We prove the existence of standing waves to the following family of nonlinear Schrödinger equations:

ih∂_tψ = -h²Δψ + V (x)ψ - ψ|ψ|^p-2, (t, x) ∈ R × Rⁿ

provided that $h > 0$ is small, $2 < p < 2n/(n − 2)$ when $n ≥ 3$, $2 < p < ∞$ when $n = 1, 2$ and $V (x) ∈ L^∞(R^n)$ is assumed to have a sublevel with positive and finite measure.

By exploiting estimates on Bloch functions obtained in a previous paper, we prove decay estimates for Klein Gordon equations with a time independent potential periodic in space in 1D and with generic mass.

We consider a system of nonlinear Klein-Gordon equations in one space dimension with quadratic nonlinearities

(∂_t²+∂_x²+ m_j²)u_j = N_j(∂u),

j = 1, . . . , l. We show the existence of solutions in an analytic function space. When the nonlinearity satisfies a strong null condition introduced by Georgiev we prove the global existence and obtain the large time asymptotic behavior of small solutions.

This paper deals with the Cauchy problem of the semilinear wave equation with a small initial data in 2-dimensional space. When the nonlinearity is cubic, we can not expect the global existence of smooth solutions, in general. However, Godin [1] showed that if the nonlinearity has the null-form, the solution exists globally. In this paper, we will show the global solvability for the other type of nonlinearities which does not have null-form.

We consider the Cauchy problem for systems of semilinear wave equations in 2D with small initial data, and introduce a sufficient condition for global existence of small solutions. Our condition is weaker than the null condition for 2D wave equations, and it is motivated by Alinhac's condition for 3D. We also show that some global solutions under our condition are not asymptotically free.

We survey a group of joint results with different authors concerning the decay properties of evolution equations with variable coefficients. The problems studied include the wave, Schrödinger and Dirac equation, perturbed with electromagnetic potentials, and the main focus of the paper is on global dispersive and Strichartz estimates when the coefficients are of low regularity and of critical decay.

We study the orbital instability of standing wave solutions for the Klein-Gordon-Schrödinger system in three space dimensions. It is proved that the standing wave is unstable if the frequency is sufficiently small.

We discuss relations between the nonlinear Klein-Gordon equation and the nonlinear Schrödinger equation in view of the global wellposedness in the energy space and L^2. In some critical cases, we show that the global wellposedness for the former equation with some uniform bounds implies that for the latter.

Let $Tf(x, t) = e^{itφ(D)f}$ be the solution of a general dispersive equation with phase function φ and initial data $f$ in a Sobolev space. When the phase φ has a suitable growth condition and the initial data f has an angular regularity, we prove global and local L^p estimates for maximal operators generated by T. Here we do not assume the radial symmetry for the initial data. These results reveal some sufficient conditions on initial data for the boundedness of maximal operators in contrast to the negative results of [28]. We also prove a weighted L^2 maximal estimate, which is an extension of [19] to nonradial initial data.

We review the proof of existence and uniqueness of solutions of the Maxwell-Schrödinger system in a neighborhood of infinity in time, with prescribed asymptotic behaviour defined in terms of asymptotic data, without any restriction on the size of those data. That result is the basic step in the construction of modified wave operators for the Maxwell-Schrödinger system.

This article is a review of [6], which is concerned with the scattering theory for the Schrödinger-improved Boussinesq system in two space dimensions.

Precise information on the lifespan sometimes tells us how the nonlinearity affects large time behavior of solutions to nonlinear evolution equations. As pointed out by John and Hörmander in 1987, there are surprising connections between the lifespan and the null condition in the wave equation case. In this paper we give a review of analogous lifespan estimates for nonlinear Schrödinger and Klein-Gordon equations. We also discuss how this viewpoint could give a unified understanding of many previous results.

We study the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrödinger equation. We show the uniqueness of solution in C([0, T]; H^s) for the critical case or L^\infty(0, T; H^s) for the subcritical case under certain assumptions on spatial dimensions and power of nonlinearity. We do not assume the solution belongs to any auxiliary spaces associated with the Strichartz estimate. For that purpose, we also prove the estimate of product between functions and distributions and the continuity of mapping: u \to |u| on the homogeneous Sobolev or Besove space.

The initial value problem for the defocusing cubic nonlinear Schrödinger equation on ${\Bbb R}^2$ is locally well-posed in Hs for s ≥ 0. The L^2-space norm is invariant under rescaling to the equation, then the critical regularity is s = 0. In this note, we prove the global well-posedness in Hs for all s > 1/2. The proof uses the almost conservation approach by adding additional (non-resonant) correction terms to the original almost conserved energy.