POSITIVITY PROPERTIES OF THE FOURIER TRANSFORM AND THE STABILITY OF PERIODIC TRAVELLING-WAVE SOLUTIONS

In this paper we establish a method to obtain the stability of periodic travelling-wave solutions for equations of Korteweg-de Vries-type u(t) + u(p)u(x) - Mu(x) = 0, with M being a general pseudodifferential operator and where p >= 1 is an integer. Our approach uses the theory of totally positive operators, the Poisson summation theorem, and the theory of Jacobi elliptic functions. In particular we obtain the stability of a family of periodic travelling waves solutions for the Benjamin Ono equation. The present technique gives a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated with the Korteweg-de Vries and modified Korteweg-de Vries equations, respectively. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

An improved class of nonlinear bidirectional Boussinesq equations of sixth order using a wave surface elevation formulation is derived. Exact travelling wave solutions for the proposed class of nonlinear evolution equations are deduced. A new exact travelling wave solution is found which is the uniform limit of a geometric series. The ratio of this series is proportional to a classical soliton-type solution of the form of the square of a hyperbolic secant function. This happens for some values of the wave propagation velocity. However, there are other values of this velocity which display this new type of soliton, but the classical soliton structure vanishes in some regions of the domain. Exact solutions of the form of the square of the classical soliton are also deduced. In some cases, we find that the ratio between the amplitude of this wave and the amplitude of the classical soliton is equal to 35/36. It is shown that different families of travelling wave solutions are associated with different values of the parameters introduced in the improved equations.

Some Travelling Wave Solutions of KdV-Burgers Equation

In this paper we study the extended Tanh method to obtain some exact solutions of KdV-Burgers equation. The principle of the Tanh method has been explained and then apply to the nonlinear KdV- Burgers evolution equation. A finnite power series in tanh is considered as an ansatz and the symbolic computational system is used to obtain solution of that nonlinear evolution equation. The obtained solutions are all travelling wave solutions.

Colombeau's theory and shock wave solutions for systems of PDEs

In this article we study the existence of shock wave solutions for systems of partial differential equations of hydrodynamics with viscosity in one space dimension in the context of Colombeau's theory of generalized functions. This study uses the equality in the strict sense and the association of generalized functions (that is the weak equality). The shock wave solutions are given in terms of generalized functions that have the classical Heaviside step function as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function that have to satisfy part of the equations in the strict sense and part of the equations in the sense of association.

On the energy transported by exact plane gravitational-wave solutions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Dispersive wave solutions of the klein-gordon equation in cosmology

L'equazione di Klein-Gordon descrive una ampia varietà di fenomeni fisici come la propagazione delle onde in Meccanica dei Continui ed il comportamento delle particelle spinless in Meccanica Quantistica Relativistica. Recentemente, la forma dissipativa di questa equazione si è rivelata essere una legge di evoluzione fondamentale in alcuni modelli cosmologici, in particolare nell'ambito dei cosiddetti modelli di k-inflazione in presenza di campi tachionici. L'obiettivo di questo lavoro consiste nell'analizzare gli effetti del parametro dissipativo sulla dispersione nelle soluzioni dell'equazione d'onda. Saranno inoltre studiati alcuni tipici problemi al contorno di particolare interesse cosmologico per mezzo di grafici corrispondenti alle soluzioni fondamentali (Funzioni di Green).

Stability and Instability of Solitary Wave Solutions of a Nonlinear Dispersive System of Benjamin-Bona-Mahony Type

2000 Mathematics Subject Classification: 35B35, 35B40, 35Q35, 76B25, 76E30.

Some non- linear problems in theoretical physics

An immense variety of problems in theoretical physics are of the non-linear type. Non~linear partial differential equations (NPDE) have almost become the rule rather than an exception in diverse branches of physics such as fluid mechanics, field theory, particle physics, statistical physics and optics, and the construction of exact solutions of these equations constitutes one of the most vigorous activities in theoretical physics today. The thesis entitled ‘Some Non-linear Problems in Theoretical Physics’ addresses various aspects of this problem at the classical level. For obtaining exact solutions we have used mathematical tools like the bilinear operator method, base equation technique and similarity method with emphasis on its group theoretical aspects. The thesis deals with certain methods of finding exact solutions of a number of non-linear partial differential equations of importance to theoretical physics. Some of these new solutions are of relevance from the applications point of view in diverse branches such as elementary particle physics, field theory, solid state physics and non-linear optics and give some insight into the stable or unstable behavior of dynamical Systems The thesis consists of six chapters.

