Stability of nonlinear Schrödinger equations on modulation spaces

We consider the Cauchy problem for a family of SchrSdinger equations with initial data in modulation spaces Mp,1^s. We develop the existence, uniqueness, blowup criterion, stability of regularity, scattering theory, and stability theory.     

Small solutions to nonlinear Schrödinger equations in the Sobolev spaces

The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrödinger equations is studied. The global well-posedness of the problem is proved in the Sobolev spaceH ^s=H^s(R ⁿ) of fractional orders>n/2 under the following assumptions. (1) Concerning the Cauchy data ?∈H ^s: ‖?;L ²‖ is relatively small with respect to ‖?;H ^σ‖ for any fixed σ withn/2<σ≤s. (2) Concerning the nonlinearityf: f(u) behaves as a conformal poweru ¹⁺⁴/n near zero and has an arbitrary growth rate at infinity.     

Local well-posedness of the nonlinear Schrödinger equations on the sphere for data in modulation spaces

In this paper, the authors discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schrödinger on the sphere S². Exploring suitable a priori estimates, the authors prove the existence of solution for data whose regularity is s = 1/4.     

Weighted Strichartz estimates for the Schrödinger and wave equations on Damek–Ricci spaces

We prove Strichartz estimates for radial solutions of the Schrödinger and wave equations on Damek–Ricci spaces, and in particular on symmetric spaces of noncompact type and rank one, using the perturbative theory with potentials. The curvature of the noncompact manifold has an influence on the dispersive properties, and indeed we obtain Strichartz estimates with weights at spatial infinity, which are stronger than the standard ones in the flat case.     

Multiple solutions for non-homogeneous degenerate Schrödinger equations in cone Sobolev spaces

The present paper deals with the study of semilinear and non-homogeneous Schrödinger equations on a manifold with conical singularity. We provide a suitable constant by Sobolev embedding constant and for p ∈ (2, 2?) with respect to non-homogeneous term g(x) ∈ L ₂ ^n/2 (B), which helps to find multiple solutions of our problem. More precisely, we prove the existence of two solutions to the problem 1.1 with negative and positive energy in cone Sobolev space H _2,0 ^1,n/2 (B). Finally, we consider p = 2 and we prove the existence and uniqueness of Fuchsian-Poisson problem.     

Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations

We investigate the existence of localized sign-changing solutions for the semiclassical nonlinear Schrödinger equation

$$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1(\mathbb {R}^N) \end{aligned}$$

where \(N\ge 2,\) \(2<p<2^*\), \(\epsilon >0\) is a small parameter, and V is assumed to be bounded and bounded away from zero. When V has a local minimum point P, as \(\epsilon \rightarrow 0\), we construct an infinite sequence of localized sign-changing solutions clustered at P and these solutions are of higher topological type in the sense that they are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. It has been an open question whether the sign-changing solutions of higher topological type can be localized and our result gives an affirmative answer. The existing results in the literature have been subject to some geometrical or topological constraints that limit the number of localized sign-changing solutions. At a local minimum point of V, Bartsch et al. (Math Ann 338:147–185, 2007) proved the existence of N pairs of localized sign-changing solutions and D’Aprile and Pistoia (Ann Inst Hénri Poincare Anal Non Linéaire 26:1423–1451, 2009) constructed 9 pairs of localized sign-changing solutions for \(N\ge 3\). Our result gives an unbounded sequence of such solutions. Our method combines the Byeon and Wang’s penalization approach and minimax method via a variant of the classical symmetric mountain pass theorem, and is rather robust without using any non-degeneracy conditions.

     

Remarks on some systems of nonlinear Schrödinger equations

We prove the existence of solutions of some nonautonomous systems of nonlinear Schr?dinger equations, by means of perturbation techniques. The work has been supported by M.U.R.S.T. under the national project “Variational methods and nonlinear differential equations”.     

A note on uniqueness of the ground state for nonlinear Schrödinger equations

We are concerned with the following nonlinear Schrödinger equation where , . For small enough and a class of , we show the uniqueness of the positive ground state under certain assumptions on asymptotic behavior of and its first derivatives. Here our results are suitable for a kind of which has different increasing rates at different directions.     

