Abstract : This mini-symposium is devoted to all research areas that are related to nonlinear differential equations and their applications in science and engineering. The main focus of this mini-symposium is on the Lie symmetry analysis, conservation laws and their applications to ordinary and partial differential equations. These differential equations could originate from mathematical models of diverse disciplines such as architecture, chemical kinetics, civil engineering, ecology, economics, engineering, fluid mechanics, biology and finance. Other approaches in finding exact solutions to nonlinear differential equations will also be discussed. This includes, but not limited to, asymptotic analysis methodologies, bifurcation theory, inverse scattering transform techniques, the Hirota method, the Adomian decomposition method, and others.
[02723] Conservation laws and variational structure of damped nonlinear wave equations
Format : Talk at Waseda University
Author(s) :
Almudena P. Márquez (University of Cadiz)
Stephen Anco (Brock University)
Tamara M. Garrido (University of Cadiz)
María L. Gandarias (University of Cadiz)
Abstract : All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light-cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether’s theorem, which is applicable because the damping term can be transformed into a time-dependent self-interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well known variation symmetries admitted by nonlinear undamped wave equations in one dimension.
[03330] Constructing mass-conserving cnoidal wave solutions for the KdV equation
Format : Talk at Waseda University
Author(s) :
Carel Petrus Olivier (North-West University)
Frank Verheest (Universiteit Gent)
Abstract : Nonlinear periodic travelling wave solutions of the Korteweg-deVries (KdV) equation in the form of cnoidal wave solutions are investigated. The general cnoidal wave solution does not ensure that the mass of the undisturbed medium is conserved. In this paper, a framework is provided to construct mass-conserving cnoidal wave solutions., and the resulting solutions are analyzed. It is shown that these solutions are consistent with linear solutions in the small amplitude limit.
[04290] Conservation laws and symmetries of a Generalized Drinfeld-Sokolov system
Format : Talk at Waseda University
Author(s) :
Tamara M. Garrido (University of Cadiz)
Rafael De La Rosa (University of Cadiz)
Elena Recio (University of Cadiz)
Almudena P. Márquez (University of Cadiz)
Abstract : The generalized Drinfeld-Sokolov system is a widely-used model that describes wave phenomena in various contexts. Many properties of this system, such as Hamiltonian formulations and integrability, have been extensively studied, and exact solutions have been derived for specific cases. In this paper, we apply the direct method of multipliers to obtain all low-order local conservation laws of the system. These laws correspond to physical quantities that remain constant over time, such as energy and momentum, and we provide a physical interpretation for each of them. Additionally, we investigate the Lie point symmetries and first-order symmetries of the system. Through the point symmetries and constructing the optimal systems of one-dimensional subalgebras, we are able to reduce the system of partial differential equations to ordinary differential systems.
[02959] Lie symmetry analysis of flow and pressure inside horizontal chamber
Format : Talk at Waseda University
Author(s) :
Tanki Motsepa (University of Mpumalanga)
Modisawatsona Lucas Lekoko (North-West University)
Gabriel Magalakwe (North-West University)
Abstract : Exact solutions improve industrial processes by giving operators greater grasp of how systems operate. The study aims to find exact momentum and pressure solutions during the unsteady filtration process. Lie symmetry analysis is used to transform a system of PDEs representing the case study into solvable ODEs. The ODEs are then solved to obtain velocity and pressure solutions. Effects of parameters resulting from the dynamics are examined to identify the parameters that yield maximum outflow.