Numerical solution of some nonlocal, nonlinear dispersive wave equations

We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, for which there is at present no analytic proof. Thus we study in particular the resolution property of arbitrary initial profiles into sequences of solitary waves for both equations and clean interaction of Benjamin-Ono solitary waves. We also verify numerically that the behaviour of the solution of the Intermediate Long Wave equation as the model parameter tends to the infinite depth limit is the one predicted by the theory.

Resonant gravity-wave drag enhancement in linear stratified flow over mountains

High-drag states produced in stratified flow over a 2D ridge and an axisymmetric mountain are investigated using a linear, hydrostatic, analytical model. A wind profile is assumed where the background velocity is constant up to a height z1 and then decreases linearly, and the internal gravity-wave solutions are calculated exactly. In flow over a 2D ridge, the normalized surface drag is given by a closed-form analytical expression, while in flow over an axisymmetric mountain it is given by an expression involving a simple 1D integral. The drag is found to depend on two dimensionless parameters: a dimensionless height formed with z_1, and the Richardson number, Ri, in the shear layer. The drag oscillates as z_1 increases, with a period of half the hydrostatic vertical wavelength of the gravity waves. The amplitude of this modulation increases as Ri decreases. This behaviour is due to wave reflection at z_1. Drag maxima correspond to constructive interference of the upward- and downward-propagating waves in the region z < z_1, while drag minima correspond to destructive interference. The reflection coefficient at the interface z = z_1 increases as Ri decreases. The critical level, z_c, plays no role in the drag amplification. A preliminary numerical treatment of nonlinear effects is presented, where z_c appears to become more relevant, and flow over a 2D ridge qualitatively changes its character. But these effects, and their connection with linear theory, still need to be better understood.

Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilization waves on eggs

Wave solutions to a mechanochemical model for cytoskeletal activity are studied and the results applied to the waves of chemical and mechanical activity that sweep over an egg shortly after fertilization. The model takes into account the calcium-controlled presence of actively contractile units in the cytoplasm, and consists of a viscoelastic force equilibrium equation and a conservation equation for calcium. Using piecewise linear caricatures, we obtain analytic solutions for travelling waves on a strip and demonstrate uiat the full nonlinear system behaves as predicted by the analytic solutions. The equations are solved on a sphere and the numerical results are similar to the analytic solutions. We indicate how the speed of the waves can be used as a diagnostic tool with which the chemical reactivity of the egg surface can be measured.

Soliton solutions for quasilinear Schrodinger equations with critical growth

In this paper we establish the existence of standing wave solutions for quasilinear Schrodinger equations involving critical growth. By using a change of variables, the quasilinear equations are reduced to semilinear one. whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration-compactness principle due to P.L. Lions together with some classical arguments used by H. Brezis and L. Nirenberg (1983) in [9]. (C) 2009 Elsevier Inc. All rights reserved.

On Modified Method of Simplest Equation for Obtaining Exact Solutions of Nonlinear PDEs: Case of Elliptic Simplest Equation

2010 Mathematics Subject Classification: 74J30, 34L30.

Symmetries and exact solutions of KP equation with an arbitrary nonlinear term

The generalized KP (GKP) equations with an arbitrary nonlinear term model and characterize many nonlinear physical phenomena. The symmetries of GKP equation with an arbitrary nonlinear term are obtained. The condition that must satisfy for existence the symmetries group of GKP is derived and also the obtained symmetries are classified according to different forms of the nonlinear term. The resulting similarity reductions are studied by performing the bifurcation and the phase portrait of GKP and also the corresponding solitary wave solutions of GKP
equation are constructed.

Neural fields with sigmoidal firing rates: approximate solutions

Many tissue level models of neural networks are written in the language of nonlinear integro-differential equations. Analytical solutions have only been obtained for the special case that the nonlinearity is a Heaviside function. Thus the pursuit of even approximate solutions to such models is of interest to the broad mathematical neuroscience community. Here we develop one such scheme, for stationary and travelling wave solutions, that can deal with a certain class of smoothed Heaviside functions. The distribution that smoothes the Heaviside is viewed as a fundamental object, and all expressions describing the scheme are constructed in terms of integrals over this distribution. The comparison of our scheme and results from direct numerical simulations is used to highlight the very good levels of approximation that can be achieved by iterating the process only a small number of times.