The three bilinearities uv, uv, uv for functions u, v : R2×[0, T ] 7−→ C are sharply estimated in function spaces Xs,b associated to the Schrödinger operator i∂t+∆. These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cu...

A numerical algorithm for the reconstruction of the permittivity of a three-dimensional penetrable object from scattering data is presented. The reconstruction algorithm is based on the local shape function method (LSF) combined with the conjugate gradient method with FFT (CGFFT). The nonlinearity due to the multiple scattering is accounted for in an iterative minimization scheme. Numerical exa...

In this paper, we consider the existence of multiple positive solutions for Kirchhoff-Schrödinger-Poisson system with the nonlinear term containing both general singularity and quasicritical nonlinearity. By combining the variational method with the perturbation method, we obtain the existence of two positive solutions with the parameter λ small enough. One of the solutions is the local minimum...

Abstract. In this article, we prove that solutions to a problem in nonlinear elasticity corresponding to small initial displacements exist globally in the exterior of a nontrapping obstacle. The medium is assumed to be homogeneous, isotropic, and hyperelastic, and the nonlinearity is assumed to satisfy a null condition. The techniques contained herein would allow for more complicated geometries...

We introduce a novel concept of surface bound states in the continuum, i.e., surface modes embedded into the linear spectral band of a discrete lattice. We suggest an efficient method for creating such surface modes and the local bounded potential necessary to support the embedded modes. We demonstrate that the surface embedded modes are structurally stable, and the position of their eigenvalue...

We study the effective third-order nonlinear susceptibility of a graded metal–dielectric composite film of anisotropic particles with weak nonlinearity by invoking the local field effects within the Maxwell–Garnett theory. We further numerically demonstrate that the film can serve as a novel optical material to produce a broad structure in both the linear and the nonlinear response and an enhan...

We study the Cauchy problem for the 1-d periodic fractional Schrödinger equation with cubic nonlinearity. In particular we prove local well-posedness in Sobolev spaces, for solutions evolving from rough initial data. In addition we show the existence of global-in-time infinite energy solutions. Our tools include a new Strichartz estimate on the torus along with ideas that Bourgain developed in ...

We consider quasilinear parabolic nondiagonal systems of equations with additional terms quadratic nonlinearity in the gradient. study local continuity solutions that are not necessarily bounded under optimal smoothness conditions on principal matrix and one-sided strongly nonlinear terms.

<p style='text-indent:20px;'>The purpose of this paper is threefold. Firstly, we establish a Gagliardo-Nirenberg inequality with optimal constant, which involves fractional norm and an inhomogeneous nonlinearity. Secondly, as application inequality, study ground state standing waves to nonlinear Schrödinger equation (NLS) mixed Laplacians nonlinearity, consider minimization problem gives ...