Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrödinger type equation on torus

We consider the time local and global well-posedness for the fourth order nonlinear Schrödinger type equation (4NLS) on the torus. The nonlinear term of (4NLS) contains the derivatives of unknown function and this prevents us to apply the classical energy method. To overcome this difficulty, we introduce the modified energy and derive an a priori estimate for the solution to (4NLS).     

Exact soliton solutions of a D-dimensional nonlinear Schrödinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schrödinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

Two-dimensional solitons in irregular lattice systems

We compute and study localized nonlinear modes (solitons) in the semi-infinite gap of the focusing two-dimensional nonlinear Schrödinger (NLS) equation with various irregular lattice-type potentials. The potentials are characterized by large variations from periodicity, such as vacancy defects, edge dislocations, and a quasicrystal structure. We use a spectral fixed-point computational scheme to obtain the solitons. The eigenvalue dependence of the soliton power indicates parameter regions of self-focusing instability; we compare these results with direct numerical simulations of the NLS equation. We show that in the general case, solitons on local lattice maximums collapse. Furthermore, we show that the Nth-order quasicrystal solitons approach Bessel solitons in the large-N limit.     

Solutions of multidimensional partial differential equations representable as a one-dimensional flow

We propose an algorithm for reducing an (M+1)-dimensional nonlinear partial differential equation (PDE) representable in the form of a one-dimensional flow u_t + $w_{x_1 } $ (u, u_x u_xx,…) = 0 (where w is an arbitrary local function of u and its xi derivatives, i = 1,…, M) to a family of M-dimensional nonlinear PDEs F(u,w) = 0, where F is a general (or particular) solution of a certain second-order two-dimensional nonlinear PDE. In particular, the M-dimensional PDE might turn out to be an ordinary differential equation, which can be integrated in some cases to obtain explicit solutions of the original (M+1)-dimensional equation. Moreover, a spectral parameter can be introduced in the function F, which leads to a linear spectral equation associated with the original equation. We present simplest examples of nonlinear PDEs together with their explicit solutions.     

Numerical study of the transverse stability of the Peregrine solution

We generalize a previously published numerical approach for the one-dimensional (1D) nonlinear Schrödinger (NLS) equation based on a multidomain spectral method on the whole real line in two ways: first, a fully explicit fourth-order method for the time integration, based on a splitting scheme and an implicit Runge-Kutta method for the linear part, is presented. Second, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the 1D NLS equation and thus a y-independent solution to the 2D NLS. It is shown that the Peregine solution is unstable agains all standard perturbations, and that some perturbations can even lead to a blow-up for the elliptic NLS equation.     

Chaotic motions for a perturbed nonlinear Schrödinger equation with the power-law nonlinearity in a nano optical fiber

Perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano optical fiber is studied with the help of its equivalent two-dimensional planar dynamic system and Hamiltonian. Via the bifurcation theory and qualitative theory, equilibrium points for the two-dimensional planar dynamic system are obtained. With the external perturbation taken into consideration, chaotic motions for the perturbed NLS equation with the power-law nonlinearity are derived based on the equilibrium points.     

Solitary waves and a stability analysis of an equation of short and long dispersive waves

A universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of Korteweg–de Vries (KdV) type is coupled to an equation of nonlinear Schrödinger (NLS) type. The system has solutions of steady form in which one component is like a solitary-wave solution of the KdV equation and the other component is like a ground-state solution of the NLS equation. We study the stability of solitary-wave solutions to an equation of short and long waves by using variational methods based on the use of energy–momentum functionals and the techniques of convexity type. We use the concentration compactness method to prove the existence of solitary waves. We prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed.     

非线性弹性杆波动方程的摄动分析

针对计入横向惯性效应后的非线性弹性杆纵向波动方程进行了分析．在小振幅、长波长的一般情况下,根据远方场简单波理论,采用约化摄动法,得到了NLS方程,并讨论了存在NLS孤子的条件．     

Nonlinear scattering theory for a class of wave equations in H

For the semi-linear (higher order) wave equation and the nonlinear (higher order) Schrödinger equation, we show that the scattering operators map a band in H^s into H^s if the nonlinearities have (sub-)critical powers in H^s. The smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing NLS equation are also shown provided that the nonlinearities have subcritical growth in H¹. Moreover, the spatial decaying behavior of solutions in energy space for the defocusing NLS equation are obtained.     

