Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations

In this Letter, He's homotopy perturbation method (HPM) is implemented for finding the solitary-wave solutions of the regularized long-wave (RLW) and generalized modified Boussinesq (GMB) equations. We obtain numerical solutions of these equations for the initial conditions. We will show that the convergence of the HPM is faster than those obtained by the Adomian decomposition method (ADM). The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

Adomian Decomposition Method and Exact Solutions of the Perturbed KdV Equation

The Adomian decomposition method is used to solve the Cauchy problem of the perturbed KdV equation.Three types of exact solitary wave solutions are reobtained via the A domian‘s approach by selecting the initial conditionsappropriately.     

Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

A detailed study of the spinless (1+1)D free‐particle Salpeter equation is presented. It involves several aspects of the topic: from the analysis of the behavior of solutions of the equation, both numerically evaluated and asymptotically approximated for definite initial conditions, to the comparison with the behavior of the corresponding solutions of the Schrödinger equation in order to both highlight the differences and to possibly understand how the latter “flow” in the former. Interesting analogies with other fields of physics, in particular with optics, are suggested.     

On the asymptotic equivalence between the Enskog and the Boltzmann equations

An asymptotic equivalence theorem is proven between the solutions of the initial value problem in all space for the Boltzmann and Enskog equations for initial data which assure global existence for the solutions to the initial value problem for one of the two equations. The proof is given starting from the solution of the Boltzmann equation, then the proof line is simply indicated when one starts from the Enskog equation. The proof holds for Knudsen numbers of the order of unity and equivalence is proven when the scale of the dimensions of the gas particles characterizing the Enskog equation tends to zero.On leave from Department of Mathematics, University of Warsaw, Poland.     

Classical electrodynamics of a point particle

Even for the simplest physical situations the Lorentz-Dirac equation, solved as an initial value problem, gives unphysical ‘run-away’ solutions. Dirac’s method for avoiding these unphysical solutions generates solutions which exhibit unphysical acausal pre-acceleration. A careful examination of the application of the conservation of momentum in the derivation of the Abraham self-force reveals a fundamental error concerning the force acting on the particle. This error, originally made by Abraham (1903), has been repeated by subsequent investigators. When corrected, a new equation of motion results. A discussion of the general properties of the new equation of motion is given, and solutions for several important special cases are presented. The behaviour of these solutions is causal, physically reasonable, and easily understood.     

Homogeneous Generalized Master Equations

It is shown that the method proposed in V. F. Los [J. Phys. A: Math. Gen. 34: 638–6403 (2001)], which allows for turning the inhomogeneous time-convolution generalized master equation (TC-GME) into homogeneous (while retaining initial correlations) time-convolution generalized master equation (TC-HGME) for the relevant part of a distribution function, is fully applicable to the quantum case and to the time-convolutionless GME (TCL-GME). It is demonstrated by rederiving the TC-HGME and showing that it works in both the classical and quantum physics cases. The time-convolutionless HGME (TCL-HGME) retaining initial correlations, which is formally the same for both the classical and quantum physics, has also been derived. Both the TC-HGME and TCL-HGME are exact equations applicable on any timescale and allow for consecutive treating the initial correlations and collisions on the equal footing. A new equation for a momentum distribution function retaining initial correlations has been obtained in the linear in the density of quantum particles approximation. Connection of this equation to the quantum Boltzmann equation is discussed.     

Further Riccati Differential Equations with Elliptic Coefficients and Meromorphic Solutions

We exhibit some families of Riccati differential equations in the complex domain having elliptic coefficients and study the problem of understanding the cases where there are no multivalued solutions. We give criteria ensuring that all the solutions to these equations are meromorphic functions defined in the whole complex plane, and highlight some cases where all solutions are, furthermore, doubly periodic.     

Shock-induced chain-branched ignition

Shock-induced ignition is considered for a three-step chain-branching mechanism at large activation energies. The nature of the ignition process depends critically on the magnitude of the initial post-shock temperature relative to the chain-branching cross-over temperature. For the limits examined, the induction region is preceded by an exponentially weak initiation zone that provides relevant initial conditions for the induction phase. When the initial post-shock temperature is sufficiently close to the chain-branching temperature, ignition is characterized by logarithmic singular behaviors in the pressure and temperature perturbations and the structure has some similarities with the one-step chemistry problem. However, for larger initial post-shock temperatures, the logarithmic singularities are replaced by linear temporal growth. Suitable non-linear Clarke equations are deduced for both of these cases and numerical solutions are presented.     

Reduced Maxwell-Bloch equations with anisotropic dipole momentum

We develop the inverse scattering transform for the recently found integrable system of reduced Maxwell-Bloch equations with two components of polarization and with an anisotropic dipole momentum. The model describes few-cycle pulses of optical or other field propagations. We find that the existence of a nontrivial group of symmetry of the corresponding Lax pair leads to a particular form of the inverse scattering transform equations. We show that solutions can be expressed in terms of the solution of a matrix Riemann-Hilbert problem formulated for the complex plane with a nontrivial group of automorphisms.     

