Related papers: Solvable cubic resonant systems
We consider a non-linear parabolic partial differential equation (PDE) on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity…
The cubic nonlinear Schr\"odinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of…
We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate $1:1$ or $1:-1$ semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite…
In this work, we investigate the modulational instability of plane wave solutions within a modified Gross-Pitaevskii equation framework. The equation features cubic and quartic nonlinearity. It models the behaviour of quasi-one-dimensional…
In quasi-exactly solvable problems partial analytic solution (energy spectrum and associated wavefunctions) are obtained if some potential parameters are assigned specific values. We introduce a new class in which exact solutions are…
We study exact solutions of the quasi-one-dimensional Gross-Pitaevskii (GP) equation with the (space, time)-modulated potential and nonlinearity and the time-dependent gain or loss term in Bose-Einstein condensates. In particular, based on…
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…
In this article we study the long-time behaviour of a system of nonlinear Partial Differential Equations (PDEs) modelling the motion of incompressible, isothermal and conducting modified bipolar fluids in presence of magnetic field. We…
We reconsider the homogeneous Faddeev-Merkuriev integral equations for three-body Coulombic systems with attractive Coulomb interactions and point out that the resonant solutions are contaminated with spurious resonances. The spurious…
We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine…
We investigate the localized nonlinear matter waves of the quasi-two dimensional Bose-Einstein condensates with spatially modulated nonlinearity in harmonic potential. It is shown that the whole Bose-Einstein condensates, similar to the…
The modulational stability of the nonlinear Schr{\"o}dinger (NLS) equation is examined in the case with a quadratic external potential. This study is motivated by recent experimental studies in the context of matter waves in Bose-Einstein…
The Gross-Pitaevskii equation is a widely used model in physics, in particular in the context of Bose-Einstein condensates. However, it only takes into account local interactions between particles. This paper demonstrates the validity of…
We consider several models (including both multidimensional ordinary differential equations (ODEs) and partial differential equations (PDEs), possibly ill-posed), subject to very strong damping and quasi-periodic external forcing. We study…
We study the well-posedness and asymptotic behaviour of selected PDE-PDE and PDE-ODE systems on one-dimensional spatial domains, namely a boundary coupled wave-heat system and a wave equation with a dynamic boundary condition. We prove…
The energy super-critical Gross--Pitaevskii equation with a harmonic potential is revisited in the particular case of cubic focusing nonlinearity and dimension d > 4. In order to prove the existence of a ground state (a positive, radially…
We consider spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant in five dimensions and analyze evolution of small perturbations of anti-de Sitter spacetime using the recently proposed resonant…
We propose a novel version of the dissipative Gross--Pitaevski equation and examine its properties. In contrast to previous proposals our approach, based on the metriplectic formulation of the dissipative system dynamics, conserves the…
Traveling waves in two-component Bose-Einstein condensates whose dynamics is described by the Manakov limit of the Gross-Pitaevskii equations are considered in general situation with relative motion of the components when their chemical…
We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results…