Weighted Topological Entropy of the Set of Generic Points in Topological Dynamical Systems

This article is devoted to the investigation of the weighted topological entropy of generic points of the ergodic measures in dynamical systems. We showed that the weighted topological entropy of generic points of the ergodic measure \(\mu \) is equal to the weighted measure entropy of \(\mu ,\) which generalized the classical result of Bowen (Trans Am Math Soc 184:125–136, 1973). As an application, we also use the result to study the dimension of generic points for a class of skew product expanding maps on high dimensional tori.     

Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass

In this paper we prove the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed \(L^2\)-norm, and the stability of associated solitary waves for two classes of coupled nonlinear dispersive equations. The first problem here describes the nonlinear interaction between two Schrödinger type short waves and a generalized Korteweg-de Vries type long wave and the second problem describes the nonlinear interaction of two generalized Korteweg-de Vries type long waves with a common Schrödinger type short wave. The results here extend many of the previously obtained results for two-component coupled Schrödinger–Korteweg-de Vries systems.     

Nonlocal Nonlinear Schrödinger Equations in R 3

This paper studies a class of nonlocal nonlinear Schrödinger equations in R ³, which occurs in the infinite ion acoustic speed limit of the Zakharov system with magnetic fields in a cold plasma. The magnetic fields induce some nonlocal effects in these nonlinear Schrödinger systems, and the main goal of this paper is to understand these effects. The key is to establish some a priori estimates on the nonlocal terms generated by the magnetic field, through which we obtain various conclusions including finite time blow-ups, sharp threshold of global existence and instability of standing waves for these equations.     

Remarks on the Generalized Reflectionless Schrödinger Potentials

We discuss certain compact, translation-invariant subsets of the set \({\mathcal {R}}\) of the generalized reflectionless potentials for the one-dimensional Schrödinger operator. We determine a stationary ergodic subset of \({\mathcal {R}}\) whose Lyapunov exponent is discontinuous at a point. We also determine an almost automorphic, non-almost periodic minimal subset of \(\mathcal {R}\).     

The bifurcation and exact travelling wave solutions of (1+2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity

By using the method of dynamical systems, this paper researches the bifurcation and the exact traveling wave solutions for a (1+2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity. Exact parametric representations of all wave solutions are given.     

Stable localized spatial solitons in \mathcal {PT}-symmetric potentials with power-law nonlinearity

We derive analytical spatial soliton solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation with power-law nonlinearity in \(\mathcal {PT}\) -symmetric potentials. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulations. Moreover, some dynamical characteristics of these solutions, such as the phase switch, the power, and the transverse power-flow density, are also examined.     

Standard decomposition of expansive ergodically supported dynamics

In this work, we introduce the notion of weak quasigroups, which are quasigroup operations defined almost everywhere on some set. Then, we prove that the topological entropy and the ergodic period of an invertible expansive ergodically supported dynamical system \((X,T)\) with the shadowing property establish a sufficient criterion for the existence of quasigroup operations defined almost everywhere outside of universally null sets and for which \(T\) is an automorphism. Furthermore, we find a decomposition of the dynamics of \(T\) in terms of \(T\) -invariant weak topological subquasigroups.     

On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

We prove that if \(f:G\rightarrow G\) is a map on a topological graph G such that the inverse limit \(\varprojlim (G,f)\) is hereditarily indecomposable, and entropy of f is positive, then there exists an entropy set with infinite topological entropy. When G is the circle and the degree of f is positive then the entropy is always infinite and the rotation set of f is nondegenerate. This shows that the Anosov-Katok type constructions of the pseudo-circle as a minimal set in volume-preserving smooth dynamical systems, or in complex dynamics, obtained previously by Handel, Herman and Chéritat cannot be modeled on inverse limits. This also extends a previous result of Mouron who proved that if \(G=[0,1]\), then \(h(f)\in \{0,\infty \}\), and combined with a result of Ito shows that certain dynamical systems on compact finite-dimensional Riemannian manifolds must either have zero entropy on their invariant sets or be non-differentiable.     

The Maslov Index and Spectral Counts for Linear Hamiltonian Systems on [0, 1]

Working with general linear Hamiltonian systems on [0, 1], and with a wide range of self-adjoint boundary conditions, including both separated and coupled, we develop a general framework for relating the Maslov index to spectral counts. Our approach is illustrated with applications to Schrödinger systems on \({\mathbb {R}}\) with periodic coefficients, and to Euler–Bernoulli systems in the same context.     

Stability of a convex order one periodic solution of unilateral asymptotic type

We study a three-coupled variable-coefficient nonlinear Schrödinger equation, which describes soliton dynamics in the three-spine \(\alpha \)-helical protein with inhomogeneous effect, and analytically obtain multi-soliton solutions, whose formation originates from the dynamical balance between the dispersion owing to the resonant interaction of intrapeptide dipole vibrations and the nonlinear interaction provided by those vibrations with the local displacements of the peptide groups. Using these analytical solutions as initial solutions, we discuss the dynamical behaviors of solitonic interactions and the influence of the protein inhomogeneity on shape-changing collisions of solitons by direct numerical simulations.     

