Related papers: Dual variational methods and nonvanishing for the …
The periodic standing wave (PSW) method for the binary inspiral of black holes and neutron stars computes exact numerical solutions for periodic standing wave spacetimes and then extracts approximate solutions of the physical problem, with…
Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behavior at infinity is established. Some generalizations to nonautonomous radial…
The periodic standing-wave method for binary inspiral computes the exact numerical solution for periodic binary motion with standing gravitational waves, and uses it as an approximation to slow binary inspiral with outgoing waves. Important…
We study the nodal set of solutions to equations of the form $$ (-\Delta)^s u = \lambda_+ (u_+)^{q-1} - \lambda_- (u_-)^{q-1}\quad\text{in $B_1$}, $$ where $\lambda_+,\lambda_->0, q \in [1,2)$, and $u_+$ and $u_-$ are respectively the…
We investigate the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+|u|^{p-1}u=0$ with $1+\frac{4}{N}<p<1+\frac{4}{N-2}$ (when $N=1, 2$, $1+\frac{4}{N}<p<\infty$) in energy space $H^1$ and study the divergent property of…
We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized…
In this paper we prove the existence of vortices, namely standing waves with non null angular momentum, for the nonlinear Klein-Gordon equation in dimension $N\geq 3$. We show with variational methods that the existence of these kind of…
In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear…
We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=\phi(u)$, with $\phi(u)=\Phi u^{3} + O(u^{5})$ odd and analytic, $\Phi\neq0$, and we construct small amplitude periodic solutions with frequency…
Periodic waves are standing wave solutions of nonlinear Schr\''odinger equations whose profile is periodic in space dimension one. We consider general nonlinearities and provide variational characterizations for the periodic wave profiles.…
We construct infinitely many real-valued, time-periodic breather solutions of power-type nonlinear wave equations. These solutions are obtained from critical points of a dual functional and they are weakly localized in space. Our abstract…
We study the asymptotic behaviour, as time goes to infinity, of the Fisher-KPP equation $\partial_t u=\Delta u +u-u^2$ in spatial dimension $2$, when the initial condition looks like a Heaviside function. Thus the solution is,…
In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…
This is a study of two-dimensional steady periodic travelling waves on the surface of an infinitely deep irrotational ocean, when the top streamline is in contact with a membrane which has a nonlinear response to stretching and bending, and…
We derive a novel two-component generalization of the nonlinear variational wave equation as a model for the director field of a nematic liquid crystal with a variable order parameter. The two-component nonlinear variational wave equation…
It is known that for some time periodic potentials $q(t, x) \geq 0$ having compact support with respect to $x$ some solutions of the Cauchy problem for the wave equation $\partial_t^2 u - \Delta_x u + q(t,x)u = 0$ have exponentially…
We prove the existence of infinitely many nontrivial solutions for time-harmonic nonlinear Maxwell's equations on bounded domains and on $\mathbb{R}^3$ using dual variational methods. In the dual setting we apply a new version of the…
We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. The variable coefficient…
In this paper, we study nonlinear Helmholtz equations (NLH) $-\Delta_{\mathbb{H}^N} u - \frac{(N-1)^2}{4} u -\lambda^2 u = \Gamma|u|^{p-2}u$ in $\mathbb{H}^N$, $N\geq 2$ where $\Delta_{\mathbb{H}^N}$ denotes the Laplace-Beltrami operator in…
We consider the periodic standing waves in the derivative nonlinear Schrodinger (DNLS) equation arising in plasma physics. By using a newly developed algebraic method with two eigenvalues, we classify all periodic standing waves in terms of…