Nonlinear Schrödinger equations: concentration on weighted geodesics in the semi-classical limit

We consider the problem

${?}^2\mathrm{\Delta }u?V(x)u+u^p=0,u>0,u\in H^1({\mathbb{R}}^2),$

where $p>1$, $?>0$ is a small parameter and V is a uniformly positive, smooth potential. Let Γ be a closed curve, nondegenerate geodesic relative to the weighted arclength ${\int }_{\Gamma }{V}^{\sigma }$, where $\sigma =\frac{p+1}{p?1}?\frac{1}{2}$. We prove the existence of a solution $u_{\mathrm{\epsilon }}$ concentrating along the whole of Γ, exponentially small in ? at any positive distance from it, provided that ? is small and away from certain critical numbers. This proves a conjecture raised in [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Commun. Math. Phys. 235 (2003) 427–466] in the two-dimensional case. To cite this article: M. del Pino et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Homogenization of Schrödinger-type equations

We consider a self-adjoint elliptic operator A_ε, ε> 0, on L₂(R^d; Cⁿ) given by the differential expression b(D)*g(x/ε)b(D). Here \(b(D) = \sum\nolimits_{j = 1}^d {b_j D_j }\) is a first-order matrix differential operator such that the symbol b(ξ) has maximal rank. The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice. We study the operator exponential \({e^{ - i\tau {A_\varepsilon }}}\), where τ ∈ R. It is shown that, as ε → 0, the operator \({e^{ - i\tau {A_\varepsilon }}}\) converges to \({e^{ - i\tau {A^0}}}\) in the norm of operators acting from the Sobolev space H^s(R^d;Cⁿ) (with suitable s) to L₂(R^d;Cⁿ). Here A⁰ is the effective operator with constant coefficients. Order-sharp error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation i?_τu_ε(x, τ) = A_εu_ε(x, τ).     

Blow-up solutions of inhomogeneous nonlinear Schrödinger equations on torus

In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation${\mathrm{\partial }}_{t}u=i(f(x)\mathrm{\Delta }u+\nabla f(x)·\nabla u+k(x)|u{|}^{2}u)$on ${\mathbb{T}}^2$. We present the $L^2$-concentration property for general initial data and investigate the $L^2$-minimality.     

Some results on a class of nonliner Schrödinger equations

By using a Liapunov-Schmidt reduction we prove an existence result for the nonlinear Schr?dinger equation in where satisfies suitable assumptions. We also provide a necessary condition for the existence of solutions. Received June 7, 1999 / in final form November 10, 1999 / Published online July 20, 2000     

Some properties of the Schröder numbers

In the paper, the authors present some properties, including convexity, complete monotonicity, product inequalities, and determinantal inequalities, of the large Schröder numbers and find three relations between the Schröder numbers and central Delannoy numbers. Moreover, the authors sketch generalizing the Schröder numbers and central Delannoy numbers and their generating functions.     

On the new critical exponent for the nonlinear Schrödinger equations

We consider the Cauchy problem for the pth order nonlinear Schrödinger equation in one space dimension $$\left\{\begin{array}{ll}iu_{t} + \frac{1}{2} u_{xx} = |u|^{p}, x \in {\bf R}, \, t > 0, \\ \qquad u(0, x) = u_{0} (x), \; x \in {\bf R},\end{array}\right.$$ where \({p > p_{s} = \frac{3 + \sqrt{17}}{2}}\) . We reveal that p = 4 is a new critical exponent with respect to the large time asymptotic behavior of solutions. We prove that if p _s < p < 4, then the large time asymptotics of solutions essentially differs from that for the linear case, whereas it has a quasilinear character for the case of p > 4.     

Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball

We establish new results for the radial nonlinear wave and Schrödinger equations on the ball in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the corresponding Gibbs measure. This complements results from Burq and Tzvetkov (2008) [8], [9] and Tzvetkov (2006, 2008) [10], [11].