Nonlinear interaction of head-on solitary waves in integrable and nonintegrable systems

This study numerically investigates the nonlinear interaction of head-on solitary waves in a granular chain(a nonintegrable system)and compares the simulation results with the theoretical results in fluid(an integrable system).Three stages(the pre-in-phase traveling stage,the central-collision stage,and the post-in-phase traveling stage)are identified to describe the nonlinear interaction processes in the granular chain.The nonlinear scattering effect occurs in the central-collision stage,which decreases the amplitude of the incident solitary waves.Compared with the leading-time phase in the incident and separation collision processes,the lagging-time phase in the separation collision process is smaller.This asymmetrical nonlinear collision results in an occurrence of leading phase shifts of time and space in the post-in-phase traveling stage.We next find that the solitary wave amplitude does not influence the immediate space-phase shift in the granular chain.The space-phase shift of the post-in-phase traveling stage is only determined by the measurement position rather than the wave amplitude.The results are reversed in the fluid.An increase in solitary wave amplitude leads to decreased attachment,detachment,and residence times for granular chains and fluid.For the immediate time-phase shift,leading and lagging phenomena appear in the granular chain and the fluid,respectively.These results offer new knowledge for designing mechanical metamaterials and energy-mitigating systems.

Uniform nonintegrability of random variables

Recently, T. K. Chandra, T. -C. Hu and A. Rosalsky [Statist. Probab. Lett., 2016, 116： 27-37] introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. We introduce a weaker definition on uniform nonintegrability （W-UNI） of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of W- UNI, one of which is a W-UNI analogue of the celebrated de La Vall6e Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised by Chandra et al.

A Variable Separation Approach to Solve the Integrable and Nonintegrable Models:Coherent Structures of the (2 + 1)-Dimensional KdV Eqnation

We study the localized coherent structures ofa generally nonintegrable (2+ 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.

A Combinatorial Optimized Knapsack Linear Space for Information Retrieval

Key information extraction can reduce the dimensional effects while evaluating the correct preferences of users during semantic data analysis.Currently,the classifiers are used to maximize the performance of web-page recommendation in terms of precision and satisfaction.The recent method disambiguates contextual sentiment using conceptual prediction with robustness,however the conceptual prediction method is not able to yield the optimal solution.Context-dependent terms are primarily evaluated by constructing linear space of context features,presuming that if the terms come together in certain consumerrelated reviews,they are semantically reliant.Moreover,the more frequently they coexist,the greater the semantic dependency is.However,the influence of the terms that coexist with each other can be part of the frequency of the terms of their semantic dependence,as they are non-integrative and their individual meaning cannot be derived.In this work,we consider the strength of a term and the influence of a term as a combinatorial optimization,called Combinatorial Optimized Linear Space Knapsack for Information Retrieval(COLSK-IR).The COLSK-IR is considered as a knapsack problem with the total weight being the“term influence”or“influence of term”and the total value being the“term frequency”or“frequency of term”for semantic data analysis.The method,by which the term influence and the term frequency are considered to identify the optimal solutions,is called combinatorial optimizations.Thus,we choose the knapsack for performing an integer programming problem and perform multiple experiments using the linear space through combinatorial optimization to identify the possible optimum solutions.It is evident from our experimental results that the COLSK-IR provides better results than previous methods to detect strongly dependent snippets with minimum ambiguity that are related to inter-sentential context during semantic data analysis.

Solitary wave for a nonintegrable discrete nonlinear Schr?dinger equation in nonlinear optical waveguide arrays

We study a nonintegrable discrete nonlinear SchriSdinger （dNLS） equation with the term of nonlinear nearest-neighbor interaction occurred in nonlinear optical waveguide arrays. By using discrete Fourier transformation, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation. The analysis of stability of stationary solitary waves is performed. It is shown that the nonlinear nearest-neighbor interaction term has great influence on the form of solitary wave. The shape of solitary wave is important in the electric field propagating. If we neglect the nonlinear nearest-neighbor interaction term, much important information in the electric field propagating may be missed. Our numerical simulation also demonstrates the difference of chaos phenomenon between the nonintegrable dNLS equation with nonlinear nearest-neighbor interaction and another nonintegrable dNLS equation without the term.

