Related papers: An explicitly solvable NLS model with discontinuou…
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 determine the $N$-particle stationary states of a staggered stochastic hopping model with reflective boundaries. It is shown that the stationary states are in fact so-called optimum ground states. Recursion relations in the particle…
We present an exact solution of an experimentally realizable and strongly interacting one-dimensional spin system which is a limiting case of a quantum Ising model with long range interaction in a transverse and longitudinal field.…
We present numerical simulations of the defocusing nonlinear Schrodinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy…
In this paper we consider the logarithmic Schr\"{o}dinger equation on a star graph. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. Then we show the…
The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state--one that minimizes the energy and satisfies the…
We consider the following nonlinear Schr\"{o}dinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+\Delta u+|u|^\frac{4}{3}u+\mu\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where…
In this work we consider the initial-value problem associated with a coupled system of generalized Korteweg-de Vries equations. We present a relationship between the best constant for a Gagliardo-Nirenberg type inequality and a criterion…
In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the…
We consider a Keller-Segel model with non-linear porous medium type diffusion and nonlocal attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen…
We look for ground state solutions to the Schr\"odinger-type system \[ \begin{cases} -\Delta u_j + \lambda_j u_j = \partial_jF(u)\\ \int_{\rn} u_j^2 \, dx = a_j^2\\ (\lambda_j,u_j) \in \mathbb{R} \times H^1(\mathbb{R}^N) \end{cases} j \in…
We consider a half-soliton stationary state of the nonlinear Schrodinger equation with the power nonlinearity on a star graph consisting of N edges and a single vertex. For the subcritical power nonlinearity, the half-soliton state is a…
We analyse the existence and the stability of the ground states of the one-dimensional nonlinear Schr\"{o}dinger equation with a focusing power nonlinearity and a defect located at the origin. In this paper a ground state is defined as a…
The nonlinear Hartree equation (NLH) in the Heisenberg picture admits steady states of the form $\gamma_f=f(-\Delta)$ representing quantum states of infinitely many particles. In this article, we consider the time evolution of perturbations…
We study standing waves for the nonlinear Schr\"odinger equation on a discrete graph. We characterize for a self-adjoint realizations of Schr\"odinger operators conditions related with the geometry of the graph that guarantee discreteness…
In this paper we show the existence of ground-state solutions for the energy-critical NLS perturbed with subcritical terms when the space dimension $d\geq4$. However in dimension three, we show that when the perturbation is small enough,…
We consider the focusing nonlinear Schr\"odinger equation with inverse square potential \[ i\partial_t u + \Delta u + c|x|^{-2} u = - |u|^\alpha u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d \geq…
We discuss the problem of establishing the existence of the Ground States for the subcritical focusing Nonlinear Schr\"odinger energy on a domain made of a line and a plane intersecting at a point. The problem is physically motivated by the…
We study the existence and stability of the standing waves for the periodic cubic nonlinear Schr\"odinger equation with a point defect determined by a periodic Dirac distribution at the origin. This equation admits a smooth curve of…
We study the nonlinear Schr\"odinger equation arising in dipolar Bose-Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is added. In both…