Related papers: On linear instability of solitary waves for the no…
We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schr\"odinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it…
We obtain exact solutions of the nonlinear Dirac equation in 1+1 dimension of the form $ \Psi(x,t) =\Phi(x) \rme^{-\rmi \omega t}$ where the nonlinear interactions are a combination of vector-vector and scalar-scalar interactions with the…
In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schr\"odinger system \[ i\partial_t u_{j}+\partial_{xx}u_{j}+ \left(\sum_{k=1}^{3} a_{kj}…
We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system proving the existence of solutions in which the Dirac wave function is of the form $\phi(x,\omega)e^{-i\omega t}$, $\omega\in(-m,\omega_*)$, with some…
We consider the following generalized derivative nonlinear Schr\"odinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2\sigma}\partial_xu=0,\ (t,x)\in\mathbb R\times\mathbb R \end{equation*} when $\sigma\in(0,1)$. The…
We consider the nonlinear Schr{\"o}dinger equation (NLSE) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{\kappa+1} (\psi^\star \psi)^{\kappa+1}$ in the presence of the external forcing terms of the form $r e^{-i(kx +…
We consider a semilinear Schr\"odinger equation, driven by the power degenerate second order differential operator $\nabla\cdot (|x|^{2a} \nabla), a\in (0,1)$. We construct the solitary waves, in the sharp range of parameters, as minimizers…
We review asymptotic stability of solitary waves for nonlinear dispersive equations set on the line. Our focus is threefold: first, the nonlinear Schrodinger equation; second, the notion of full asymptotic stability (which states that…
We study the orbital stablity and instability of solitary wave solutions for nonlinear Schr\"odinger equations of derivative type.
We consider the focusing nonlinear Schr\"odinger equation on a large class of rotationally symmetric, noncompact manifolds. We prove the existence of a solitary wave by perturbing off the flat Euclidean case. Furthermore, we study the…
We discuss the (in)stability of solitary waves for a quasi-linear Schr{\"o}dinger equation. The equation contains a quasi-linear term, responsible for a saturation effect, as well as a power nonlinearity. For different exponents of the…
In this paper, we study the following nonlinear Dirac equations \begin{align*} \begin{cases} -i\sum\limits_{k=1}^3\alpha_k\partial_k u+m\beta u=f(x,|u|)u+\omega u, \displaystyle \int_{\mathbb{R}^3} |u|^2dx=a^2, \end{cases} \end{align*}…
We begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schr\"odinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a {point} (or contact)…
We study the point spectrum of the nonlinear Dirac equation in any spatial dimension, linearized at one of the solitary wave solutions. We prove that, in any dimension, the linearized equation has no embedded eigenvalues in the part of the…
We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar self-interaction in the presence of external forces as well as damping of the form $\gamma^0 f(x,t) - i \mu \gamma^0 \Psi$, where both $f, \{f_j = r_i e^{i…
We show global asymptotic stability of solitary waves of the nonlinear Schr\"odinger equation in space dimension 1. Furthermore, the radiation is shown to exhibit long range scattering if the nonlinearity is cubic at the origin, or standard…
We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_t\psi = ic\hbar\sum_{k=1}^3\alpha_k\partial_k\psi - mc^2\beta \psi - M(x)\psi + f(|\psi|)\psi,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] where $V$ is a…
In this paper, we present new results regarding the orbital stability of solitary standing waves for the general fourth-order Schr\"odinger equation with mixed dispersion. The existence of solitary waves can be determined both as minimizers…
In this work, we provide two complementary perspectives for the (spectral) stability of solitary traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. One…
The stability and dynamical properties of the so-called resonant nonlinear Schr\"odinger (RNLS) equation, are considered. The RNLS is a variant of the nonlinear Schr\"odinger (NLS) equation with the addition of a perturbation used to…