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

Completeness of the cubic and quartic H\'enon-Heiles Hamiltonians

The quartic H\'enon-Heiles Hamiltonian $H = (P_1^2+P_2^2)/2+(\Omega_1 Q_1^2+\Omega_2 Q_2^2)/2 +C Q_1^4+ B Q_1^2 Q_2^2 + A Q_2^4 +(1/2)(\alpha/Q_1^2+\beta/Q_2^2) - \gamma Q_1$ passes the Painlev\'e test for only four sets of values of the constants. Only one of these, identical to the traveling wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the three others are not yet integrated in the generic case $(\alpha,\beta,\gamma)\not=(0,0,0)$. We integrate them by building a birational transformation to two fourth order first degree equations in the classification (Cosgrove, 2000) of such polynomial equations which possess the Painlev\'e property. This transformation involves the stationary reduction of various partial differential equations (PDEs). The result is the same as for the three cubic H\'enon-Heiles Hamiltonians, namely, in all four quartic cases, a general solution which is meromorphic and hyperelliptic with genus two. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painlev\'e integrability (completeness property).Comment: 10 pages, To appear, Theor.Math.Phys. Gallipoli, 34 June--3 July 200

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Backlund transformations for the equations considered. We are also able to derive Backlund transformations onto other ODEs in the Painleve classification.Comment: To appear in Nonlinearity (22 pages

Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation

In evolution equations for a complex amplitude, the phase obeys a much more intricate equation than the amplitude. Nevertheless, general methods should be applicable to both variables. On the example of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation (CGL5), we explain how to overcome the difficulties arising in two such methods: (i) the criterium that the sum of residues of an elliptic solution should be zero, (ii) the construction of a first order differential equation admitting the given equation as a differential consequence (subequation method).Comment: 12 pages, no figure, to appear, Theoretical and Mathematical Physic

Treatment of osteoporosis: Recognizing and managing cutaneous adverse reactions and drug-induced hypersensitivity

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Optical Solitary Waves in the Higher Order Nonlinear Schrodinger Equation

We study solitary wave solutions of the higher order nonlinear Schrodinger equation for the propagation of short light pulses in an optical fiber. Using a scaling transformation we reduce the equation to a two-parameter canonical form. Solitary wave (1-soliton) solutions exist provided easily met inequality constraints on the parameters in the equation are satisfied. Conditions for the existence of N-soliton solutions (N>1) are determined; when these conditions are met the equation becomes the modified KdV equation. A proper subset of these conditions meet the Painleve plausibility conditions for integrability.Comment: REVTeX, 4 pages, no figures. To appear in Phys. Rev. Let

Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation

We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie's lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica

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

New variable separation approach: application to nonlinear diffusion equations

The concept of the derivative-dependent functional separable solution, as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach. As a consequence, a complete list of canonical forms for such equations which admit the derivative-dependent functional separable solutions is obtained and some exact solutions to the resulting equations are described.Comment: 19 pages, 2 fig

The application of real-time PCR to the analysis of T cell repertoires

The diversity of T-cell populations is determined by the spectrum of antigen-specific T-cell receptors (TCRs) that are heterodimers of α and β subunits encoded by rearranged combinations of variable (AV and BV), joining (AJ and BJ), and constant region genes (AC and BC). We have developed a novel approach for analysis of β transcript diversity in mice with a real-time PCR-based method that uses a matrix of BV- and BJ-specific primers to amplify 240 distinct BV–BJ combinations. Defined endpoints (Ct values) and dissociation curves are generated for each BV–BJ combination and the Ct values are consolidated in a matrix that characterizes the β transcript diversity of each RNA sample. Relative diversities of BV–BJ combinations in individual RNA samples are further described by estimates of scaled entropy. A skin allograft system was used to demonstrate that dissection of repertoires into 240 BV–BJ combinations increases efficiency of identifying and sequencing β transcripts that are overrepresented at inflammatory sites. These BV–BJ matrices should generate greater investigation in laboratory and clinical settings due to increased throughput, resolution and identification of overrepresented TCR transcripts