Wave propagation in 1D elastic solids in presence of long-range central interactions

In this paper wave propagation in non-local elastic solids is examined in the framework of the mechanically based non-local elasticity theory established by the author in previous papers. It is shown that such a model coincides with the well-known Kröner-Eringen integral model of non-local elasticity in unbounded domains. The appeal of the proposed model is that the mechanical boundary conditions may easily be imposed because the applied pressure at the boundaries of the solid must be equilibrated by the Cauchy stress. In fact, the long-range forces between different volume elements are modelled, in the body domain, as central body forces applied to the interacting elements. It is shown that the shape change of travelling disturbances coalesces with those predicted by the non-local integral theory of elasticity in unbounded domains, but several differences arise in the case of bounded domains. The wave propagation problem has been formulated by means of the Hamiltonian functional of the proposed mechanically based model of non-local elasticity, introducing an additional term to the elastic potential energy that accounts for elastic long-range interactions. In this way, the wave equation may be obtained in a weak formulation and be further used to provide approximate analytical solutions to the governing equation in the context of standing wave analysis. An equivalent discrete point-spring model, similar to lattice-type networks, has also been introduced to show the mechanical equivalence of the non-local elastic model as well as to provide a mechanical scheme suitable for the numerical treatment of pressure waves travelling in non-local bounded domains.     

具5次强非线性项的波方程新的孤波解

提出一种新的函数变换法，并与直接积分法相结合简便地求出了Lienard方程、广义PC方程以及力学中重要的一类非线性波方程等几类具5次强非线性项的波方程的四类显示精确孤波解.本方法同样适用于求解其他具有更高次非线性项的非线性方程. 关键词： 函数变换法 孤波解 直接积分法 非线性波方程     

Unified treatment of the bound states of the Schioberg and the Eckart potentials using Feynman path integral approach

We obtain analytical expressions for the energy eigenvalues of both the Schioberg and Eckart potentials using an approximation of the centrifugal term.In order to determine the e-states solutions,we use the Feynman path integral approach to quantum mechanics.We show that by performing nonlinear space-time transformations in the radial path integral,we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system.The explicit expression of bound state energy is obtained and the associated eigenfunctions are given in terms of hypergeometric functions.We show that the Eckart potential can be derived from the Schioberg potential.The obtained results are compared to those produced by other methods and are found to be consistent.     

Auto-solitary solutions in a generalized model of gas discharge contraction

Stable auto-solitary solutions were found on the basis of three-dimensional numerical simulations within the simplest model under global constraint. The model involves a diffusion equation with a nonlinear source term containing both local and non-local nonlinearity. The source term was chosen so as to describe qualitatively the most fundamental peculiarities of discharge physics, namely local nonlinear increase in heating and ionization rate and non-local attenuation of electric field strength with plasma density growth. The properties of the autosolitons created by the model have been investigated employing the different parameters as control parameter. Therefore the results of calculations can be used to construct a process of plasma contraction in gas discharge. Received 26 July 1999 and Received in final form 5 February 2000     

Optical beams in lossy non-local Kerr media

It is discussed that optical beams propagate in non-local Kerr medium waveguides with losses. A variational principle is carried out for the 1 + 1-D non-local non-linear Schrödinger equation in the presence of the losses. In the strongly non-local case, the approximate analytical solutions are obtained. The lossy soliton solution shows that, Unlike its local counterpart, such lossy strongly non-local soliton does not possess the adiabatic property anymore. In addition, the general approximate results for non-soliton cases are gained. The comparisons between our approximate analytic solutions and numerical simulations confirm our variational approximate solutions.     

Light propagation in media with a highly nonlinear response: An analytical study

The problem of light propagation in highly nonlinear media is studied with the help of a recently introduced systematic approach to the analytical solution of equations of nonlinear optics [L.L. Tatarinova, M.E. Garcia, Exact solutions of the eikonal equations describing self-focusing in highly nonlinear geometrical optics, Phys. Rev. A 78 (2008) 021806(R)(1—4)]. Numerous particular cases of media exhibiting high-order nonlinear refractive indices are considered. We obtain analytical expressions for determining the self-focusing position and a new exact expression for calculating the filament intensity. The constructed solutions allowed us to revise a so-called self-focusing scaling law, i.e., the functional dependence of the self-focusing position on the initial light peak intensity. It was demonstrated that this dependence is governed by the form of the nonlinear refractive index and not by the laser beam shape at the boundary.     

Parameter design of a liquid-filled sloshing system

<正>The nonlinear governing equations of the liquid sloshing modals in a cylindrical storage tank are established. Through analytical analysis,the analytical expressions of the solutions of this kind of system are obtained.With different parameters,the dynamical behaviors of the solutions are different from the trivial ones.To prevent system instability,two selection principles that the stiffness equations are positive-definite and the nonlinear terms of the system are not regenerative elements are given.Meanwhile,numerical simulations are also given,which confirm the analytical results.     

