Asymptotic behavior of weakly collapsing solutions of the nonlinear Schrödinger equation

The generic asymptotic behavior of a three-parameter weakly collapsing solution of a nonlinear Schrödinger equation is examined. A discrete set of zero-energy states is shown to exist. In the (A, C ₁) parameter space, there are two close lines along which the amplitude of oscillating terms is exponentially small in the parameter C ₁.     

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

Self-similar solutions of equations of the nonlinear Schrödinger type

Experimental data are presented for the temperature dependence of the conductivity of Cu: SiO₂ metal-insulator composite films containing 3-nm Cu granules. At low temperatures in the concentration range 17–33 vol % Cu, all of the conductivity curves have a temperature dependence of the form σ ∝ exp{ (T ₀/T)^1/2}, while at higher temperatures a transition is observed to an activational dependence. A numerical simulation of the conduction in a composite material shows that an explanation of the observed temperature dependence must include the Coulomb interaction and the presence of a rather large random potential. The simulation also yields the size dependence and temperature dependence of the mesoscopic scatter of the conductivities of composite conductors. It is shown that a self-selecting percolation channel of current flow is formed in the region of strong mesoscopic scatter.     

Analyticity and global existence of small solutions to some nonlinear Schrödinger equations

In this paper we will study the nonlinear Schrödinger equations: $$\begin{gathered} i\partial _t u + \tfrac{1}{2}\Delta u = \left| u \right|^2 u, (t,x) \in \mathbb{R} \times \mathbb{R}_x^n , \hfill \\ u(0,x) = \phi (x), x \in \mathbb{R}_x^n \hfill \\ \end{gathered} $$ . It is shown that the solutions of (*) exist and are analytic in space variables fort∈??{0} if φ(x) (∈H ^2n+1,2(? _x ⁿ )) decay exponentially as |x|→∞ andn≧2.     

Nondegenerate soliton solutions in certain coupled nonlinear Schrödinger systems

In this paper, we report a more general class of nondegenerate soliton solutions, associated with two distinct wave numbers in different modes, for a certain class of physically important integrable two component nonlinear Schrödinger type equations through bilinearization procedure. In particular, we consider coupled nonlinear Schrödinger (CNLS) equations (both focusing as well as mixed type nonlinearities), coherently coupled nonlinear Schrödinger (CCNLS) equations and long-wave-short-wave resonance interaction (LSRI) system. We point out that the obtained general form of soliton solutions exhibit novel profile structures than the previously known degenerate soliton solutions corresponding to identical wave numbers in both the modes. We show that such degenerate soliton solutions can be recovered from the newly derived nondegenerate soliton solutions as limiting cases.     

Properties of weakly collapsing solutions to the nonlinear Schrödinger equation

It is shown that one of the conditions for a weakly collapsing solution with zero energy produces an infinite number of functionals I _N identically vanishing on the regular solutions to the corresponding differential equation. On the parameter plane {A, C₁}, there are at least two singular lines. Along one of these lines (A/C₁=1/6), are located weakly collapsing solutions with zero energy. It is assumed that, along the second line (A/C₁=α_c), another family of weakly collapsing solutions with zero energy is located. In the domain of large values of the parameters C₁, α=A/C₁, there exists a domain of an intermediate asymptotic form, where the amplitude of oscillations of the function U grows in a large domain relative to the ξ coordinate.     

Existence of solutions for Schrödinger evolution equations

We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equationiu/t=(–1/2)u+V(t,x)u,u(0)=u ₀. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u ₀C ¹(,L ²) C ⁰(H ²( ⁿ )) foru ₀H ²( ⁿ ). The conditions are general enough to accommodate moving singularities of type x^–2+(n4) or x^–n/2+(n3).     

Optical soliton solutions of the generalized higher-order nonlinear Schrödinger equations and their applications

The propagation of the optical solitons is usually governed by the higher order nonlinear Schrödinger equations (NLSE). In optics, the NLSE modelizes light-wave propagation in an optical fiber. In this article, modified extended direct algebraic method with add of symbolic computation is employed to construct bright soliton, dark soliton, periodic solitary wave and elliptic function solutions of two higher order NLSEs such as the resonant NLSE and NLSE with the dual-power law nonlinearity. Realizing the properties of static and dynamic for these kinds of solutions are very important in various many aspects and have important applications. The obtaining results confirm that the current method is powerful and effectiveness which can be employed to other complex problems that arising in mathematical physics.     

