Related papers: Quasi-Periodic Solutions For Nonlinear Klein-Gordo…
For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global…
Quasi-periodic solutions with Liouvillean frequency of forced nonlinear Schr\"odinger equation are constructed. This is based on an infinite dimensional KAM theory for Liouvillean frequency.
We consider a quasi-linear elliptic equation with Dirac source terms arising in a generalized self-dual Chern-Simons-Higgs gauge theory. In this paper, we study doubly periodic vortices with arbitrary vortex configuration. First of all, we…
We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein-Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter…
This paper is concerned with a one dimensional (1D) derivative nonlinear Schr\"odinger equation with periodic boundary conditions \begin{equation*} \mi u_t+u_{xx}+\mi |u|^2u_x=0, \ \ x\in \mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}.…
In this paper, we prove almost global existence of solutions to certain quasilinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides with Neumann boundary conditions. We use a Galerkin method to expand the…
We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-\Delta u+2\alpha \partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{align*} where $\alpha>0$, $2\leq d\leq 5$ and energy…
Consider the Klein-Gordon-Zakharov equations in $\mathbb{R}^{1+2}$, and we are interested in establishing the small global solution to the equations and in investigating the pointwise asymptotic behavior of the solution. The…
We prove existence and multiplicity of Cantor families of small amplitude analytic in time periodic solutions of the completely resonant cubic nonlinear Klein-Gordon equation on $\mathbb{S}^3$ for an asymptotically full measure set of…
From the work on the weak-null condition by Lindblad and Rodnianski, it is well-known that `bad' quadratic sourcing terms are allowed to appear in coupled semilinear wave equations in three spatial dimensions, provided that such terms…
It is proved that the KAM tori (thus quasi-periodic solutions) are long time stable for infinite dimensional Hamiltonian systems generated by nonlinear wave equation, by constructing a partial normal form of higher order around the KAM…
For the one-dimensional nonlinear damped Klein-Gordon equation \[ \partial_{t}^{2}u+2\alpha\partial_{t}u-\partial_{x}^{2}u+u-|u|^{p-1}u=0 \quad \mbox{on $\mathbb{R}\times\mathbb{R}$,}\] with $\alpha>0$ and $p>2$, we prove that any global…
We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The…
We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equation in the case of strong nonlinearities using a method based on the Linear Delta Expansion. All the results obtained in this article are fully…
Based on a Riemann theta function and the super-Hirota bilinear form, we propose a key formula for explicitly constructing quasi-periodic wave solutions of the supersymmetric Ito's equation in superspace $\mathbb{C}_{\Lambda}^{2,1}$. Once a…
We consider a nonlinear system of two-dimensional Klein-Gordon equations with masses satisfying the resonance relation. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at…
In 2015, M. Canadell and R. de la Llave consider a time-dependent perturbation of a vector field having an invariant torus supporting quasiperiodic solutions. Under a smallness assumption on the perturbation and assuming the perturbation…
It has been known that if the initial data decay sufficiently fast at space infinity, then 1D Klein-Gordon equations with quadratic nonlinearity admit classical solutions up to time $e^{C/\epsilon^2}$ while $e^{C/\epsilon^2}$ is also the…
We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schroedinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our…
Nonexistence of global weak solutions of Klein-Gordon equations with gauge variant semilinear terms are considered in Friedmann-Lema\^itre-Robertson-Walker spacetimes. Effects of spatial expansion or contraction on the solutions are studied…