Related papers: Dual variational methods and nonvanishing for the …
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and…
We formulate a problem that can be viewed as a natural variation of the so-called Pompeiu or Schiffer problem in the context of scattering of plane waves for the Linear Helmholtz equation. For the two dimensional version of this variation,…
In this paper, we consider the weighted fourth order equation $$\Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u=|x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq…
We present a numerical study of spatially quasi-periodic gravity-capillary waves of finite depth in both the initial value problem and traveling wave settings. We adopt a quasi-periodic conformal mapping formulation of the Euler equations,…
We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the…
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the…
This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \[ \left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) =…
In this paper, we use Hirota's bilinear method to directly construct periodic wave solutions of nonlinear equations. The asymptotic property of periodic wave solutions are analyzed. It is shown that well-known soliton solutions can be…
Motivated by the viewpoint of integrable systems, we study commuting flows of 2-component quasilinear equations, reducing to investigate the solutions of the wave equation with non-constant speed. In this paper, we apply the reduction…
In this paper, we consider the following Klein-Gordon-Maxwell equations \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u+ V(x)u-(2\omega+\phi)\phi u=f(x,u)+h(x)&\mbox{in $\mathbb{R}^{3}$},\\ -\Delta \phi+ \phi u^2=-\omega u^2&\mbox{in…
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation…
In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space \[ \partial_t^2 u - (\Delta_{{\mathbb H}^n} + \rho^2) u = - |u|^{p-1} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}; \] and introduce a…
The aim of this paper is to study negative classical solutions to a $k$-Hessian equation involving a nonlinearity with a general weight \begin{equation} \label{Eq:Ma:0} \tag{$P$} \begin{cases} S_k(D^2u)= \lambda \rho(|x|) (1-u)^q &\mbox{in…
In this paper, by combing the variational methods and Trudinger-Moser inequality, we study the existence and multiplicity of the positive standing wave for the following Chern-Simons-Schr\"odinger equation \begin{equation} -\Delta u+u…
This paper studies the inhomogeneous fractional Sch\"odinger equation $$i\dot u-(-\Delta)^s u=\pm(I_\alpha *|\cdot|^b|u|^p)|x|^b|u|^{p-2}u.$$ In the mass super-critical and energy sub-critical regimes, using a Gagliardo-Nirenberg adapted to…
A wave front and a wave back that spontaneously connect two hyperbolic equilibria, known as a heteroclinic wave loop, give rise to periodic waves with arbitrarily large spatial periods through the heteroclinic bifurcation. The nonlinear…
For $q \in (0, \infty)$, we consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_\xi f(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} in a…
The nonlinear selfdual variational principle established in a preceeding paper [8] -- though good enough to be readily applicable in many stationary nonlinear partial differential equations -- did not however cover the case of nonlinear…
New travelling wave solutions to the Fornberg-Whitham equation are investigated. They are characterized by two parameters. The expresssions for the periodic and solitary wave solutions are obtained.
In this article, we first employ the concentration compactness techniques to prove existence and stability results of standing waves for nonlinear fractional Schr\"{o}dinger-Choquard equation \[ i\partial_t\Psi + (-\Delta)^{\alpha}\Psi = a…