相关论文: Classification of radial non-radiative solutions t…

We carry out a constructive review of non-standard solutions of relativistic wave equations. Such solutions are obtained via splitting of relativistic wave equations written in spinor form. All these solutions are also solutions of the…

This paper reports results on the classification of traveling wave solutions, including nonnegative weak sense, in the spatial 1D degenerate parabolic equation. These are obtained through dynamical systems theory and geometric approaches…

We analyze a system of reacting elements harmonically coupled to nearest neighbors in the continuum limit. An analytic solution is found for traveling waves. The procedure is used to find oscillatory as well as solitary waves. A comparison…

Consider the focusing energy-critical wave equation in space dimension 3, 4 or 5. We prove that any global solution which is bounded in the energy space converges in the exterior of wave cones to a radiation term which is a solution of the…

We solve here the so called division problem for wave equations with generic quadratic non-linearities in high dimensions. Specifically, we show that semilinear wave equations which can be written as systems involving quadratic derivative…

Consider the energy-critical focusing wave equation in odd space dimension $N\geq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded…

We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order.…

A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations $\Delta u +f(u)=0$ in $\mathbb{R}^n$ must be radial and radially decreasing. In this paper, we consider…

Nonlinear waves in a liquid with gas bubbles are studied. Higher order terms with respect to the small parameter are taken into account in the derivation of the equation for nonlinear waves. A nonlinear differential equation is derived for…

We consider the focusing wave equation outside a ball of $\R^3$, with Dirichlet boundary condition and a superquintic power nonlinearity. We classify all radial stationary solutions, and prove that all radial global solutions are…

The analysis of nonlinear wave equations has experienced a dramatic growth in the last ten years or so. The key factor in this has been the transition from linear analysis, first to the study of bilinear and multilinear wave interactions,…

Large-scale simulations of the wave equation in electromagnetism, seismology, and acoustics, can be solved efficiently by finite difference methods. The accuracy of these numerical solutions usually depends on the minimization of…

The focusing critical wave equation in three dimensions exhibits a special class of static solutions which are linearly unstable. These solutions decay like an inverse first power. We construct small codimension one stable manifolds in the…

We consider the nonlinear wave equation, with a large exponent, power-like non-linearity, outside a ball of the Euclidean 3-dimensional space. In a previous article, we have proved that any global solution converges, up to a radiation term,…

In this paper we consider global and non-global radial solutions of the focusing energy--critical wave equation on $\mathbb{R} \times \mathbb{R}^N$ where $N \geq 5$ is odd. We prove that if the solution remains bounded in the energy space…

We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}<p<\frac4{d-3}$ (which is energy-supercritical) and dimensions $3\leq d\leq 6$; we also…

As well known, an electron will produce radiative reaction force when the electron emits electromagnetism radiation. The electron radiative effect had not been considered in Schr\"{o}dinger wave equation. In this paper, we give the…

We present full classification of Q-conditional symmetries for the two-dimensional nonlinear wave equation.

We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in 4 to 6 dimensional spaces with radial initial data. We define $w=r^{(d-1)/2} u$, reduce the equation above to one-dimensional…

A class of nonlocal nonlinear wave equation arises from the modeling of a one dimensional motion in a nonlinearly, nonlocally elastic medium. The equation involves a kernel function with nonnegative Fourier transform. We discretize the…