A conservative numerical method for the fractional nonlinear Schrödinger equation in two dimensions

Science China Mathematics - This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrödinger (NLS) equation involving fractional Laplacian....     

Exact soliton solutions of a D-dimensional nonlinear Schr?dinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schr?dinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.     

Stability of Classes of Mappings and Holder Continuity of Higher Derivatives of Elliptic Solutions to Systems of Nonlinear Differential Equations

Nirenberg published the following well-known result in 1954: Let a function z be a twice continuously differentiable solution to a nonlinear second-order elliptic equation. Suppose that the function F defining the equation is continuous and has continuous first-order partial derivatives with respect to all of its arguments (i.e., independent together with z and the symbols of all first- and second-order partial derivatives of z). Then the partial derivatives of z are locally Holder continuous. Simultaneously with Nirenberg, Morrey obtained an analogous result for elliptic systems of second-order nonlinear equations. In this article, we get the same result for the higher derivatives of elliptic solutions to systems of nonlinear partial differential equations of arbitrary order and a rather general shape. The proof is based on the results of the author's recent research on the study of the stability phenomena in the C ^l-norm of classes of mappings.     

Statistical Approach to Modulational Instability in Nonlinear Discrete Systems

We use a statistical approach to investigate the modulational instability (Benjamin-Feir instability) in several nonlinear discrete systems: the discrete nonlinear Schrodinger (NLS) equation, the Ablowitz-Ladik equation, and the discrete deformable NLS equation. We derive a kinetic equation for the two-point correlation function and use a Wigner-Moyal transformation to write it in a mixed space-wave-number representation. We perform a linear stability analysis of the resulting equation and discuss the obtained integral stability condition using several forms of the initial unperturbed spectrum (Lorentzian and δ-spectrum). We compare the results with the continuum limit (the NLS equation) and with previous results.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 56–63, July, 2005.     

Relaxation of solitons in nonlinear Schrödinger equations with potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L²-unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.     

Schwarz symmetrization and comparison results for nonlinear elliptic equations and eigenvalue problems

We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the existence and bound of solutions for some nonlinear equation. Moreover, for some nonlinear problems, we show that if the first p-eigenvalue of a domain is big, the supremum of a solution related to this domain is close to zero. For that we obtain L ^∞ estimates for solutions of nonlinear and eigenvalue problems in terms of other L ^p norms.     

Exactly Solvable Model for Nonlinear Pulse Propagation in Optical Fibers

The nonlinear Schrödinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation, which was first derived by means of bi-Hamiltonian methods in [ 1 ]. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as a model for nonlinear pulse propagation in monomode optical fibers when certain higher-order nonlinear effects are taken into account; (b) We show that the equation is equivalent, up to a simple change of variables, to the first negative member of the integrable hierarchy associated with the derivative NLS equation; (c) We analyze traveling-wave solutions.     

Two-gap elliptic solutions to integrable nonlinear equations

We study spectral surfaces associated with elliptic two-gap solutions to the nonlinear Schrödinger equation (NLS), the Korteweg-de Vries equation (KdV), and the sine-Gordon equation (SG). It is shown that elliptic solutions to the NLS and SG equations, as well as solutions to the KdV equation elliptic with respect tot, can be assigned to any hyperelliptic surface of genus 2 that forms a covering over an elliptic surface.     

Nonlinear semigroup approach to transport equations with delayed neutrons

This paper deal with a nonlinear transport equation with delayed neutron and general boundary conditions. We establish, via the nonlinear semigroups approach, the existence and uniqueness of the mild solution, weak solution, strong solution and local solution on L~p-spaces(1 ≤ p +∞). Local and non local evolution problems are discussed.