Explicit Form Of Solution Research Articles

Exact gray multi-soliton solutions for nonlinear Schrödinger equation with variable coefficients

In this paper, the generalized nonlinear Schrödinger equation with variable coefficients is studied. First, exact gray one-soliton solution in explicit form is obtained by using the complex amplitude ansatz method. Second, the multi-soliton solutions are constructed through suitable mapping and the explicit two-soliton solution is generated. These solutions are useful in studying the propagation characteristics and interaction of dark (gray) solitons in inhomogeneous fiber system or in soliton management system. Finally, the stabilities of these solutions are discussed numerically.

Interaction between a screw dislocation and an elastic elliptical inhomogeneity with interfacial cracks

The elastic interaction between a screw dislocation and an elliptical inhomogeneity with interfacial cracks is studied. The screw dislocation may be located outside or inside the inhomogeneity. An efficient complex variable method for the complex multiply connected region is developed, and the general solutions to the problem are derived. As illustrative examples, solutions in explicit series form for complex potentials are presented in the case of one or two interfacial cracks. Image forces on the dislocation are calculated by using the Peach-Koehler formula. The influence of crack geometries and material properties on the image forces is evaluated and discussed. It is shown that the interfacial crack has a significant effect on the equilibrium position of the dislocation near an elliptical-arc interface. The main results indicate, when the length of the crack goes up to a critical value, the presence of the interfacial crack can change the interaction mechanism between a screw dislocation and an elliptical inclusion. The present solutions can include a number of previously known results as special cases.

Structure of a class of traveling waves in delayed cellular neural networks

This work investigates the structure of a class of traveling wave solutions of delayed cellular neural networks distributed in the one-dimensional integer lattice $\mathbb Z^1$. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with its two left neighbors in which one is instantaneous and the other is distributively delayed due to, for example, finite switching speed and finite velocity of signal transmission. Applying the method of step with the aid of positive roots of the corresponding characteristic function of the profile equation, we can directly figure out the solution in explicit form. We then partition the parameter space $(\alpha, \beta)$-plane into four regions such that the qualitative properties of traveling waves can be completely determined for each region. In addition to the existence of monotonic traveling wave solutions, we also find that, for certain parameters, there exist non-monotonic traveling wave solutions such as camel-like waves with many critical points.

The effect of long-range forces on the dynamics of a bar

The one-dimensional dynamic response of an infinite bar composed of a linear “microelastic material” is examined. The principal physical characteristic of this constitutive model is that it accounts for the effects of long-range forces. The general theory that describes our setting, including the accompanying equation of motion, was developed independently by Kunin (Elastic Media with Microstructure I, 1982), Rogula (Nonlocal Theory of Material Media, 1982) and Silling (J. Mech. Phys. Solids 48 (2000) 175), and is called the peridynamic theory. The general initial-value problem is solved and the motion is found to be dispersive as a consequence of the long-range forces. The result converges, in the limit of short-range forces, to the classical result for a linearly elastic medium. Explicit solutions in elementary form are given in a broad class of special cases. The most striking observations arise in the Riemann-like problem corresponding to a constant initial displacement field and a piecewise constant initial velocity field. Even though, initially, the displacement field is continuous, it involves a jump discontinuity for all later times, the Lagrangian location of which remains stationary. For some materials the magnitude of the discontinuity-jump oscillates about an average value, while for others it grows monotonically, presumably fracturing the material when it exceeds some critical level.

Dark soliton solution for higher-order nonlinear Schrödinger equation with variable coefficients

In this paper, we consider the generalized higher-order nonlinear Schrödinger equation model with variable coefficients and obtain an exact dark soliton solution in explicit form for specified soliton management conditions. As an example, we consider a soliton control system, and present the main features of the dark soliton solution. Further, the stability of the dark soliton solution is numerically studied under the finite perturbations. Especially, we discuss the integrable conditions and the results reveal that the soliton control system may relax the limitations to parametric conditions. Finally, we investigate the interaction between neighboring femtosecond dark solitons under the combined control of the group velocity dispersion, third-order dispersion and the nonlinearity.

A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients

In this paper, the generalized nonlinear Schrödinger equation with variable coefficients is considered from the integrable point of view, and an exact multi-soliton solution is presented by employing the simple, straightforward Darboux transformation based on the Lax Pair, and then one- and two-soliton solutions in explicit forms are generated. As an example, we consider the distributed amplification system, and some main features of solutions are shown. The results reveal that the combined effects of controlling both the group velocity dispersion distribution and the nonlinearity distribution can restrict the interaction between the neighboring solitons. Also, by simulating numerically, the stability of the neighboring solitons with respect to the finite perturbations is discussed in detail. Finally, under nonintegrable condition the evolution of soliton is in detail discussed.

