Similar oscillations on both sides of a shock. Part I. Even-odd alternative dispersions and general assignments of Fourier dispersions towards a unfication of dispersion models

We consider assigning different dispersions for different dynamical modes, particularly with the distinguishment and alternation of opposite signs for alternative Fourier components. The Korteweg-de Vries (KdV) equation with periodic boundary condition and longest-wave sinusoidal initial field, as used by N. Zabusky and M. D. Kruskal, is chosen for our case study with such alternating-dispersion of the Fourier modes of (normalized) even and odd wavenumbers. Numerical results verify the capability of our new model to produce two-sided (around the shock) oscillations, as appear on both sides of some ion-acoustic and quantum shocks, not admitted by models such as the KdV(-Burgers) equation, but also indicate even more, including singular zero-dispersion limit or non-convergence to the classical shock (described by the entropy solution), non-thermalization (of the Galerkin-truncated models) and applicability to other models (showcased by the modified KdV equation with cubic nonlinearity). A unification of various dispersive models, keeping the essential mathematical elegance (such as the variational principle and Hamiltonian formulation) of each, for phenomena with complicated dispersion relation is thus suggested with a further explicit example of two even-order dispersions (from the Hilbert transforms) extending the Benjamin-Ono model. The most general situation can be simply formulated by the introduction of the dispersive derivative, the indicator function and the Fourier transform, resulting in an integro-differential dispersion equation. Other issues such as the real-number order dispersion model and the transition from non-thermalization to thermalization and, correspondingly, from regularization to non-regularization for untruncated models are also briefly remarked

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

An Extended Auxiliary Function Method and Its Application in mKdV Equation

An extended auxiliary function method is presented for constructing exact traveling wave solutions to nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions to the elliptic equation to construct exact traveling wave solutions for nonlinear partial differential equations. mKdV equation is chosen to illustrate the application of the extended auxiliary function method. Consequently, more new exact traveling wave solutions are derived that are not obtained by the previously known methods

Dispersive shock waves and modulation theory

There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G. B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs

Inverse Scattering Transform Method for Lattice Equations

The main original contribution of this thesis is the development of a fully discrete inverse scattering transform (IST) method for nonlinear partial difference equations. The equations we solve are nonlinear partial difference equations on a quad-graph, also called lattice equations, which are known to be multidimensionally consistent in N dimensions for arbitrary N. Such equations were discovered by Nijhoff, Quispel and Capel and Adler and later classified by Adler, Bobenko and Suris. The main equation solved by our IST framework is the Q3δ lattice equation. Our approach also solves all of its limiting cases, including H1, known as the lattice potential KdV equation. Our results provide the discrete analogue of the solution of the initial value problem on the real line. We provide a rigorous justification that solves the problem for wide classes of initial data given along initial paths in a multidimensional lattice. Moreover, we show how soliton solutions arise from the IST method and also utilise asymptotics of the eigenfunctions to construct infinitely many conservation laws

CONSERVATION LAWS OF A NONLINEAR INCOMPRESSIBLE TWO-FLUID MODEL

We study the conservation laws of the Choi-Camassa two-fluid model (1999) which is developed by approximating the two-dimensional (2D) Euler equations for incompressible motion of two non-mixing fluids in a channel. As preliminary work of this thesis, we compute the basic local conservation laws and the point symmetries of the 2D Euler equations for the incompressible fluid, and those of the vorticity system of the 2D Euler equations. To serve the main purpose of this thesis, we derive local conservation laws of the Choi-Camassa equations with an explicit expression for each locally conserved density and corresponding spatial flux. Using the direct conservation law construction method, we have constructed seven conservation laws including the conservation of mass, total horizontal momentum, energy, and irrotationality. The conserved quantities of the Choi-Camassa equations are compared with those of the full 2D Euler equations of incompressible fluid. We review periodic solutions, solitary wave solutions and kink solutions of the Choi-Camassa equations. As a result of the presence of Galilean symmetry for the Choi-Camassa model, the solitary wave solutions, the kink and the anti-kink solutions travel with arbitrary constant wave speed. We plot the local conserved densities of the Choi-Camassa model on the solitary wave and on the kink wave. For the solitary waves, all the densities are finite and decay exponentially, while for the kink wave, all the densities except one are finite and decay exponentially

An analysis of symmetries and conservation laws of some classes of PDEs that arise in mathematical physics and biology.

Doctor of Philosophy in Applied Mathematics. University of KwaZulu-Natal, Durban, 2016.In this thesis, the symmetry properties and the conservation laws for a number of well-known PDEs which occur in certain areas of mathematical physics are studied. We focus on wave equations that arise in plasma physics, solid physics and fluid mechanics. Firstly, we carry out analyses for a class of non-linear partial differential equations, which describes the longitudinal motion of an elasto-plastic bar and anti-plane shearing deformation. In order to systematically explore the mathematical structure and underlying physics of the elasto-plastic flow in a medium, we generate all the geometric vector fields of the model equations. Using the classical Lie group method, it is shown that this equation does not admit space dilation type symmetries for a speci fic parameter value. On the basis of the optimal system, the symmetry reductions and exact solutions to this equation are derived. The conservation laws of the equation are constructed with the help of Noether's theorem We also consider a generalized Boussinesq (GB) equation with damping term which occurs in the study of shallow water waves and a system of variant Boussinesq equations. The conservation laws of these systems are derived via the partial Noether method and thus demonstrate that these conservation laws satisfy the divergence property. We illustrate the use of these conservation laws by obtaining several solutions for the equations through the application of the double reduction method, which encompasses the association of symmetries and conservation laws. A similar analysis is performed for the generalised Gardner equation with dual power law nonlinearities of any order. In this case, we derive the conservation laws of the system via the Noether approach after increasing the order and by the use of the multiplier method. It is observed that only the Noether's approach gives a uni ed treatment to the derivation of conserved vectors for the Gardner equation and can lead to local or an in finite number of nonlocal conservation laws. By investigating the solutions using symmetry analysis and double reduction methods, we show that the double reduction method yields more exact solutions; some of these solutions cannot be recovered by symmetry analysis alone. We also illustrate the importance of group theory in the analysis of equations which arise during investigations of reaction-diffusion prey-predator mechanisms. We show that the Lie analysis can help obtain different types of invariant solutions. We show that the solutions generate an interesting illustration of the possible behavioural patterns