13,401 research outputs found
Traveling wave solutions of the (2+1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method
AbstractThe modified simple equation (MSE) method is promising for finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this letter, we investigate solutions of the (2+1)-dimensional Zoomeron equation and the (2+1)-dimensional Burgers equation by using the MSE method and the Exp-function method. The competence of the methods for constructing exact solutions has been established
Meromorphic solutions of nonlinear ordinary differential equations
Exact solutions of some popular nonlinear ordinary differential equations are
analyzed taking their Laurent series into account. Using the Laurent series for
solutions of nonlinear ordinary differential equations we discuss the nature of
many methods for finding exact solutions. We show that most of these methods
are conceptually identical to one another and they allow us to have only the
same solutions of nonlinear ordinary differential equations
Traveling wave fronts and the transition to saturation
We propose a general method to study the solutions to nonlinear QCD evolution
equations, based on a deep analogy with the physics of traveling waves. In
particular, we show that the transition to the saturation regime of high energy
QCD is identical to the formation of the front of a traveling wave. Within this
physical picture, we provide the expressions for the saturation scale and the
gluon density profile as a function of the total rapidity and the transverse
momentum. The application to the Balitsky-Kovchegov equation for both fixed and
running coupling constants confirms the effectiveness of this method.Comment: 9 pages, 3 figures, references adde
Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB)
equations, i.e. scalar conservation laws with diffusive-dispersive
regularization. We review the existence of traveling wave solutions for these
two classes of evolution equations. For classical equations the traveling wave
problem (TWP) for a local KdVB equation can be identified with the TWP for a
reaction-diffusion equation. In this article we study this relationship for
these two classes of evolution equations with nonlocal diffusion/dispersion.
This connection is especially useful, if the TW equation is not studied
directly, but the existence of a TWS is proven using one of the evolution
equations instead. Finally, we present three models from fluid dynamics and
discuss the TWP via its link to associated reaction-diffusion equations
Hamiltonian description and traveling waves of the spatial Dysthe equations
The spatial version of the fourth-order Dysthe equations describe the
evolution of weakly nonlinear narrowband wave trains in deep waters. For
unidirectional waves, the hidden Hamiltonian structure and new invariants are
unveiled by means of a gauge transformation to a new canonical form of the
evolution equations. A highly accurate Fourier-type spectral scheme is
developed to solve for the equations and validate the new conservation laws,
which are satisfied up to machine precision. Further, traveling waves are
numerically investigated using the Petviashvili method. It is found that their
collision appears inelastic, suggesting the non-integrability of the Dysthe
equations.Comment: Research report. 17 pages, 7 figures, 38 references. Other author's
papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/ . arXiv
admin note: substantial text overlap with arXiv:1110.408
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