Exact solutions of the Boltzmann—Maxwell kinetic equation

Exact solutions of the following topics are reviewed: locally Maxwell solutions, spatially uniform relaxation of a binary mixture of gases (moment solutions and a generalization of the Bobylev—Krook—Wu solutions), homoenergetic affine flows (the Galkin—Truesdell class of exact solutions), spherical outflow-inflow (the Nikol'skii transformation), dominant solutions and power solutions. New results are obtained on the second and third topics. Particular attention is devoted to the qualitative properties of the solutions, which are of fairly general interest. We mean by an exact solution a solution in explicit form, i.e. in terms of elementary or transcendental functions, of the non-linear Boltzmann—Maxwell kinetic equation (the distribution function) or the Maxwell transport equations (the moments of the distribution functions). The latter are also called moment solutions of the kinetic equation. The molecules of the gas are mainly assumed to be Maxwellian, when the coefficient of viscosity and thermal conductivity depend linearly on the temperature.

Large Uniform Expansions of Periodic Solutions to Strongly Non-Linear Evolution Equations with Odd Polynomial Non-Linearity

A new method of uniform expansions of periodic solutions to ordinary differential equations with arbitrary odd polynomial non-linearity is constructed to study quasi-harmonic processes in non-linear dynamical systems, in particular when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter. In the particular case of a single-periodic solution, this small parameter appears due to descending the amplitudes of harmonics monotonically with increasing their number. Due to uniform expansion the amplitudes of higher harmonics turn out to be rational and fractional functions in the amplitude of the first harmonic and the frequency of oscillations. We show that the method of uniform expansions is an effective tool for obtaining convergent expansions of periodic solutions in explicit form all over the domain, where periodic solutions exist, independently of the magnitude of non-linearity. In each subsequent approximation, one more higher harmonic is taken into account, with all the other harmonics being corrected. We demonstrate the effectiveness of the method on the examples of the harmonically forced Duffing oscillator; free vibrations of the oscillator with fifth-power non-linearity and mathematical pendulum.

Soliton interaction under the influence of higher-order effects

In this paper, we present exact N-soliton solution by employing simple, straightforward Darboux transformation based on the Lax pair for Hirota equation, a higher-order nonlinear Schrödinger (HNLS) equation. As examples, one- and two-soliton solutions in explicit forms are given and their properties are also analyzed. A bound solution without interaction will be theoretically predicted if one can adjust frequency shift for each soliton appropriately. Further, we obtain the approximate eigenvalues by employing two-soliton solution and discuss analytically the interaction between neighboring solitons under the influence of the higher-order effects. It is shown that the combined effects of the higher-order effects can restrain the interaction between neighboring solitons to some extent. The results are proved by directly solving HNLS equation numerically.

Potential symmetries and invariant solutions for the inhomogeneous nonlinear diffusion equation

The work reported here is a sequel to our paper (Physica A 304 (2002) 395), wherein we have obtained the invariant solutions of the generalized one-dimensional Fokker–Planck equation via the so-called potential symmetry method. Herein, the application of the same method has been carried over to the inhomogeneous nonlinear diffusion (INLD) equation to obtain the determining equations in most general form, which determine the one-parameter transformation group. To get the invariant solutions in explicit form, we confine our attention to the physically interesting situations. This study has led us to some new exact solutions, that does not seem to have been reported in the literature.

Analytical study of bifurcation of a flow of a gas between coaxial circular cylinders with evaporation and condensation

Time-independent behavior of a gas between two coaxial circular cylinders made of the condensed phase of the gas, where the cylinders are rotating around their common axis and evaporation or condensation is taking place, is considered with special attention given to bifurcation of the flow. The problem is studied analytically for small values of the speeds of rotation of the cylinders and the Knudsen number on the basis of the Boltzmann equation, and the solution is obtained explicitly. The bifurcation of flow occurs even in a simple case where the gas is axially symmetric and uniform (or the flow field depends only on the radial coordinate). The comprehensive feature of the bifurcation of flow is clarified with the explicit forms of solutions and the bifurcation diagram.

Solution of equations in Boolean algebra

The explicit form of solutions of Boolean equations with one unknown is obtained. The effectiveness of the method is demonstrated for a number of equations whose solution previously has been found only in “tabular” form. The proposed approach leads to a method for solving systems of equations in Boolean set algebra. We use it to analyze the famous paradoxes of set theory, such as the barber paradox and the liar paradox, as well as Russell's and Cantor's paradoxes.