Periodic solution of the generalized Rayleigh equation

The periodic solutions of a strongly cubic nonlinear oscillator whose motion is described with the generalized Rayleigh equation are studied. Approximate analytic solving methods are introduced. A new method based on homotopy and averaging is developed to determine the limit cycle motion. The obtained analytical solutions are compared with those calculated by the elliptic harmonic balance method with generalized Fourier series and Jacobian elliptic functions. Three types of cubic nonlinearity are considered: the coefficients of the linear and cubic terms are positive, the coefficient of the linear term is positive and that of the cubic term is negative and the opposite case. Comparisons of the analytical solution and numerical solution, obtained by using the Runge-Kutta method, are illustrated with examples.     

各向异性海森伯自旋链中的超椭圆函数波解

在霍尔斯坦-普里马科夫表象中研究了各向异性海森伯自旋链模型.在半经典近似条件下,考虑高阶非线性项和周期性边界条件,应用相干态求出了用雅可比椭圆函数的反函数的组合表示的超椭圆函数波解,并讨论了解的物理意义.     

New exact solutions of Burgers’ type equations with conformable derivative

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.     

Solitons of two-component Bose-Einstein condensates modulated in space and time

In this Letter we present soliton solutions of two coupled nonlinear Schrödinger equations modulated in space and time. The approach allows us to obtain solitons for a large variety of solutions depending on the nonlinearity and potential profiles. As examples we show three cases with soliton solutions: a solution for the case of a potential changing from repulsive to attractive behavior, and the other two solutions corresponding to localized and delocalized nonlinearity terms, respectively.     

Non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane

The non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane was investigated using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the jumps in displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of crack length, the distance between two collinear cracks and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement or magnetic flux singularities are present at the crack tips in a magnetoelectroelastic composite material plane. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

Zero-range interaction in arbitrary dimensions and in the presence of external forces

We analyze zero-range interactions in arbitrary dimensions and in the presence of external forces in terms of non-local separable potentials. We show that in one-, two- and three-dimensional cases this approach is equivalent to the standard method with a renormalization operator. Our analysis indicates that the method of a projection operator is also useful for studying processes occurring under the influence of additional forces. As an example we present exact solutions for a particle interacting with a static electric field.     

非定常可压等熵流非线性方程显式解析解的推导

本文对作者以前凭试凑、灵感、运气与经验得出的一系列非定常可压流动显式解析解,寻找线索,总结出其可能的推导途径,并以非定常可压等熵一维流为例,具体给出了四种新的求解方法。这些方法会对今后寻找工程热物理领域的非线性主控方程的解析解有所帮助。本文同时还给出了两个新的解析解。     

Chirped and chirpfree soliton solutions of generalized nonlinear Schrödinger equation with distributed coefficients

Using the F-expansion method, we systematically present exact solutions of the generalized nonlinear nonlinear Schrödinger equation with varying intermodal dispersion and nonlinear gain or loss. This approach allows us to obtain large variety of solutions in terms of Jacobi-elliptical and Weierstrass-elliptical functions. The chirped and unchirped spatiotemporal soliton solutions and trigonometric-function solutions have been also obtained as limiting cases. The dynamics of these spatiotemporal soliton is discussed in context of optical fiber communication. To visualize the propagation characteristics of chirp and unchirped dark-bright soliton solutions, few numerical simulations are given. It is found that wave profile of solitons depend on the group velocity dispersion and the gain or loss functions.     

Multi-cut solutions of Laplacian growth

A new class of solutions to Laplacian growth (LG) with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hele-Shaw cells. Integrals of motion for the multi-cut LG solutions in terms of singularities of the Schwarz function are found, and the dynamics of densities (jumps) on the cuts are derived. The subclass of these solutions with linear Cauchy densities on the cuts of the Schwarz function is of particular interest, because in this case the integral equation for the conformal map becomes linear. These solutions can also be of physical importance by representing oil/air interfaces, which form oil fjords with a constant opening angle, in accordance with recent experiments in a Hele-shaw cell.     

Nonlinear Double-well Schrödinger Equations in the Semiclassical Limit

We consider time-dependent Schrödinger equations with a double well potential and an external nonlinear, both local and non-local, perturbation. In the semiclassical limit, the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system. In the case of local nonlinear perturbation, we assume the spatial dimension d=1 or d=2.     

Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations

In this paper, we prove the convergence of homotopy analysis method (HAM). We also apply the homotopy analysis method to obtain approximate analytical solutions of systems of the second kind Volterra integral equations. The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method.     

Quantum random walks and piecewise deterministic evolutions

In the continuous space and time limit, we show that the probability density to find the quantum random walk (QRW) driven by the Hadamard "coin" solves a hyperbolic evolution equation similar to the one obtained for a random two-velocity evolution with spatially inhomogeneous transition rates between the velocity states. In spite of the presence of a nonlinear drift term, it is remarkable that the QRW position can easily be described in simple analytical terms. This allows us to derive the quadratic time dependence of the variance typical for the QRW.     

Optical beams in sub-strongly non-local nonlinear media: A variational solution

Discussed is the propagation of optical beams in non-local nonlinear media modelled by 1 + 1D non-local nonlinear Schrödinger equation (NNLSE). In the sub-strongly non-local case, an approximate analytical solution is obtained for an arbitrary response function by a variational approach. Described by a combination of the Jacobian elliptic functions, the solution is periodic, and its period depends on not only the input power but also the initial beam width, which is confirmed by the numerical simulation of the NNLSE.