Related papers: Existence of standing waves for the complex Ginzbu…
We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t = e^{i\theta} \Delta \phi + e^{i\gamma} |\phi |^\alpha \phi $ with periodic boundary conditions. Our result includes all values of…
In this paper, we are concerned with the standing waves for the following nonlinear Schr\"{o}dinger equation $$i\partial_{t}\psi=-\Delta \psi+b^2(x_1^2+x_2^2)\psi+\frac{\lambda_1}{|x|}\psi+ \lambda_2(|\cdot|^{-1}\ast |\psi|^2)\psi-…
This work comes as the second part in a series of investigations into the dynamics of rotating waves as solutions to lattice dynamical systems. Such nonlinear waves as solutions to mathematical equations are of great interest throughout the…
We study strong instability of standing waves $e^{i\omega t} \phi_{\omega}(x)$ for nonlinear Schr\"odinger equations with $L^2$-supercritical nonlinearity and a harmonic potential, where $\phi_{\omega}$ is a ground state of the…
We reformulate the one-dimensional complex Ginzburg-Landau equation as a fourth order ordinary differential equation in order to find stationary spatially-periodic solutions. Using this formalism, we prove the existence and stability of…
In this paper, we study the standing wave solutions of Klein--Gordon equation with logarithmic nonlinearity. The existence of the standing wave solution related to the ground state $\phi_0(x)$ is obtained. Further, we prove the instability…
We study the instability of standing-wave solutions $e^{i\omega t}\phi_{\omega}(x)$ to the inhomogeneous nonlinear Schr\"{o}dinger equation $$i\phi_t=-\triangle\phi+|x|^2\phi-|x|^b|\phi|^{p-1}\phi, \qquad \in\mathbb{R}^N, $$ where $ b > 0 $…
Consider the focussing cubic nonlinear Schr\"odinger equation in $R^3$: $$ i\psi_t+\Delta\psi = -|\psi|^2 \psi. $$ It admits special solutions of the form $e^{it\alpha}\phi$, where $\phi$ is a Schwartz function and a positive ($\phi>0$)…
There have been several existence results for the standing waves of FitzHugh-Nagumo equations. Such waves are the connecting orbits of an autonomous second-order Lagrangian system and the corresponding kinetic energy is an indefinite…
We consider standing waves of the nonlinear Schr\"odinger equation $i\partial_t u = -\Delta_\alpha u + |u|^{p-1}u$ in the defocusing case in dimensions $N=2$ and $N=3$. Here, $-\Delta_\alpha$ denotes the Laplacian with a point interaction.…
We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities and a damping term. After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial…
In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven for $N\geq 2$ and…
We study the strong instability of standing waves $e^{i\omega t}\phi_\omega(x)$ for nonlinear Schr\"{o}dinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $\omega\in\mathbb{R}$ is a…
It is known that von Neumann-Landau wave equation can present a mathematical formalism of motion of quantum mechanics, that is an extension of Schr\"{o}dinger's wave equation. In this paper, we concern with the Dirichlet problem of the…
We study the strong instability of ground-state standing waves $e^{i\omega t}\phi_\omega(x)$ for $N$-dimensional nonlinear Schr\"odinger equations with double power nonlinearity. One is $L^2$-subcritical, and the other is…
Rotating waves are a fascinating feature of a wide array of complex systems, particularly those arising in the study of many chemical and biological processes. With many rigorous mathematical investigations of rotating waves relying on the…
We study existence and stability of standing waves for coupled nonlinear Hartree type equations \[ -i\frac{\partial}{\partial t}\psi_j=\Delta \psi_j+\sum_{k=1}^m \left(W\star |\psi_k|^p \right)|\psi_j|^{p-2}\psi_j, \] where…
This paper is concerned with the Klein-Gordon-Maxwell system in a bounded spatial domain. We discuss the existence of standing waves $\psi=u(x)e^{-i\omega t}$ in equilibrium with a purely electrostatic field $\mathbf{E}=-\nabla\phi(x)$. We…
We consider a complex Ginzburg-Landau equation, corresponding to a Gross-Pitaevskii equation with a small dissipation term. We study an asymptotic regime for long-wave perturbations of constant maps of modulus one. We show that such…
Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg-Landau equation (CGL), $$ \partial_t u = (a + i\alpha) \Delta u - (b + i \beta) |u|^\sigma u + k u, \,\,…