Analytical Solution for Plane Trusses with Equidistant Supports

The type of plane truss considered in this paper is the continuous Warren truss resting on equidistant roller supports, subjected to transverse nodal loads. In order to uncouple the governing equation, we need to replace the original structure, which does not possess cyclic periodicity, by an equivalent system that is cyclic biperiodic. By applying the U-transformation twice, the governing equation for the equivalent system can be uncoupled and become a set of single degree of freedom equations, which leads to the explicit form solution for the supporting reaction and nodal displacement. The expressions of the solution include two numbers of substructures and supports. As an example, a Warren truss with six substructures and four supports subjected to a concentrated load acting at its center node in the symmetric line of the original structure is worked out by means of the formulas obtained in the present study. It is shown that the result is exact.

Self-Similar Solutions for Hypersonic Source With Variable Mass Loss Rate

For the 1D radial hypersonic flow the development of the configuration with two shocks and contact discontinuity is considered. At small and large moments of time solutions in explicit form are found. As follows from these solutions the contact surface accelerates in time. This acceleration makes possible the Rayleigh-Taylor instability to develop. The 2D numerical investigation of the problem has confirmed the instability of the 1D solution.

On the heat impulse method for deducing sap flow

Speed of sap flow in plants and trees is of interest to botanists and environmentalists because of its connection with the rate of utilisation of nutrients in the soil. An established method uses the transport of heat where an impulsive heat source is introduced along a radial line by a probe in the trunk sapwood. The temperature is monitored, upstream and downstream, and, by solving the heat flow equation in the moving fluid, the sap velocity may be deduced indirectly under some simplifying assumptions which chiefly render the method most useful when applied to trees of relatively large diameter. Transform methods are used to obtain the appropriate threedimensional time-dependent solution in explicit form and values for the resulting sap velocity are compared with the existing two-dimensional theory.

Nonplanar interfacial cracks in anisotropic bimaterials

This paper presents a general approach for the two-dimensional elastic problem of a crack lying along an elliptical interface seperating two dissimilar anisotropic materials. The analysis is based upon the use of the Eshelby–Stroh formalism of anisotropic elasticity theory and a special conformal mapping technique devised by Lekhniskii. The resulting elastic fields are fully described by a pair of function vectors whose components are holomorphic functions. These function vectors define the two-phase potentials of the bi-material. The associated expressions are universal in the sense of being applicable to any applied load. As in the case of a planar interface crack, the crack tip stress field is free of oscillation if the bimaterial matrix H is real. The general results are applied to specific examples and explicit forms of solutions are obtained.

MEMBRANE, FOUR-BRANE AND DUAL COORDINATES IN THE M(ATRIX) THEORY COMPACTIFIED ON TORI

In the M(atrix) theory by making the expansions of the matrices around the infinite membrane and four-brane solutions, we derive the three- and five-dimensional gauge theories in the dual space–times. The explicit forms of solutions yield the dual coordinates and each expansion is related to a toroidal compactification of the M(atrix) theory. We also analyze the expansion around the finite membrane. From the derived Lorentz and gauge-invariant actions, the gauge coupling constants are shown to be characterized by the volumes of tori.

Quantum projection. Integration of the open Tod quantum chain

A quantum projection method is developed on the basis of noncommutative integration of linear differential equations and the results of M. A. Ol’shanetskii and A. M. Perelomov on the integration of classical Hamiltonian systems (projection method). The method proposed makes it possible to obtain in explicit form solutions of the quantum equations whose classical analogs can be integrated by projection. Then the semisimplicity property of the symmetry algebra of the original equation is no longer a factor. The solution basis of a Schrodinger equation with the potential of an open three-particle Tod chain is constructed as a nontrivial example.

Diffraction of a plane wave by a narrow cone

The problem of diffraction of a plane scalar wave by a narrow cone is considered. The shape of the cone is arbitrary. The boundary condition is the Dirichlet or Neumann one. The wave scattered by the cone vertex arises as a result of the diffraction process. The subject of this paper is to calculate the wave amplitude. If the cone is narrow, it is possible to obtain simpler approximate formulas in comparison with Smyshlayev's one. The exactness of the approximate formulas is checked numerically. The etalon is a solution in explicit form in the axially symmetric case. The calculation shows that our formula is more exact in the case of the Dirichlet boundary condition than Felsen's formula. The approximate formula is a generalization of Felsen's one for circular cone to an arbitrary narrow cone in the case of the Neumann boundary condition. Bibliography: 6 titles.

On the orthogonal representation of generalized random fields

An orthogonal representation for a class of generalized random fields defined on an infinite-dimensional separable Hilbert space is studied. This representation is an extension of the expansion studied in Ruiz-Medina and Angulo (1995). The results are applied to obtain the orthogonal expansion for a linear functional of a zero-mean, second-order random field satisfying certain regularity conditions. Finally, some applications of the above representation in obtaining linear prediction estimates, and in obtaining explicit-form solutions to stochastic partial differential equations, are discussed.