We prove a local in time well-posedness result for quasi-linear Hamiltonian Schr\"odinger equations on $\mathbb{T}^d$ any $d\geq 1$. For initial condition the Sobolev space $H^s$, with $s$ large, we existence and unicity of classical solutions Cauchy problem associated to equation. The lifespan such solution depends only size datum. Moreover continuity map.

In [BOV20] Barutello, Ortega, and Verzini introduced a non-local functional which regularizes the free fall. This has critical point at infinity therefore does not satisfy Palais-Smale condition. this article we study $L^2$ gradient flow gives rise to heat flow. We construct rich cascade Morse complex one generator in each degree $k\ge 1$. Calculation reveals rather poor homology having just ge...

We prove a unique continuation property for the fractional Laplacian $(-\Delta)^s$ when $s \in (-n/2,\infty)\setminus \mathbb{Z}$. In addition, we study Poincar\'e-type inequalities operator $s\geq 0$. apply results to show that one can uniquely recover, up gauge, electric and magnetic potentials from Dirichlet-to-Neumann map associated higher order Schr\"odinger equation. also equation with si...

In this paper we consider the Cauchy initial value problem for defocusing quintic nonlinear Schr\"odinger equation in $\mathbb{R}^2$ with general data critical space $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$. We show that if a solution remains bounded (\mathbb{R}^2)$ its maximal interval of existence, then is infinite and scatters.

This paper is devoted to studying the following nonlinear biharmonic Schr\"odinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} \Delta^{2}u-\lambda u=\mu|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\ \mathbb{R}^{N}, \end{aligned} \end{equation*} where $N\geq5$, $\mu>0$, $2<q<2+\frac{8}{N}$, $4^*=\frac{2N}{N-4}$ $H^2$-critical Sobolev exponent, and $\lambda$ appea...

We consider the nonlinear Schr\odinger equation with combined nonlinearities, where leading term is an intracritical focusing power-type nonlinearity, and perturbation given by a defocusing one. completely answer question wether ground state energy, which threshold between global existence formation of singularities, achieved. For any prescribed mass, for mass-supercritical or mass-critical per...

The observation of a wave group persisting for more than 200 periods in the direct numerical simulation nonlinear unidirectional irregular water waves deep is discussed. conditions are characterized by parameters realistic broad-banded sea. identified solution associated scattering problem Schr\"odinger equation as intense envelope soliton with remarkably stable parameters. Most extreme occur o...

We build string backgrounds for Yang-Baxter deformations of the $AdS_4\times\mathbb{CP}^3$ superstring generated by $r$-matrices satisfying classical equation. obtain metric and NS-NS two-form gravity dual corresponding to noncommutative dipole ABJM theory, as well a deformed background with Schr\"odinger symmetry. The first two may also be found TsT transformations while last we get new family...

The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ is considered. Cauchy problem shown to be globally well-posed for small initial data critical spaces and it proved that solutions scatter free as $t \to \pm \infty$. proof based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schr\"odinger equation, ii) transversal bilinear rest...