Dark two-soliton solutions for nonlinear Schrödinger equations in inhomogeneous optical fibers

In this paper, the variable coefficient nonlinear Schrödinger equation is investigated analytically. With the bilinear method, the bilinear forms and analytic soliton solutions are obtained. Based on the obtained analytic solutions, the effect of free parameters on the control of soliton transmission is studied. Influences of second-order and third-order dispersion coefficients on dark solitons are discussed. Results in this paper would be of great significance in the generation of dark solitons.     

Limit behavior of saturated approximations of nonlinear Schrödinger equation

We consider the solutionu _?(t) of the saturated nonlinear Schrödinger equation (1) $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} u + \varepsilon \left| u \right|^{q - 1} uandu(0,.) = \varphi (.)$$ where \(N \geqslant 2,\varepsilon > 0,1 + 4/N< q< (N + 2)/(N - 2),u:\mathbb{R} \times \mathbb{R}^N \to \mathbb{C},\varphi \) , ? is a radially symmetric function inH ¹(R ^N ). We assume that the solution of the limit equation is not globally defined in time. There is aT>0 such that \(\mathop {\lim }\limits_{t \to T} \left\| {u(t)} \right\|_{H^1 } = + \infty \) , whereu(t) is solution of (1) $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} uandu(0,.) = \varphi (.)$$ For ?>0 fixed,u _?(t) is defined for all time. We are interested in the limit behavior as ?→0 ofu _?(t) fort≥T. In the case where there is no loss of mass inu _? at infinity in a sense to be made precise, we describe the behavior ofu _? as ? goes to zero and we derive an existence result for a solution of (1) after the blow-up timeT in a certain sense. Nonlinear Schrödinger equation with supercritical exponents are also considered.     

Rapidly decaying solutions of the nonlinear Schrödinger equation

We consider global solutions of the nonlinear Schrödinger equation

On concentration of positive bound states of nonlinear Schrödinger equations

We study the concentration behavior of positive bound states of the nonlinear Schrödinger equation $$ih\frac{{\partial \psi }}{{\partial t}} = \frac{{ - h^2 }}{{2m}}\Delta \psi + V\left( x \right)\psi - \gamma \left| \psi \right|^{p - 1} \psi .$$ Under certain condition ofV, we show that positive ground state solutions must concentrate at global minimum points ofV ash→0⁺; moreover, a point at which a sequence of positive bound states concentrates must be a critical point ofV. In cases thatV is radial, we prove that the positive radial solutions with least energy among all nontrivial radial solutions must concentrate at the origin ash→0⁺.     

Nature of transitions in augmented discrete nonlinear Schrödinger equations

We investigate the nature of the transitions between free and self-trapping states occurring in systems described by augmented forms of the discrete nonlinear Schr?dinger equation. These arise from an interaction between a moving quasiparticle (such as an electron or an exciton) and lattice vibrations, when the effects of nonlinearities in interaction potential and restoring force are included. We derive analytic conditions for the stability of the free state and the crossover between first- and second-order transitions. We demonstrate our results for different types of nonlinearities in the interaction potential and restoring force. We find that, depending on the type of nonlinearity, it is possible to have both first- and second-order transitions. We discuss possible hysteresis effects.     

Periodic wavetrains for systems of coupled nonlinear Schrödinger equations

Exact, periodic wavetrains for systems of coupled nonlinear Schrödinger equations are obtained by the Hirota bilinear method and theta functions identities. Both the bright and dark soliton regimes are treated, and the solutions involve products of elliptic functions. The validity of these solutions is verified independently by a computer algebra software. The long wave limit is studied. Physical implications will be assessed.     

Focusing and multi-focusing solutions of the nonlinear Schrödinger equation

A method of dynamic rescaling of variables is used to investigate numerically the nature of the focusing singularities of the cubic and quintic Schrödinger equations in two and three dimensions and describe their universal properties. The same method is applied to simulate the multi-focusing phenomena produced by simple models of saturating nonlinearities.     

Periodic solutions of nonlinear Schro^edinger type equations

By using the solutions of an auxiliary elliptic equation, a direct algebraic method is proposed to construct the exact solutions of nonlinear Schrfdinger type equations. It is shown that many exact periodic solutions of some nonlinear Schro^edinger type equations are explicitly obtained with the aid of symbolic computation, including corresponding envelope solitary and shock wave solutions.