Related papers: Approximation theorems for the Schr\"odinger equat…
Using two innovations, smooth, but distinctly different, scaling laws for the numerical reconnection of pairs of initially orthogonal and anti-parallel quantum vortices are obtained using the three-dimensional Gross-Pitaevskii equations,…
It is shown, that any sufficiently smooth periodic solution of the self-focusing Nonlinear Schr\"odinger equation can be approximated by periodic finite-gap ones with an arbitrary small error. As a corollary an analogous result for the…
We statistically study vortex reconnections in quantum fluids by evolving different realizations of vortex Hopf links using the Gross--Pitaevskii model. Despite the time-reversibility of the model, we report a clear evidence that the…
Since the Ginzburg-Landau theory is concerned with macroscopic phenomena, and gravity affects how objects interact at the macroscopic level. It becomes relevant to study the Ginzburg-Landau theory in curved space, that is, in the presence…
We prove the convergence in Zhidkov spaces of the first-order Lie-Trotter and the second-order Strang splitting schemes for the time integration of the Gross-Pitaesvkii equation with a time-dependent potential and non-zero boundary…
We study numerically the reconnection of quantized vortices and the concurrent acoustic emission by the analysis of the Gross-Pitaevskii equation. Two quantized vortices reconnect following the process similar to classical vortices; they…
We investigate the asymptotic behavior at time infinity of solutions close to a non-zero constant equilibrium for the Gross-Pitaevskii (or Ginzburg-Landau Schroedinger) equation. We prove that, in dimensions larger than 3, small…
In this paper, we consider the concentration property of solutions to the dispersive Ginzburg-Landau (or Gross-Pitaevskii) equation in three dimensions. On a spatial domain, it has long been conjectured that such a solution concentrates…
We justify the validity of the discrete nonlinear Schrodinger equation for the tight-binding approximation in the context of the Gross-Pitaevskii equation with a periodic potential. Our construction of the periodic potential and the…
We develop a modified semi-classical approach to the approximate solution of Schrodinger's equation for certain nonlinear quantum oscillations problems. At lowest order, the Hamilton-Jacobi equation of the conventional semi-classical…
We build a smooth time-dependent real potential on the two-dimensional torus, decaying as time tends to infinity in Sobolev norms along with all its time derivative, and we exhibit a smooth solution to the associated Schr\"odinger equation…
An insight into vortex reconnections in superfluids is presented making use of analytical results and numerical simulations of the Gross--Pitaevskii model. Universal aspects of the reconnection process are investigated by considering…
In this paper, we mainly consider the global solvability of smooth solutions for the Cauchy problem of the three-dimensional Landau-Lifshitz-Slonczewski equation in the Morrey space. We derive the covariant complex Ginzburg-Landau equation…
The extended Painlev\'e P.D.E. system $\Delta y -x_1 y - 2 |y|^2y=0$, $(x_1,\ldots,x_n)\in \mathbb{R}^n$, $y:\mathbb{R}^n\to\mathbb{R}^m$, is obtained by multiplying by $-x_1$ the linear term of the Ginzburg-Landau equation $\Delta…
Quantum turbulence that exhibits vortex creation, annihilation and interactions is demonstrated as an exact solution of the time-dependent, free-particle Schr\"odinger equation evolved from a smooth random-phased initial condition. Relaxed…
Quantized vortices in a complex wave field described by a defocusing nonlinear Schr\"odinger equation with a space-varying dispersion coefficient are studied theoretically and compared to vortices in the Gross-Pitaevskii model with external…
We present numerical solutions of the Gross--Pitaevskii equation corresponding to reconnecting vortex lines. We determine the separation of vortices as a function of time during the approach to reconnection, and study the formation of…
We prove almost global existence for supercritical nonlinear Schr\"odinger equations on the $d$-torus ($d$ arbitrary) on the good geometry selected in part I. This is seen as the Cauchy consequence of I, since the known invariant measure of…
We study the solution theory of the nonlinear Schr\"odinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in $H^1$. Our…
In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schr\"odinger equations driven by a multiplicative $Q$-Wiener process. Beyond the uniform…