The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation

Abstract In [3] Dias and Figueira have reported that the square of the solution for the nonlinear Dirac equation satisfies the linear wave equation in one space dimension. So the aim of this paper is to proceed with their work and to clarify a structure of the nonlinear Dirac equation. The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation are obtained. Keywords: Nonlinear Dirac equation, Dirac-Klein-Gordon equation, Pauli matrix Mathematics Subject Classification (2000): 35C05, 35L45     

Exact solutions of coupled nonlinear Klein-Gordon equation

In this paper, we employ the general integral method for traveling wave solutions of coupled nonlinear Klein-Gordon equations. Based on the idea of the exact Jacobi elliptic function, a simple and efficient method is proposed for obtaining exact solutions of nonlinear evolution equations. The solutions obtained include solitons, periodic solutions and Jacobi elliptic function solutions.     

Exact solutions of nonlinear generalizations of the wave equation

Exact solutions of nonlinear generalizations of the wave equation are constructed. In some cases these solutions are solitary waves or solitions. Thus, by explicit construction solitons or solitary waves are shown to exist in dispersionless systems. In contrast to previous solitary wave solutions, these solutions are limiting cases of solutions of nonlinear partial differential equations with dispersion.     

Finite-zone solutions in terms of automorphic functions the Kadomtsev-petviashvili equation

One obtains an expression for the finite-zone solutions of the Kadomtsev-Petviashvili equation in terms of the Poincaré theta-series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 5–16, 1983.The author expresses his sincere gratitude to A. R. Its for useful discussions and for his constant interest in the paper.     

Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation

We prove endpoint Strichartz estimates for the Klein-Gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions. As an application, global wellposedness of the nonlinear Dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable.     

Exact envelope-soliton solutions of a two-dimensional nonlinear wave equation

ExactN-envelope-soliton solutions are obtained, by extending Hirota's procedure, for the twodimensional nonlinear wave Eqn. (1) withq>0, which describes the evolution of the envelope of a train of surface gravity waves on deep water. They are shown to propagate in directions making an angle greater than tan^–1/2 with the propagation direction of the underlying carrier waves. We also point out and discuss the limitations of Hirota's procedure for generating solitonsolutions to problems of more than one spatial dimensions. Envelope-soliton solutions to Eqn. (1) withq<0 are also discussed.

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Sommaire On généralise la méthode de Hirota pour obtenir des solutions exactes àN-solitons et on l'applique à l'équation des ondes nonlinéaires à deux dimensions (1) avecq>0, qui décrit l'évolution de l'enveloppe d'un train d'ondes de gravitation dans un fluide de grande profondeur. Ces solutions se propagent en directions formant un angle plus grand que tan^–1/2 avec la direction de propagation des ondes porteuses fondamentales. On montre aussi que la méthode de Hirota n'est pas capable de produire, en deux dimensions, des solutions exactes aussi générales que dans le cas d'une seule dimension. Enfin on étudie les solutions de l'équation (1) avecq<0.

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Research supported by the Natural Sciences and Engineering Research Council of Canada. The author thanks Dr. G. Tenti for valuable discussions.     

Exact solutions for two nonlinear wave equations with nonlinear terms of any order

In this paper, based on a variable-coefficient balancing-act method, by means of an appropriate transformation and with the help of Mathematica, we obtain some new types of solitary-wave solutions to the generalized Benjamin–Bona–Mahony (BBM) equation and the generalized Burgers–Fisher (BF) equation with nonlinear terms of any order. These solutions fully cover the various solitary waves of BBM equation and BF equation previously reported.     

Convergent expansions of the Bessel functions in terms of elementary functions

We consider the Bessel functions J _ν (z) and Y _ν (z) for R ν > ?1/2 and R z ≥ 0. We derive a convergent expansion of J _ν (z) in terms of the derivatives of \((\sin z)/z\), and a convergent expansion of Y _ν (z) in terms of derivatives of \((1-\cos z)/z\), derivatives of (1 ? e ^?z )/z and Γ(2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments.     

Exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres

First, by using the generally projective Riccati equation method, many kinds of exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres are obtained in a unified way. Then, some relations among these solutions are revealed.     

Exact solutions of a two-dimensional nonlinear Schrödinger equation

In the present study, we converted the resulting nonlinear equation for the evolution of weakly nonlinear hydrodynamic disturbances on a static cosmological background with self-focusing in a two-dimensional nonlinear Schrödinger (NLS) equation. Applying the function transformation method, the NLS equation was transformed to an ordinary differential equation, which depended only on one function ξ and can be solved. The general solution of the latter equation in ζ leads to a general solution of NLS equation. A new set of exact solutions for the two-dimensional NLS equation is obtained.     

On the expansion of confluent hypergeometric functions in terms of Bessel functions

For the confluent hypergeometric functions U (a, b, z) and M (a, b, z) asymptotic expansions are given for a → ∞. The expansions contain modified Bessel functions. For real values of the parameters rigorous error bounds are given.     

Exact traveling wave solutions and bifurcations in a nonlinear elastic rod equation

The exact parametric representations of the traveling wave solutions for a nonlinear elastic rod equation are considered. By using the method of planar dynamical systems, in different parameter regions, the phase portraits of the corresponding traveling wave system are given. Exact explicit kink wave solutions, periodic wave solutions and some unbounded wave solutions are obtained.     

Exact and explicit solutions for a nonlinear extended iterative differential equation

This paper is concerned with a nonlinear iterative functional differential equation x′(z) = 1/x(p(z) + bx′(z)). By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only in the general case, but also in critical cases, especially for α given in Schröder transformation is a root of the unity. And in case (H4), we dealt with the equation under the Brjuno condition, which is weaker than the Diophantine condition. Moreover, the exact and explicit solution of the original equation has been investigated for the first time. Such equations are important in both applications and the theory of iterations.     

Various exact solutions of nonlinear Schrödinger equation with two nonlinear terms

Four kinds of exact solutions to nonlinear Schrödinger equation with two higher order nonlinear terms are obtained by a subsidiary ordinary differential equation method (sub-equation method for short). They are the bell type solitary waves, the kink type solitary waves, the algebraic solitary waves and the sinusoidal waves.