Direct search for exact solutions to the nonlinear Schrödinger equation

A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schrödinger equation is clearly exhibited during the solution process.     

Well-posedness for the nonlocal nonlinear Schrödinger equation

We establish local well-posedness for small initial data in the usual Sobolev spaces Hs(R), s?1, and global well-posedness in H¹(R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equation

     

The correct traveling wave solutions for the high-order dispersive nonlinear Schrödinger equation

The high-order dispersive nonlinear Schrödinger equation is considered. The exact solutions were obtained by Zhang et al. [J.L. Zhang, M.L. Wang, X.Z. Li, Phys. Lett. A 357 (2006) 188-195] are analyzed. We can demonstrate that some solutions do not satisfy this equation. To obtain the correct solutions, the F-expansion method is applied to solve it.     

Evolution of solitary waves for a perturbed nonlinear Schrödinger equation

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of solitary wave tails occur, the location of zeros and the location and amplitude of peaks, in the solitary wave tail. Two examples, the near-continuum limit of a discrete NLS equation and an explicit numerical scheme for the NLS equation, are considered in detail. For the discrete NLS equation it is found that three qualitatively different types of solitary wave tail can occur, while for the explicit finite-difference scheme, only one type of solitary wave tail occurs. An excellent comparison between the perturbation solution and numerical simulations, for the solitary wave tail, is found for both examples.     

A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations

In an abstract setting we prove a nonlinear superposition principle for zeros of equivariant vector fields that are asymptotically additive in a well-defined sense. This result is used to obtain multibump solutions for two basic types of periodic stationary Schrödinger equations with superlinear nonlinearity. The nonlinear term may be of convolution type. If the superquadratic term in the energy functional is convex, our results also apply in certain cases if 0 is in a gap of the spectrum of the Schrödinger operator.     

Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension

It is shown that there are plenty of quasi-periodic solutions of nonlinear Schrödinger equations of higher spatial dimension, where the dimension of the frequency vectors of the quasi-periodic solutions are equal to that of the space.     

Exact solutions to nonlinear Schrödinger equation with variable coefficients

According to Ma-Fuchsseiter’s idea, a trial equation method was proposed to find the exact envelop traveling wave solutions to some nonlinear differential equations with variable coefficients. As an application, combining with the complete discrimination system for polynomial, some exact envelop traveling wave solutions to Schrödinger equation with variable coefficients were obtained. At the same time, the physical meanings of the obtained solutions are discussed, and the problem needed to further study is pointed out.     

Stability of small periodic waves for the nonlinear Schrödinger equation

The nonlinear Schrödinger equation possesses three distinct six-parameter families of complex-valued quasiperiodic traveling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x−ct for some c∈R. In this paper we investigate the stability of the small amplitude traveling waves, both in the defocusing and the focusing case. Our first result shows that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude traveling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability.     

Classification of optical wave solutions to the nonlinearly dispersive Schrödinger equation

Different kinds of optical wave solutions to the nonlinearly dispersive Schrödinger equation are given according to different parameters’ regions. Those solutions include looped wave solutions, cusped wave solutions, peaked wave solutions, compacted wave solutions. The looped and cusped forms have not been reported in the literature regarding to the study of the nonlinear Schrödinger equation. We also study the limiting behavior of all periodic solutions as the parameters trend to some special values.     

Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε²Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions u_ε concentrate along minimum points of the unbounded manifold of critical points of V.     

Exact explicit traveling wave solutions for two nonlinear Schrödinger type equations

The traveling wave solutions of the generalized nonlinear derivative Schrödinger equation and the high-order dispersive nonlinear Schrödinger equation are studied by using the approach of dynamical systems and the theory of bifurcations. With the aid of Maple, all bifurcations and phase portraits in the parametric space are obtained. All possible explicit parametric representations of the bounded traveling wave solutions (solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions) are given.     

Exact solutions of the nonlinear Schrödinger equation by the first integral method

The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the nonlinear Schrödinger equation.     

Characterization of soliton solutions in 2D nonlinear Schrödinger lattices by using the spatial disorder

In this paper, the pattern of the soliton solutions to the discrete nonlinear Schrödinger (DNLS) equations in a 2D   lattice is studied by the construction of horseshoes in _l∞$l_{\infty }$-spaces. The spatial disorder of the DNLS equations is the result of the strong amplitudes and stiffness of the nonlinearities. The complexity of this disorder is log(N+1)$\log (N+1)$ where N   is the number of turning points of the nonlinearities. For the case N=1$N=1$, there exist disjoint intervals _I0$I_0$ and _I1$I_1$, for which the state _um,n$u_{m,n}$ at site (m,n)$(m,n)$ can be either dark (_um,n∈_I0$u_{m,n}\in I_0$) or bright (_um,n∈_I1$u_{m,n}\in I_1$) that depends on the configuration _km,n=0$k_{m,n}=0$ or 1, respectively. Bright soliton solutions of the DNLS equations with a cubic nonlinearity are also discussed.     

Symbolic computation on the Darboux transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation from fiber optics

Considering the propagation of ultrashort pulse in the realistic fiber optics, a generalized variable-coefficient higher-order nonlinear Schrödinger equation is investigated in this paper. Under certain constraints, a new 3×3 Lax pair for this equation is obtained through the Ablowitz-Kaup-Newell-Segur procedure. Furthermore, with symbolic computation, the Darboux transformation and nth-iterated potential transformation formula for such a model are explicitly derived. The corresponding features of ultrashort pulse in inhomogeneous optical fibers are graphically discussed by the one- and two-soliton-like solutions.     

Strichartz estimates for the magnetic Schrödinger equation

We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n?6.     

New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation

The repeated homogeneous balance is used to construct a new exact traveling wave solution of the Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can be applied to other nonlinear evolution equations.     

On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations

The existence and concentration behavior of a nodal solution are established for the equation

     

Blow-up solutions to the nonlinear Schrödinger equation with oscillating nonlinearities

In this article we investigate the possibility of finite time blow-up in H¹(R²) for solutions to critical and supercritical nonlinear Schrödinger equations with an oscillating nonlinearity. We prove that despite the oscillations some solutions blow up in finite time. Conversely, we observe that for a given initial data oscillations can extend the local existence time of the corresponding solution.     

Exact boundary controllability of the nonlinear Schrödinger equation

This paper studies the exact boundary controllability of the semi-linear Schrödinger equation posed on a bounded domain Ω⊂Rn with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if