The use of Volterra series in the analysis of the nonlinear Schrödinger equation

A Volterra series analysis is used to analyse the dispersive behaviour in the frequency domain for the non-linear Schrödinger equation (NLS). It is shown that the solution of the initial value problem for the nonlinear Schrödinger equation admits a local multi-input Volterra series representation. Higher order spatial frequency responses of the nonlinear Schrödinger equation can therefore be defined in a similar manner as for lumped parameter non-linear systems. A systematic procedure is presented to calculate these higher order spatial frequency response functions analytically. The frequency domain behaviour of the equation, subject to Gaussian initial waves, is then investigated to reveal a variety of non-linear phenomena such as self-phase modulation (SPM), cross-phase modulation (CPM), and Raman effects modelled using the NLS.     

Shadowing Property, Weak Mixing and Regular Recurrence

We show that a non-wandering dynamical system with the shadowing property is either equicontinuous or has positive entropy and that in this context uniformly positive entropy is equivalent to weak mixing. We also show that weak mixing together with the shadowing property imply the specification property with a special kind of regularity in tracing (a weaker version of periodic specification property). This in turn implies that the set of ergodic measures supported on the closures of orbits of regularly recurrent points is dense in the space of all invariant measures (in particular, invariant measures in such a system form the Poulsen simplex, up to an affine homeomorphism).     

Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations

The Bäcklund transformation from the Riccati form of inverse method is presented for the Perturbed Nonlinear Schrödinger Equation. Consequently, the exact solutions for Perturbed Nonlinear Schrödinger equation can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS; that is, a linear eigenvalues problem in the form of a system of PDE. Moreover, we construct a new soliton solution from the old one and its wave function.     

On the nonlinear interactions and existence of breathers in a chain of coupled pendulums

The paper considers a chain of linearly coupled pendulums. Continues first order system equations are treated via time and space multiple scale method which lead to nonlinear Schrödinger equation. Further investigations on the nonlinear Schrödinger equation detects systems responses in the form of propagated nonlinear waves as functions of their envelope and phases. This provides information about localization of nonlinear waves and their directions in space and time.     

Long-time Instability and Unbounded Sobolev Orbits for Some Periodic Nonlinear Schrödinger Equations

We study the energy cascade problematic for some nonlinear Schrödinger equations on ${\mathbb{T}^2}$ in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schrödinger nonlinearities that are equal or closely related to the standard polynomial one ${|u|^2u}$ . Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the first Birkhoff normal form).     

Quasi-Periodic Solutions with Prescribed Frequency in Reversible Systems

In this paper, we obtain a family of small-amplitude real analytic quasi-periodic solutions for a class of derivative nonlinear Schrödinger equations, subject to Dirichlet boundary conditions, which correspond to infinite-dimensional reversible systems with critical unbounded perturbations. We prove that the frequencies of the quasi-periodic solutions, accordingly, the tangential frequencies of the invariant tori for these reversible systems can be in a fixed direction.     

Rotation Number and Exponential Dichotomy for Linear Hamiltonian Systems: From Theoretical to Numerical Results

The paper considers the rotation number for a family of linear nonautonomous Hamiltonian systems and its relation with the exponential dichotomy concept. We propose numerical techniques to compute the rotation number and we employ them to infer when a given system enjoys or not an exponential dichotomy. Comparisons with QR-based techniques for exponential dichotomy will give new insights on the structure of the spectrum for the one-dimensional quasi-periodic Schrödinger operator. Experiments on the two dimensional Schrödinger equation will be presented as well.     

Governing equations of envelopes created by nearly bichromatic waves on deep water

In this paper, the author derives the modified Schrödinger equation that governs the envelope created by nearly bichromatic waves, which are defined by the waves whose energy is almost concentrated in two closely approached wavenumbers. The stability of the solution of the modified Schrödinger equation for nearly bichromatic waves on deep water is discussed and the fact that the Benjamin–Feir instability occurs in a condition is shown. Moreover, the solutions of the modified Schrödinger equation for nearly bichromatic waves on deep water are obtained and, in a special case, the solution becomes the standing wave solution is shown.     

Universal Systems for Entropy Intervals

A topological system is universal for a class of ergodic measure-theoretic systems if its simplex of invariant measures contains, up to an isomorphism, all elements of this class and no elements from outside the class. We construct universal systems for classes given by the combination of three properties: measure-theoretic entropy belonging to a nondegenerate interval of the extended nonnegative real halfline, invertibility and aperiodicity. For classes consisting of aperiodic systems the universal system can be made minimal.