The Peculiarity of Numerical Solving the Euler and Navier-Stokes Equations

The analysis of integrability of the Euler and Navier-Stokes equations shows that these equations have the solutions of two types: 1) solutions that are defined on the tangent nonintegrable manifold and 2) solutions that are defined on integrable structures (that are realized discretely under the conditions related to some degrees of freedom). Since such solutions are defined on different spatial objects, they cannot be obtained by a continuous numerical simulation of derivatives. To obtain a complete solution of the Euler and Navier-Stokes equations by numerical simulation, it is necessary to use two different frames of reference.

Rogue Waves in Nonintegrable KdV-Type Systems

It is proved that rogue waves can be found in Korteweg de-Vries（KdV） systems if real nonintegrable effects, higher order nonlinearity and nonlinear diffusion are considered. Rogue waves can also be formed without modulation instability which is considered as the main formation mechanism of the rogue waves.

Some Remarks to Numerical Solutions of the Equations of Mathematical Physics

The equations of mathematical physics, which describe some actual processes, are defined on manifolds (tangent, a companying or others) that are not integrable. The derivatives on such manifolds turn out to be inconsistent, i.e. they don’t form a differential. Therefore, the solutions to equations obtained in numerical modelling the derivatives on such manifolds are not functions. They will depend on the commutator made up by noncommutative mixed derivatives, and this fact relates to inconsistence of derivatives. (As it will be shown, such solutions have a physical meaning). The exact solutions (functions) to the equations of mathematical physics are obtained only in the case when the integrable structures are realized. So called generalized solutions are solutions on integrable structures. They are functions (depend only on variables) but are defined only on integrable structure, and, hence, the derivatives of functions or the functions themselves have discontinuities in the direction normal to integrable structure. In numerical simulation of the derivatives of differential equations, one cannot obtain such generalized solutions by continuous way, since this is connected with going from initial nonintegrable manifold to integrable structures. In numerical solving the equations of mathematical physics, it is possible to obtain exact solutions to differential equations only with the help of additional methods. The analysis of the solutions to differential equations with the help of skew-symmetric forms [1,2] can give certain recommendations for numerical solving the differential equations.

Spontaneous Emergence of Physical Structures and Observable Formations: Fluctuations, Waves, Turbulent Pulsations and So on

As it is known, the closed inexact exterior form and associated closed dual form make up a differential-geometrical structure. Such a differential-geometrical structure describes a physical structure, namely, a pseudostructure on which conservation laws are fulfilled (A closed dual form describes a pseudostructure. And a closed exterior form, as it is known, describes a conservative quantity, since the differential of closed form is equal to zero). It has been shown that closed inexact exterior forms, which describe physical structures, are obtained from the equations of mathematical physics. This process proceeds spontaneously under realization of any degrees of freedom of the material medium described. Such a process describes an emergence of physical structures and this is accompanied by an appearance of observed formations such as fluctuations, waves, turbulent pulsations and so on.

A unified model with a generalized gauge symmetry and its cosmological implications

A unified model is based on a generalized gauge symmetry with groups [Sg3c]color×（SU2×U1）X[U1b×U11]. It implies that all interactions should preserve conservation laws of baryon number, lepton number, and electric charge, etc. The baryonie U1b, leptonie U11 and color SU3o gauge transformations are generalized to involve nonintegrable phase factors. One has gauge invariant fourth-order equations for massless gauge fields, which leads to linear potentials in the [U1b × U11] and color [SUao] sectors. We discuss possible cosmological implications of the new baryonie gauge field. It can produce a very small constant repulsive force between two baryon galaxies （or between two anti-baryon galaxies）, where the baryon force can overcome the gravitational force at very large distances and leads to an accelerated cosmic expansion. Based on conservation laws in the unified model, we discuss a simple rotating dumbbell universe with equal amounts of matter and anti-matter, which may be pictured as two gigantic rotating clusters of galaxies. Within the gigantic baryonie cluster, a galaxy will have an approximately linearly accelerated expansion due to the effective force of constant density of all baryonie matter. The same expansion happens in the gigantic anti-baryonic cluster. Physical implications of the generalized gauge symmetry on charmonium confining potentials due to new SUac field equations, frequency shift of distant supernovae Ia and their experimental tests are discussed.