Symbolic computation of solitary wave solutions and solitons through homogenization of degree

A simplified version of Hirota's method for the computation of solitary waves and solitons of nonlinear PDEs is presented. A change of dependent variable transforms the PDE into an equation that is homogeneous of degree. Solitons are then computed using a perturbation-like scheme involving linear and nonlinear operators in a finite number of steps. The method is applied to a class of fifth-order KdV equations due to Lax, Sawada-Kotera, and Kaup-Kupershmidt. The method works for non-quadratic homogeneous equations for which the bilinear form might not be known. Furthermore, homogenization of degree allows one to compute solitary wave solutions of nonlinear PDEs that do not have solitons. Examples include the Fisher and FitzHugh-Nagumo equations, and a combined KdV-Burgers equation. When applied to a wave equation with a cubic source term, one gets a bi-soliton solution describing the coalescence of two wavefronts. The method is largely algorithmic and is implemented in Mathematica.Comment: Proceedings Conference on Nonlinear and Modern Mathematical Physics (NMMP-2022) Springer Proceedings in Mathematics and Statistics, 60pp, Springer-Verlag, New York, 202

Similarity Reductions and Integrable Lattice Equations

In this thesis I extend the theory of integrable partial difference equations (PAEs) and reductions of these systems under scaling symmetries. The main approach used is the direct linearization method which was developed previously and forms a powerful tool for dealing with both continuous and discrete equations. This approach is further developed and applied to several important classes of integrable systems. Whilst the theory of continuous integrable systems is well established, the theory of analogous difference equations is much less advanced. In this context the study of symmetry reductions of integrable (PAEs) which lead to ordinary difference equations (OAEs) of Painleve type, forms a key aspect of a more general theory that is still in its infancy. The first part of the thesis lays down the general framework of the direct linearization scheme and reviews previous results obtained by this method. Most results so far have been obtained for lattice systems of KdV type. One novel result here is a new approach for deriving Lax pairs. New results in this context start with the embedding of the lattice KdV systems into a multi-dimensional lattice, the reduction of which leads to both continuous and discrete Painleve hierarchies associated with the Painleve VI equation. The issue of multidimensional lattice equations also appears, albeit in a different way, in the context of the lattice KP equations, which by dimensional reduction lead to new classes of discrete equations. This brings us in a natural way to a different class of continuous and discrete systems, namely those which can be identified to be of Boussinesq (BSQ) type. The development of this class by means of the direct linearization method forms one of the major parts of the thesis. In particular, within this class we derive new differential-difference equations and exhibit associated linear problems (Lax pairs). The consistency of initial value problems on the multi-dimensional lattice is established. Furthermore, the similarity constraints and their compatibility with the lattice systems guarantee the consistency of the reductions that are considered. As such the resulting systems of lattice equations are conjectured to be of Painleve type. The final part of the thesis contains the general framework for lattice systems of AKNS type for which we establish the basic equations as well as similarity constraints

Extensions of the General Solution to the Inverse Problem of the Calculus of Variations, and Variational, Perturbative and Reversible Systems Approaches to Regular and Embedded Solitary Waves

In the first part of this Dissertation, hierarchies of Lagrangians of degree two, three or four, each only partly determined by the choice of leading terms and with some coefficients remaining free, are derived. These have significantly greater freedom than the most general differential geometric criterion currently known for the existence of a Lagrangian and variational formulation since our existence conditions are for individual coefficients in the Lagrangian. For different choices of leading coefficients, the resulting variational equations could also represent traveling waves of various nonlinear evolution equations. Families of regular and embedded solitary waves are derived for some of these generalized variational ODEs in appropriate parameter regimes. In the second part, an earlier approach based on soliton perturbation theory is significantly generalized to obtain an analytical formula for the tail amplitudes of nonlocal solitary waves of a perturbed generalized fifth-order Korteweg-de Vries (FKdV) equation. On isolated curves in the parameter space, these tail amplitudes vanish, producing families of localized embedded solitons in large regions of the space. Off these curves, the tail amplitudes of the nonlocal waves are shown to be exponentially small in the small wavespeed limit. These seas of delocalized solitary waves are shown to be entirely distinct from those derived in that earlier work. These perturbative results are also discussed within the framework of known reversible systems results for various families of homoclinic orbits of the corresponding traveling-wave ordinary differential equation of our generalized FKdV equation. The third part considers a variety of dynamical behaviors in a multiparameter nonlinear Mathieu equation with distributed delay. A slow flow is derived using the method of averaging, and the predictions from that are then tested against direct numerical simulations of the nonlinear Mathieu system. Both areas of agreement and disagreement between the averaged and full numerical solutions are considered

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd

Front propagation into unstable states

This paper is an introductory review of the problem of front propagation into unstable states. Our presentation is centered around the concept of the asymptotic linear spreading velocity v*, the asymptotic rate with which initially localized perturbations spread into an unstable state according to the linear dynamical equations obtained by linearizing the fully nonlinear equations about the unstable state. This allows us to give a precise definition of pulled fronts, nonlinear fronts whose asymptotic propagation speed equals v*, and pushed fronts, nonlinear fronts whose asymptotic speed v^dagger is larger than v*. In addition, this approach allows us to clarify many aspects of the front selection problem, the question whether for a given dynamical equation the front is pulled or pushed. It also is the basis for the universal expressions for the power law rate of approach of the transient velocity v(t) of a pulled front as it converges toward its asymptotic value v*. Almost half of the paper is devoted to reviewing many experimental and theoretical examples of front propagation into unstable states from this unified perspective. The paper also includes short sections on the derivation of the universal power law relaxation behavior of v(t), on the absence of a moving boundary approximation for pulled fronts, on the relation between so-called global modes and front propagation, and on stochastic fronts.Comment: final version with some added references; a single pdf file of the published version is available at http://www.lorentz.leidenuniv.nl/~saarloo

Random Matrix Theories in Quantum Physics: Common Concepts

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.Comment: 178 pages, Revtex, 45 figures, submitted to Physics Report

Radio Communications

In the last decades the restless evolution of information and communication technologies (ICT) brought to a deep transformation of our habits. The growth of the Internet and the advances in hardware and software implementations modified our way to communicate and to share information. In this book, an overview of the major issues faced today by researchers in the field of radio communications is given through 35 high quality chapters written by specialists working in universities and research centers all over the world. Various aspects will be deeply discussed: channel modeling, beamforming, multiple antennas, cooperative networks, opportunistic scheduling, advanced admission control, handover management, systems performance assessment, routing issues in mobility conditions, localization, web security. Advanced techniques for the radio resource management will be discussed both in single and multiple radio technologies; either in infrastructure, mesh or ad hoc networks

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described