Instabilities in the Chapman-Enskog Expansion and Hyperbolic Burnett Equations

It is well-known that the classical Chapman-Enskog procedure does not work at the level of Burnett equations (the next step after the Navier-Stokes equations). Roughly speaking, the reason is that the solutions of higher equations of hydrodynamics (Burnett's, etc.) become unstable with respect to short-wave perturbations. This problem was recently attacked by several authors who proposed different ways to deal with it. We present in this paper one of possible alternatives. First we deduce a criterion for hyperbolicity of Burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases. Then we discuss in more detail the problem of truncation of the Chapman-Enskog expansion and show that the way of truncation is not unique. The general idea of changes of coordinates (based on analogy with the theory of dynamical systems) leads finally to nonlinear Hyperbolic Burnett Equations (HBEs) without using any information beyond the classical Burnett equations. It is proved that HBEs satisfy the linearized H-theorem. The linear version of the problem is studied in more detail, the complete Chapman-Enskog expansion is given for the linear case. A simplified proof of the Slemrod identity for Burnett coefficients is also given.     

Solution to the nonlinear boltzmann equation for maxwell models for nonisotropic initial conditions

The known solution to the spatially homogeneous nonlinear Boltzmann equation for Maxwell models in a series of Laguerre polynomials is extended to include nonisotropic initial conditions. Existence proofs for a class of solutions are supplied. The equations for the generalized (nonisotropic Laguerre) moments are derived in explicit form for two- and three-dimensional models. Further it is shown that the ordinary moments satisfy the same set of equations as the (Hermite) polynomial moments.     

Solution of a linearized kinetic model for an ultrarelativistic gas

A linearized model of the Boltzmann equation for a relativistic gas is shown to be reducible, in the ultrarelativistic limit and for (1 + 1)-dimensional problems, to a system of three uncoupled transport equations, one of which is well known. A general method for solving these equations is recalled, with a few new details, and applied to the solution of two boundary value problems. The first of these describes the propagation of an impulsive change in a half space and is shown to give an explicit example of the recently proved result that no signal can propagate with speed larger than the speed of light, according to the relativistic Boltzmann equation. The second problem deals with steady oscillations in a half space and illustrates the meaning of certain recent results concerning the dispersion relation for linear waves in relativistic gas.     

A new method based on the harmonic balance method for nonlinear oscillators

The harmonic balance (HB) method as an analytical approach is widely used for nonlinear oscillators, in which the initial conditions are generally simplified by setting velocity or displacement to be zero. Based on HB, we establish a new theory to address nonlinear conservative systems with arbitrary initial conditions, and deduce a set of over-determined algebraic equations. Since these deduced algebraic equations are not solved directly, a minimization problem is constructed instead and an iterative algorithm is employed to seek the minimization point. Taking Duffing and Duffing-harmonic equations as numerical examples, we find that these attained solutions are not only with high degree of accuracy, but also uniformly valid in the whole solution domain.     

Gradient catastrophes and sawtooth solutions for a generalized Burgers equation on an interval

The asymptotic behavior of solutions of the Burgers equation and its generalizations with initial value-boundary problem on a finite interval with constant boundary conditions is studied. Since it describes a dissipative medium, any initial profile will evolve to a time-invariant solution with the same boundary values. Yet there are three distinctive asymptotic processes: the initial profile may regularly decay to a smooth invariant solution; or a Heaviside-type gap develops through a dispersive shock and multi-oscillations; or an asymptotic limit is a stationary ‘sawtooth’ solution with periodical breaks of derivative.     

Riemann-Hilbert problems for the Ernst equation and fibre bundles

Riemann-Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary-value problems. The solution of a boundary-value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved, it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems.     

General initial value problem for the nonlinear shallow water equations: Runup of long waves on sloping beaches and bays

We formulate a new approach to solving the initial value problem of the shallow water-wave equations utilizing the famous Carrier–Greenspan transformation (Carrier and Greenspan (1957) [9]). We use a Taylor series approximation to deal with the difficulty associated with the initial conditions given on a curve in the transformed space. This extends earlier solutions to waves with near shore initial conditions, large initial velocities, and in more complex U-shaped bathymetries; and allows verification of tsunami wave inundation models in a more realistic 2-D setting.     

Travelling Waves for a Density Dependent Diffusion Nagumo Equation over the Real Line

We consider the density dependent diffusion Nagumo equation, where the diffusion coefficient is a simple power function. This equation is used in modelling electrical pulse propagation in nerve axons and in population genetics (amongst other areas). In the present paper, the δ-expansion method is applied to a travelling wave reduction of the problem, so that we may obtain globally valid perturbation solutions (in the sense that the perturbation solutions are valid over the entire infinite domain, not just locally; hence the results are a generalization of the local solutions considered recently in the literature). The resulting boundary value problem is solved on the real line subject to conditions at z → ±∞. Whenever a perturbative method is applied, it is important to discuss the accuracy and convergence properties of the resulting perturbation expansions. We compare our results with those of two different numerical methods (designed for initial and boundary value problems, respectively) and deduce that the perturbation expansions agree with the numerical results after a reasonable number of iterations. Finally, we are able to discuss the influence of the wave speed c and the asymptotic concentration value α on the obtained solutions. Upon recasting the density dependent diffusion Nagumo equation as a two-dimensional dynamical system, we are also able to discuss the influence of the nonlinear density dependence (which is governed by a power-law parameter m) on oscillations of the travelling wave solutions.     

A Note on Similarity Solutions of Navier-Stokes Equations

This note looks at the two similarity solutions of the Navier Stokes equations in polar coordinates. In the second solution an initial value problem is reduced into generalized stationary KDV and hence integrable.     

Relationship Among Solutions of a Generalized Riccati Equation

In this paper, some solutions of a generalized Riccati equation are investigated, which are given in the recent articles [Chaos, Solitons ＆ Fractals 24 （2005） 257; Phys. Lett. A 336 （2005） 463], and the relationship among the solutions is revealed.