Existence of breathers for discrete nonlinear Schrödinger equations

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.     

Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations

$$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$

(0.1)

where \(N \ge 3\), \(L^{\hbar }_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential A and a scalar continuous (electric) potential V defined by

$$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$

(0.2)

Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that

$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$

and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\hbar >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as \(\hbar \rightarrow 0\).

     

Normalized solutions of nonlinear Schrödinger equations

We consider the problem $$\left\{\begin{array}{ll}-\Delta u - g(u) = \lambda u,\\ u \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} u^2 = 1, \lambda \in \mathbb{R},\end{array}\right.$$ in dimension N ≥ 2. Here g is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the L ²-unit sphere, and we show the existence of infinitely many solutions.     

Global solutions of nonlinear Schrödinger equations

We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.     

Classical solutions of nonlinear Schrödinger equations

We prove the existence of global classical solutions to the initial value problem for the nonlinear Schrödinger equation, iu_t–u+q(|u|²)u=0 in iu_t - u + (|u|²)u = in (t, x)xⁿ for 6n11.     

Shock waves in discrete nonlinear Schrödinger equations

It is shown that in nonlinear differential–difference equations, which in the continuum limit are reduced to the nonlinear Schrödinger equation, localized excitations, reminding shock waves in liquids and gasses can propagate. Such waves may have either the conventional profile of shock waves or a shape of dark pulses evolving against a background. At the initial stages of evolution the shock waves are described by the equation u_t+uu_x=0 and split out in a train of soliton-like pulses after the shock is developed.     

Standing waves for discrete nonlinear Schrödinger equations: sign-changing nonlinearities

We consider the discrete nonlinear Schrödinger equation with infinitely growing potential and sign-changing power nonlinearity. Making use of critical point theory, we prove an existence and multiplicity result for standing wave solutions.     

Blow-up solutions of inhomogeneous nonlinear Schrödinger equations

In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation $ \partial_t u = i ( f(x) \Delta u + \nabla f(x) \cdot \nabla u +k(x)|u|^2u) $ on ${\mathbb{R}}^2In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schr?dinger equation on . We present existence and non-existence results and investigate qualitative properties of the solutions when they exist. Mathematics Subject Classification (2000) 35Q55, 35G25 Dedicated respectfully to Professor Weiyue Ding on the occasion of his sixtieth birthday.     

Standing wave solutions for the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials

We investigate the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials. Making use of the critical point theory, we obtain a new result concerning the existence of a standing wave solution.     

Non-periodic discrete Schrödinger equations: ground state solutions

In this paper, we study a class of non-periodic discrete Schrödinger equations with superlinear non-linearities at infinity. Under conditions weaker than those previously assumed, we obtain the existence of ground state solutions, i.e., non-trivial solutions with least possible energy. In addition, an example is given to illustrate our results.     

Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations

Science China Mathematics - In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation $$left{ {matrix{ { -...     

Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations

We prove a Nekhoroshev type result [26,27] for the nonlinear Schr?dinger equation with vanishing or periodic boundary conditions on ; here is a parameter and is a function analytic in a neighborhood of the origin and such that , . More precisely, we consider the Cauchy problem for (0.1) with initial data which extend to analytic entire functions of finite order, and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum. The proof is obtained by a method which applies to Hamiltonian perturbations of linear PDE's with the following properties: (i) the linear dynamics is periodic (ii) there exists a finite order Birkhoff normal form which is integrable and quasi convex as a function of the action variables. Eq. (0.1) satisfies (i) and (ii) when restricted to a level surface of , which is an integral of motion. The main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here. Received June 23, 1997; in final form June 1, 1998     

Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

We study the nonlinear Schrödinger equations: \(-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).\) where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has “critical frequency” \(\inf_{x\in{\bf R}^{N}} V(x)=0\). We also assume that V(x) satisfies \(0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty\) and V(x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V(x). Their limiting profiles—which depend on the local behavior of the potential V(x)—are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u _ε(x) as follows: rescaled function \(w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)\) converges to a least energy solution of ?Δw + V ₀(y) w = w ^p , w > 0 in Ω₀, \(w\in H^{1}_0(\Omega_0)\). Here g(ε), V ₀(x) and Ω₀ depend on the local behaviors of V(x).     

Almost periodic solutions for a class of higher dimensional Schrödinger equations

In this paper, we show that there are almost periodic solutions corresponding to full dimensional invariant tori for higher dimensional Schr?odinger equations with Fourier multiplier iu_t-Δu+M_ξu+f(|u|²)u = 0, subject to periodic boundary conditions, where the nonlinearity f is a realanalytic function near u = 0 with f(0) = 0.The proof is based on an improved infinite dimensional KAM theorem.     

Multiple solutions to logarithmic Schrödinger equations with periodic potential

We study a class of logarithmic Schrödinger equations with periodic potential which come from physically relevant situations and obtain the existence of infinitely many geometrically distinct solutions.     

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.     

Soliton solutions for generalized quasilinear Schrödinger equations

By introducing a new variable replacement, we study the existence of nontrivial solutions for generalized quasilinear Schrödinger equations which appear from plasma physics, as well as high-power ultrashort laser in matter.