The tanh and the sine-cosine methods for the complex modified K dV and the generalized K dV equations

AbstractThe complex modified K dV (CMK dV) equation and the generalized K dV equation are investigated by using the tanh method and the sine-cosine method. A variety of exact travelling wave solutions with compact and noncompact structures are formally obtained for each equation. The study reveals the power of the two schemes where each method complements the other

Soliton and periodic wave solutions to the osmosis K(2, 2) equation

In this paper, two types of traveling wave solutions to the osmosis K(2, 2) equation are investigated. They are characterized by two parameters. The expresssions for the soliton and periodic wave solutions are obtained.Comment: 14 pages, 16 figure

Compactons and kink-like solutions of BBM-like equations by means of factorization

In this work, we study the Benjamin-Bona-Mahony like equations with a fully nonlinear dispersive term by means of the factorization technique. In this way we find the travelling wave solutions of this equation in terms of the Weierstrass function and its degenerated trigonometric and hyperbolic forms. Then, we obtain the pattern of periodic, solitary, compacton and kink-like solutions. We give also the Lagrangian and the Hamiltonian, which are linked to the factorization, for the nonlinear second order ordinary differential equations associated to the travelling wave equations.Comment: 10 pages, 8 figure

Shock-peakon and shock-compacton solutions for K(p,q) equation by variational iteration method

AbstractBy variational iteration method, we obtain new solitary solutions for non-linear dispersive equations. Particularly, shock-peakon solutions in K(2,2) equation and shock-compacton solutions in K(3,3) equation are found by this simple method. These two types of solutions are new solitary wave solutions which have the shapes of shock solutions and compacton solutions (or peakon solutions)

Peakon, Cuspon, Compacton, and Loop Solutions of a Three-Dimensional 3DKP(3, 2) Equation with Nonlinear Dispersion

We study peakon, cuspon, compacton, and loop solutions for the three-dimensional Kadomtsev-Petviashvili equation (3DKP(3,2) equation) with nonlinear dispersion. Based on the method of dynamical systems, the 3DKP(3,2) equation is shown to have the parametric representations of the solitary wave solutions such as peakon, cuspon, compacton, and loop solutions. As a result, the conditions under which peakon, cuspon, compacton, and loop solutions appear are also given

Dynamical systems : mechatronics and life sciences

Proceedings of the 13th Conference „Dynamical Systems - Theory and Applications" summarize 164 and the Springer Proceedings summarize 60 best papers of university teachers and students, researchers and engineers from whole the world. The papers were chosen by the International Scientific Committee from 315 papers submitted to the conference. The reader thus obtains an overview of the recent developments of dynamical systems and can study the most progressive tendencies in this field of science

Methods of symmetry reduction and their application

In this thesis methods of symmetry reduction are applied to several physically relevant partial differential equations. The first chapter serves to acquaint the reader with the symmetry methods used in this thesis. In particular the classical method of Lie, an extension of it by Bluman and Cole [1969], known as the nonclassical method, and the direct method of Clarkson and Kruskal [1989] are described. Other known extensions of these methods are outlined, including potential symmetries, introduced by Bluman, Kumei and Reid [1988]. Also described are the tools used in practice to perform the calculations. The remainder of the thesis is split into two parts. In Part One the classical and nonclassical methods are applied to three classes of scalar equation: a generalised Boussinesq equation, a class of third order equations and a class of fourth order equations. Many symmetry reductions and exact solutions are found. In Part Two each of the classical, nonclassical and direct methods are applied to various systems of partial differential equations. These include shallow water wave systems, six representations of the Boussinesq equation and a reaction-diffusion equation written as a system. In Chapters Five and Six both the actual application of these methods and their results is compared and contrasted. In such applications, remarkable phenomena can occur, in both the nonclassical and direct methods. In particular it is shown that the application of the direct method to systems of equations is not as conceptually straightforward as previously thought, and a way of completing the calculations of the nonclassical method via hodograph transformations is introduced. In Chapter Seven it is shown how more symmetry reductions may be found via nonclassical potential symmetries, which are a new extension on the idea of potential symmetries. In the final chapter the relationship between the nonclassical and direct methods is investigated in the light of the previous chapters. The thesis is concluded with some general remarks on its findings and on possible future work

Studies on Lattice Systems Motivated by PT-Symmetry and Granular Crystals

This dissertation aims to study some nonlinear lattice dynamical systems arising in various areas, especially in nonlinear optics and in granular crystals. At first, we study the 2-dimensional PT-symmetric square lattices (of the discrete non-linear Schr¨odinger (dNLS) type) and identify the existence, stability and dynamical evolu- tion of stationary states, including discrete solitons and vortex configurations. To enable the analytical study, we consider the so-called anti-continuum (AC) limit of lattices with uncoupled sites and apply the Lyapunov–Schmidt reduction. Numerical experiments will also be provided accordingly. Secondly, we investigate the nonlinear waves in the granular chains of elastically inter- acting (through the so-called Hertzian contacts) beads. Besides the well-understood standard one-component granular chain, the traveling waves and dynamics of its variants such as heterogeneous granular chains and locally resonant granular crystals (otherwise known as mass-in-mass (MiM) or mass-with-mass (MwM) systems) are also studied. One of our goals is to systematically understand the propagation of traveling waves with an apparently non-decaying oscillating tail in MiM/MwM systems and the antiresonance mechanisms that lead to them, where Fourier transform and Fourier series are utilized to obtain integral reformulations of the problem. Additionally, we study finite mixed granular chains with strong precompression and classify different families of systems based on their well-studied linear limit. Motivated by the study of isospectral spring-mass systems (i.e., spring-mass systems that bear the same eigenfrequencies), we present strategies for building granular chains that are isospectral in the linear limit and their nonlinear dynamics will also be considered

Mathematical analysis for tumor growth model of ordinary differential equations

Special functions occur quite frequently in mathematical analysis and lend itself rather frequently in physical and engineering applications. Among the special functions, gamma function seemed to be widely used. The purpose of this thesis is to analyse the various properties of gamma function and use these properties and its definition to derive and tackle some integration problem which occur quite frequently in applications. It should be noted that if elementary techniques such as substitution and integration by parts were used to tackle most of the integration problems, then we will end up with frustration. Due to this, importance of gamma function cannot be denied