and the existence of solution of the initial value problem for the given equation are also discussed.     

轴向变速运动弦线的非线性振动的稳态响应及其稳定性

研究具有几何非线性的轴向运动弦线的稳态横向振动及其稳定性．轴向运动速度为常平均速度与小简谐涨落的叠加．应用Hamilton原理导出了描述弦线横向振动的非线性偏微分方程．直接应用于多尺度方法求解该方程．建立了避免出现长期项的可解性条件．得到了近倍频共振时非平凡稳态响应及其存在条件．给出数值例子说明了平均轴向速度、轴向速度涨落的幅值和频率的影响．应用Liapunov线性化稳定性理论,导出倍频参数共振时平凡解和非平凡解的不稳定条件．给出数值算例说明相关参数对不稳定条件的影响．     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.     

The Third-Order Nonlinear Evolution Equation Governing Wave Propagation in Relaxing Media

Initial value problem for the third-order nonlinear evolution equation governing wave propagation in relaxing media is considered for the case of two space dimensions and small initial data. Existence and uniqueness of the classical solution is established and the solution itself is constructed in the form of a series in the small parameter present in the initial conditions. Long time asymptotic representation is found, which shows that the nonlinearity does not contribute to its major term. The latter consists of two parts corresponding to isotropic and nonisotropic transfer of small perturbations in space.     

Global existence of a restoring force realizing a prescribed half-period

We consider an inverse problem to determine a nonlinearity of an autonomous differential equation so that each solution of the equation has a half-period that is a prescribed function of a half-amplitude of the solution. A global existence of the nonlinearity is proved under the assumption that the prescribed function is Lipschitz continuous. A reconstructive way of the nonlinearity is also given.     

Propagation of perturbations in thin capillary film equations with nonlinear diffusion and convection

We study the evolution of the support of an arbitrary strong generalized solution to the Cauchy problem for the thin film equation with nonlinear diffusion and convection. We find an upper bound exact (in a sense) for the propagation speed of the support of this solution.     

Conditions for the Time Global Existence and Nonexistence of a Compact Support of Solutions to the Cauchy Problem for Quasilinear Degenerate Parabolic Equations

We study a quasilinear degenerate parabolic equation with a nonhomogeneous density. We establish that depending on the behavior of the density at infinity either a solution to the Cauchy problem possesses the property that the speed of propagation of perturbations is finite or the support blows up in finite time.     

Wave propagation through a 2D lattice

Nonlinear wave propagation through a 2D lattice is investigated. Using reductive perturbation method, we show that this can be described by Kadomtsev–Petviashvili (KP) equation for quadratic nonlinearity and modified KP equation for cubic nonlinearity, respectively. With quadratic and cubic nonlinearities together, the system is governed by an integro-differential equation. We have also checked the integrability of these equations using singularity analysis and obtained solitary wave solutions.     

Propagation of Perturbations of Solutions for a Doubly Degenerate Nonlinear Parabolic Equation

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Stability of Front Solutions of the Bidomain Equation

The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen‐Cahn equation) in two spatial dimensions. In the bidomain Allen‐Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen‐Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen‐Cahn equation in striking contrast to the classical or anisotropic Allen‐Cahn equations. We identify two types of instabilities, one with respect to long‐wavelength perturbations, the other with respect to medium‐wavelength perturbations. Interestingly, whether the front is stable or unstable under long‐wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate‐wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate‐wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.© 2016 Wiley Periodicals, Inc.     

Weakly Nonlinear Waves in Nonideal Fluids

We study the propagation of weakly nonlinear waves in nonideal fluids, which exhibit mixed nonlinearity. A method of multiple scales is used to obtain a transport equation from the Navier–Stokes equations, supplemented by the equation of state for a van der Waals fluid. Effects of van der Waals parameters on the wave evolution, governed by the transport equation, are investigated.