On elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation

The Conte-Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a priory restriction is to simplify calculations by means of the use of a few Laurent series solutions instead of one and the use of the residue theorem. The application of our approach to the quintic complex one-dimensional Ginzburg-Landau equation (CGLE5) allows to find elliptic solutions in the wave form. We also find restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions for the CGLE5. Using the investigation of the CGLE5 as an example, we demonstrate that to find elliptic solutions the analysis of a system of differential equations is more preferable than the analysis of the equivalent single differential equation.Comment: LaTeX, 21 page

On traveling waves in lattices: The case of Riccati lattices

The method of simplest equation is applied for analysis of a class of lattices described by differential-difference equations that admit traveling-wave solutions constructed on the basis of the solution of the Riccati equation. We denote such lattices as Riccati lattices. We search for Riccati lattices within two classes of lattices: generalized Lotka - Volterra lattices and generalized Holling lattices. We show that from the class of generalized Lotka - Volterra lattices only the Wadati lattice belongs to the class of Riccati lattices. Opposite to this many lattices from the Holling class are Riccati lattices. We construct exact traveling wave solutions on the basis of the solution of Riccati equation for three members of the class of generalized Holing lattices.Comment: 17 pages, no figure

Solitary and periodic solutions of the generalized Kuramoto-Sivashinsky equation

The generalized Kuramoto-Sivashinsky equation in the case of the power nonlinearity with arbitrary degree is considered. New exact solutions of this equation are presented

The Nikolaevskiy equation with dispersion

The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral, Goldstone mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilise some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram (Busse balloon) for the traveling waves can be remarkably complicated.Comment: 24 pages; accepted for publication in Phys. Rev.

A Lagrangian Description of the Higher-Order Painlev\'e Equations

We derive the Lagrangians of the higher-order Painlev\'e equations using Jacobi's last multiplier technique. Some of these higher-order differential equations display certain remarkable properties like passing the Painlev\'e test and satisfy the conditions stated by Jur\'a$\check{s}$, (Acta Appl. Math. 66 (2001) 25--39), thus allowing for a Lagrangian description.Comment: 16 pages, to be published in Applied Mathematics and Computatio

Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions

We show how to generate coupled KdV hierarchies from Staeckel separable systems of Benenti type. We further show that solutions of these Staeckel systems generate a large class of finite-gap and rational solutions of cKdV hierarchies. Most of these solutions are new.Comment: 15 page

Solitary waves of nonlinear nonintegrable equations

Our goal is to find closed form analytic expressions for the solitary waves of nonlinear nonintegrable partial differential equations. The suitable methods, which can only be nonperturbative, are classified in two classes. In the first class, which includes the well known so-called truncation methods, one \textit{a priori} assumes a given class of expressions (polynomials, etc) for the unknown solution; the involved work can easily be done by hand but all solutions outside the given class are surely missed. In the second class, instead of searching an expression for the solution, one builds an intermediate, equivalent information, namely the \textit{first order} autonomous ODE satisfied by the solitary wave; in principle, no solution can be missed, but the involved work requires computer algebra. We present the application to the cubic and quintic complex one-dimensional Ginzburg-Landau equations, and to the Kuramoto-Sivashinsky equation.Comment: 28 pages, chapter in book "Dissipative solitons", ed. Akhmediev, to appea

On elliptic solutions of the cubic complex one-dimensional Ginzburg-Landau equation

The cubic complex one-dimensional Ginzburg-Landau equation is considered. Using the Hone's method, based on the use of the Laurent-series solutions and the residue theorem, we have proved that this equation has neither elliptic standing wave nor elliptic travelling wave solutions. This result amplifies the Hone's result, that this equation has no elliptic travelling wave solutions.Comment: LaTeX, 12 page

Surface texturing of steel by femtosecond laser and accompanying structure/ phase transformations

Topography, structure, and phase composition of surface layers of AISI 321 stainless steel textured by 1030-nm 320-fs-laser pulses were studied by scanning electron microscopy and X-ray diffraction analysis. Variation in single-pulse fluence and the number of pulses was found to change the laser-produced surface texture from onedimensional quasi-periodic nanograting to microrelief of various roughnes

Residual stresses in Ti6Al4V alloy after surface texturing by femtosecond laser pulses

Surface topography and residual stresses in surface layers of α + β titanium alloy Ti6Al4V textured by 1030-nm, 320-fs-laser pulses were studied by scanning electron microscopy and X-ray diffraction analysis. It was found that multipulse laser processing leads to the formation of laser-induced periodic surface structures (LIPSS) on the surface of Ti6Al4V alloy. XRD studies showed that depending on the laser pulse fluence, both tensile and compressive residual stresses are formed in thin near-surface layer