Chaudry Masood Khalique (North-West University, South Africa)
Abstract : In this presentation, we examine the dynamics of soliton waves associated with higher-order nonlinear partial differential equations, which have applications in various fields of science and engineering. Our focus is on the fifth-order strain wave equation, for which we employ the Lie group theory of differential equations to obtain analytic solutions. Specifically, we use this technique to systematically generate the Lie point symmetries spanned by the equation, which we then use to reduce it to ordinary differential equations that can be solved to obtain closed-form solutions. The ordinary differential equations are solved by direct integration and the engagement of two methods, the simplest method and generalized tanh-function method. We successfully identify soliton solutions, including dark and singular period solitons, and depict them graphically to better understand their physical meaning. We then use the multiplier method to obtain conserved vectors. Our analysis sheds light on the wave structures associated with the strain wave equation and provides insight into the physical implications of the soliton solutions.
[02308] Symmetry solutions and conservation laws of the derivative nonlinear Schrodinger equation
Chaudry Masood Khalique (North-West University, South Africa)
Abstract : In this talk we study the derivative nonlinear Schrodinger equation.This equation has many applications, for example in the propagation of circular polarized nonlinear Alfven waves in plasmas. We present general and special solutions of this equation using Lie group theory. We also derive conservation laws for the underlying equation.
[03449] Lie symmetry analysis of new 3-D fifth-order nonlinear Wazwaz equation
Abstract : In this talk, we present the analytical examination of a new (3+1)-dimensional fifth-order nonlinear Wazwaz equation with third-order dispersion terms in ocean physics and other nonlinear sciences. We apply Lie group analysis to obtain various infinitesimal generators admitted by the equation. The generators are used to reduce the understudy equation to achieve copious group-invariant solutions. Thus, various closed-form solutions are obtained for the equation. We further construct its conservation laws.
[03451] Lie group analysis of the nonlinear 3D KP-BBM equation
Format : Online Talk on Zoom
Author(s) :
Jonathan Lebogang Bodibe (North-West University, Mafikeng Campus)
Chaudry Masood Khalique (North-West University, South Africa)
Abstract : In this talk, we present Lie group analysis of the nonlinear (3+1)-dimensional Kadomtsev Petviashvili Benjamin Bona Mahony equation. We find exact solutions of the equation using Lie symmetry method together with Kudryashov's and (G’/G)-expansion methods. Moreover, we derive the conservation laws for the equation using the multiplier and Ibragimov’s methods.
[04488] Integrable equations and Riemann-Hilbert problems
Format : Talk at Waseda University
Author(s) :
Wen-Xiu Ma (University of South Florida)
Abstract : This talk covers the zero curvature formulation and the Riemann-Hilbert technique. Nonlocal integrable equations are derived from conducting group reductions. The associated matrix spectral problems are used to build a kind of Riemann-Hilbert problems, whose reflectionless cases generate soliton solutions.
[04064] A study of 3D generalized nonlinear wave equation in fluids
Format : Talk at Waseda University
Author(s) :
Chaudry Masood Khalique (North-West University, South Africa)
Abstract : In this talk we study a (3+1)-dimensional generalized nonlinear wave equation of fluids. Using Lie symmetry methods, we transform the underlying equation into a nonlinear ordinary differential equation. We deduce periodic, trigonometric bright soliton together with singular soliton solutions. Moreover, some soliton solutions are secured via the simplest equation method in the form of Jacobi elliptic functions. The dynamics of the solutions are depicted using suitable graphs. Furthermore, we construct conservation laws of the equation by employing Ibragimov's theorem.
Abstract : We present some recent advances of the applications of one-parameter Lie group transformations of famous partial differential equations. In particular, we discuss the Burgers' equation, which are often the benchmarks in the study of differential equations. We exploit the link between the equation and a recursion operator and show how the full hierarchy may be solved.
[03471] Nonclassical Potential Symmetries for the transient heat transfer equation
Format : Talk at Waseda University
Author(s) :
Mpho Nkwanazana (Sefako Makgatho Health Science Universitye)
Raseelo Moitsheki (University of the Witwatersrand)
Abstract : In this article we consider the one dimensional transient heat conduction
equation. The diffusivity term and internal heat generator are given by the
power law. The objective is to employ nonclassical and nonlocal approach to
generate nonclassical potential symmetries.
