Related papers: Validity of the hyperbolic Whitham modulation equa…
We establish global-in-time frequency localized local smoothing estimates for Schr\"odinger equations on hyperbolic space $\mathbb{H}^d$. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent,…
In the framework of Gurevich and Pitaevskii approach [1] we construct modulated by Whitham [2] solution of nonlinear Shrodinger (NS) equation partially saturating the modulational instability. This solution describes new scenario of…
We present a large family of {\it{exact}} solitary wave solutions of the one dimensional Gross-Pitaevskii equation, with time-varying scattering length and gain/loss, in both expulsive and regular parabolic confinement regimes. The…
We prove asymptotic completeness in the energy space for the nonlinear Schrodinger equation posed on hyperbolic space in the radial case, in space dimension at least 4, and for any energy-subcritical, defocusing, power nonlinearity. The…
We establish the local Hadamard well-posedness of a certain third-order nonlinear Schr\"odinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schr\"odinger equation, formulated…
The existence of hyperbolic orbits is proved for a class of restricted three-body problems with a fixed energy by taking limit for a sequence of periodic solutions which are obtained by variational methods.
We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO-type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to…
It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics --- amplitude, phase, wave number, etc. --- slowly…
We establish the unique solvability of solutions in Sobolev spaces to linear parabolic equations in a more general form than those in the literature. A distinguishing feature of our equations is the inclusion of a half-order time derivative…
We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces ${\mathrm{H}^s}$, ${ s > 0 }$, to a class of nonlinear, dispersive evolution equations of the form \begin{equation*} u_t + \left(Lu+ n(u)\right)_x =…
It is often argued that measurable predictions of Bohmian mechanics cannot be distinguished from those of a theory with arbitrarily modified particle velocities satisfying the same equivariance equation. By considering the wave function of…
We prove the existence of quasi-periodic solutions for Schroedinger equations with a multiplicative potential on T^d, d \geq 1, merely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The…
We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases} &(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases} \end{equation*} on the…
We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the…
The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity $f(u) = \tfrac{1}{2}u^2$ of the Korteweg-de Vries equation and the full linear dispersion relation $\Omega(k) = \sqrt{k\tanh…
We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small…
The paper considers a class of linear Boltzmann transport equations which models a charged particle transport. The equation is an approximation of the original exact transport equation which involves hyper-singular integrals in their…
We consider small amplitude wave packet-like solutions to the 3D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Formal multiscale calculations suggest that the modulation of such a…
We consider the small time semi-classical limit for nonlinear Schrodinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To…
We consider nonlinear perturbations of the hyperbolic equation in the Hilbert space. Necessary and sufficient conditions for the existence of solutions of boundary-value problem for the corresponding equation and iterative procedures for…