Conserved energies for the one dimensional Gross-Pitaevskii equation

We prove the global-in-time well-posedness of the one dimensional Gross-Pitaevskii equation in the energy space, which is a complete metric space equipped with a newly introduced metric and with the energy norm describing the $H^s$ regularities of the solutions. We establish a family of conserved energies for the one dimensional Gross-Pitaevskii equation, such that the energy norms of the solutions are conserved globally in time. This family of energies is also conserved by the complex modified Korteweg-de Vries flow

The well-posedness issue in endpoint spaces for an inviscid low-Mach number limit system

The present paper is devoted to the well-posedness issue for a low-Mach number limit system with heat conduction but no viscosity. We will work in the framework of general Besov spaces $B^s_{p,r}(\R^d)$, $d\geq 2$, which can be embedded into the class of Lipschitz functions. Firstly, we consider the case of $p\in[2,4]$, with no further restrictions on the initial data. Then we tackle the case of any $p\in\,]1,\infty]$, but requiring also a finite energy assumption. The extreme value $p=\infty$ can be treated due to a new a priori estimate for parabolic equations. At last we also briefly consider the case of any $p\in ]1,\infty[$ but with smallness condition on initial inhomogeneity. A continuation criterion and a lower bound for the lifespan of the solution are proved as well. In particular in dimension 2, the lower bound goes to infinity as the initial density tends to a constant.Comment: This work was superseded by arXiv:1403.0960 and arXiv:1403.096

Conserved energies for the one dimensional Gross–Pitaevskii quation: low regularity case

We construct a family of conserved energies for the one dimensional Gross-Pitaevskii equation, but in the low regularity case (in [14] we have constructed conserved energies in the high regularity situation). This can be done thanks to regularization procedures and a study of the topological structure of the finite-energy space. The asymptotic (regularised conserved) phase change on the real line with values in $\mathbb{R}/2\pi\mathbb{Z}$ is studied. We also construct a conserved quantity, the renormalized momentum $H_1$ (see Theorem 1.3), on the universal covering space of the finite-energy space

Eigenvalue analysis of the Lax operator for the one-dimensional cubic nonlinear defocusing Schr\"odinger equation

We analyze the eigenvalues of the Lax operator associated to the one-dimensional cubic nonlinear defocusing Schr\"odinger equation. With the help of a newly discovered unitary matrix, it reduces to the study of a unitarily equivalent operator, which involves only the amplitude and the phase velocity of the potential. For a specific kind of potentials which satisfy nonzero boundary conditions, the eigenvalues of the Lax operator are characterized via a family of compact operators