Accelerating dynamical peakons and their behaviour

A wide class of nonlinear dispersive wave equations are shown to possess a novel type of peakon solution in which the amplitude and speed of the peakon are time-dependent. These novel dynamical peakons exhibit a wide variety of different behaviours for their amplitude, speed, and acceleration, including an oscillatory amplitude and constant speed which describes a peakon breather. Examples are presented of families of nonlinear dispersive wave equations that illust rate various interesting behaviours, such as asymptotic travelling-wave peakons, dissipating/anti-dissipating peakons, direction-reversing peakons, runaway and blow up peakons, among others.Comment: Typos corected. To appear in DCDS

Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs

Dynamical PDEs that have a spatial divergence form possess conservation laws that involve an arbitrary function of time. In one spatial dimension, such conservation laws are shown to describe the presence of an x-independent source/sink; in two and more spatial dimensions, they are shown to produce a topological charge. Two applications are demonstrated. First, a topological charge gives rise to an associated spatial potential system, allowing non-local conservation laws and symmetries to be found for a given dynamical PDE. This type of potential system has a different form and different gauge freedom compared to potential systems that arise from ordinary conservation laws. Second, when a topological charge arises from a conservation law whose conserved density is non-trivial off of solutions to the dynamical PDE, then this relation yields a constraint on initial/boundary data for which the dynamical PDE will be well posed. Several examples of nonlinear PDEs from applied mathematics and integrable system theory are used to illustrate these results

Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations

A class of generalized nonlinear p-Laplacian evolution equations is studied. These equations model radial diffusion–reaction processes in n ≥ 1 dimensions, where the diffusivity depends on the gradient of the flow. For this class, all local conservation laws of low-order and all Lie symmetries are derived. The physical meaning of the conservation laws is discussed, and one of the conservation laws is used to show that the nonlinear equation can be mapped invertibly into a linear equation by a hodograph transformation in certain cases. The symmetries are used to derive exact group-invariant solutions from solvable three-dimensional subgroups of the full symmetry group, which yields a direct reduction of the nonlinear equation to a quadrature. The physical and analytical properties of these exact solutions are explored, some of which describe moving interfaces and Green’s functions

Conserved norms and related conservation laws for multi-peakon equations

All nonlinear dispersive wave equations in the general class m_t + f (u, u_x)m + (g(u, u_x)m)_x = 0 are known to possess multi-peakon weak solutions. A classification is presented for families of multi-peakon equations in this class that possess conserved momentum; conserved H^1 norm of u; conserved H^2 norm of u; conserved L^2 norm of m; related conservation laws. The results yield, among others, two interesting wide families of equations: m_t + 2u_xh(u, u_x)m + u(h(u, u_x)m)_x = 0 for which the H^1 norm of u is conserved; m_t − (1/2)*u_xh'(u)m + (h(u)m)_x = 0 for which the L^2 norm of m is conserved. The overlap of these two families yields a singular equation which is nevertheless found to possess both smooth solitary wave solutions and peakon travelling wave solutions

A general family of multi-peakon equations and their properties

A general family of peakon equations is introduced, involving two arbitrary functions of the wave amplitude and the wave gradient. This family contains all of the known breaking wave equations, including the integrable ones: Camassa–Holm equation, Degasperis–Procesi equation, Novikov equation, and FORQ/modified Camassa–Holm equation. One main result is to show that all of the equations in the general family possess weak solutions given by multi-peakons which are a linear superposition of peakons with timedependent amplitudes and positions. In particular, neither an integrability structure nor a Hamiltonian structure is needed to derive N-peakon weak solutions for arbitrary N > 1. As a further result, single peakon travellingwave solutions are shown to exist under a simple condition on one of the two arbitrary functions in the general family of equations, and when this condition fails, generalized single peakon solutions that have a time-dependent amplitude and a time-dependent speed are shown to exist. An interesting generalization of the Camassa–Holm and FORQ/modified Camassa–Holm equations is obtained by deriving the most general subfamily of peakon equations that possess the Hamiltonian structure shared by the Camassa–Holm and FORQ/ modified Camassa–Holm equations. Peakon travelling-wave solutions and their features, including a variational formulation (minimizer problem), are derived for these generalized equations. A final main result is that two-peakon weak solutions are investigated and shown to exhibit several novel kinds of behaviour, including the formation of a bound pair consisting of a peakon and an anti-peakon that have a maximum finite separation

Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation

For a generalized KdV-Burgers-Kuramoto equation we have studied conservation laws by using the multiplier method, and investigated its first-level and second level potential systems. Furthermore, the Lie point symmetries of the equation and the Lie point symmetries associated with the conserved vectors are determined. We obtain travellingwave reductions depending on the form of an arbitrary function. We present some explicit solutions: soliton solutions, kinks and antikinks

Travelling wave solutions on a non-zero background for the generalized Korteweg–de Vries equation

For the generalized p-power Korteweg–deVries equation, all non-periodic travelling wave solutions with non-zero boundary conditions are explicitly classified for all integer powers p >=1. These solutions are shown to consist of: bright solitary waves and static humps on a non-zero background for odd p; dark solitary waves on a non-zero background and kink (shock) waves for even p in the defocusing case; pairs of bright/dark solitary waves on a non-zero background, and also bright and dark heavy-tail waves (with power decay) on a non-zero background, for even p in the focussing case.An explicit physical parameterization is given for each type of solution in terms of the wave speed c, background size b, and wave height/depth h. The allowed kinematic region for existence of the solutions is derived, and other main kinematic features are discussed. Analytical formulas are presented in the higher power cases p = 3, 4, which are compared to the integrable cases p = 1, 2.51 página