Related papers: Logarithmic Gross-Pitaevskii equation
This paper investigates the nonlinear Schr\"{o}dinger equation with a singular convolution potential. It demonstrates the local well-posedness of this equation in a modified Sobolev space linked to the energy. Additionally, we derive…
We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schr\"odinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the…
We study the Cauchy problem for the cubic nonlinear Schroedinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we…
Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schr\"odinger equation, with certain nonlinearities, are not unique. For any $s<0$ there exist nonzero generalized solutions varying continuously in the…
We study the Cauchy problem for a system of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the order…
We study the logarithmic Schr\"odinger equation with finite range potential on $\mathbb{R}^{\mathbb{Z}^d}$. Through a ground-state representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic…
We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized…
We consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast…
We investigate the Cauchy problem for the half wave Schr\"odinger equation in the energy space. We derive the local well-posedness in the energy space for the odd power type nonlinearities under certain additional assumption for the initial…
We consider the Cauchy problem of the two-dimensional Schr\"odinger-Poisson system in the energy class. Though the Newtonian potential diverges at the spatial infinity in the logarithmic order, global well-posedness is proven in both…
We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schr\"odinger equations with nonzero conditions at infinity (includindg the Gross-Pitaevskii and the so-called "cubic-quintic" equations) in…
We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global well-posedness in the natural energy space was proved by G\'erard \cite{Gerard}. In this paper we prove scattering for small data in the same space with some…
For a large class of nonlinear Schr\"odinger equations with nonzero conditions at infinity and for any speed $c$ less than the sound velocity, we prove the existence of nontrivial finite energy traveling waves moving with speed $c$ in any…
We consider the fractional Schrodinger equation with a logarithmic nonlinearity, when the power of the Laplacian is between zero and one. We prove global existence results in three different functional spaces: the Sobolev space…
We analyze dynamical properties of the logarithmic Schr{\"o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a…
We consider non-gauge-invariant cubic nonlinear Schr\"odinger equations in one space dimension. We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^\infty$ decay up to time…
This paper explores the cubic-quintic Schr\"odinger equation in the entire Euclidean space. Our objectives are twofold: first, to advance the understanding of unresolved issues related to this equation, which are well known in the…
The Cauchy problem for the Gross-Pitaevskii equation in three space dimensions is shown to have an unconditionally unique global solution for data of the form 1 + H^s for 5/6 < s < 1, which do not have necessarily finite energy. The proof…
We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein-Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space $H^1(\mathbb{R})…
We revisit the Cauchy problem for the logarithmic Schr\"odinger equation and construct strong solutions in $H^1$, the energy space, and the $H^2$-energy space. The solutions are provided in a constructive way, which does not rely on…