Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations

We derive explicit formulas for the characteristic curves associated with the multipeakon solutions of the Camassa-Holm, Degasperis-Procesi and Novikov equations. Such a curve traces the path of a fluid particle whose instantaneous velocity equals the elevation of the wave at that point (or the square of the elevation, in the Novikov case). The peakons themselves follow characteristic curves, and the remaining characteristic curves can be viewed as paths of 'ghostpeakons' with zero amplitude; hence, they can be obtained as solutions of the ODEs governing the dynamics of multipeakon solutions. The previously known solution formulas for multipeakons only cover the case when all amplitudes are nonzero, since they are based upon inverse spectral methods unable to detect the ghostpeakons. We show how to overcome this problem by taking a suitable limit in terms of spectral data, in order to force a selected peakon amplitude to become zero. Moreover, we use direct integration to compute the characteristic curves for the solution of the Degasperis-Procesi equation where a shockpeakon forms at a peakon-antipeakon collision. In addition to the theoretical interest in knowing the characteristic curves, they are also useful for plotting multipeakon solutions, as we illustrate in several examples

Formation and Dynamics of Shock Waves in the Degasperis-Procesi Equation

Solutions of the Degasperis–Procesi nonlinear wave equation may develop discontinuities in finite time. As shown by Coclite and Karlsen, there is a uniquely determined entropy weak solution which provides a natural continuation of the solution past such a point. Here we study this phenomenon in detail for solutions involving interacting peakons and antipeakons. We show that a jump discontinuity forms when a peakon collides with an antipeakon, and that the entropy weak solution in this case is described by a "shockpeakon" ansatz reducing the PDE to a system of ODEs for positions, momenta, and shock strengths

Integrable and non-integrable equations with peaked soliton solutions

This thesis explores a number of nonlinear PDEs that have peaked soliton solutions, to apply reductions to such PDEs and solve the resultant equations. Chapter 1 provides a brief history of peakon equations, where they come from and the different viewpoints of various authors. The rest of the chapter is then devoted to detailing the mathematical tools that will be used throughout the rest of the thesis. Chapter 2 concerns a coupling of two integrable peakon equations, namely the Popowicz system, which itself is not integrable. The 2-peakon dynamics are studied, and an explicit solution to the 2-peakon dynamics is given alongside some features of the interaction. In chapter 3 a reduction from two integrable peakon equations with quadratic nonlinearity to the third Painlev´e equation is given. B¨acklund transformations and solutions for the Painlev´e equations are expressed, and then used to find solutions of the original PDEs. A general peakon family, the b-family, is also explored, giving a more general result. Chapter 4 examines two peakon equations with cubic nonlinearity, and their reductions to Painlev´e equations. A link is shown between these cubic nonlinear peakon equations and the quadratic nonlinear equations in chapter 3. Chapter 5 has conclusions and outlook in the area

Numerical superposition of Gaussian beams over propagating domain for high frequency waves and high-order invariant-preserving methods for dispersive waves

This thesis is devoted to efficient numerical methods and their implementations for two classes of wave equations. The first class is linear wave equations in very high frequency regime, for which one has to use some asymptotic approach to address the computational challenges. We focus on the use of the Gaussian beam superposition to compute the semi--classical limit of the Schr {o}dinger equation. The second class is dispersive wave equations arising in modeling water waves. For the Whitham equation, so-called the Burgers--Poisson equation, we design, analyze, and implement local discontinuous Galerkin methods to compute the energy conservative solutions with high-order of accuracy. Our Gaussian beam (GB) approach is based on the domain-propagation GB superposition algorithm introduced by Liu and Ralston [Multiscale Model. Simul., 8(2), 2010, 622--644]. We construct an efficient numerical realization of the domain propagation-based Gaussian beam superposition for solving the Schr odinger equation. The method consists of several significant steps: a semi-Lagrangian tracking of the Hamiltonian trajectory using the level set representation, a fast search algorithm for the effective indices associated with the non-trivial grid points that contribute to the approximation, an accurate approximation of the delta function evaluated on the Hamiltonian manifold, as well as efficient computation of Gaussian beam components over the effective grid points. Numerical examples in one and two dimensions demonstrate the efficiency and accuracy of the proposed algorithms. For the Burgers--Poisson equation, we design, analyze and test a class of local discontinuous Galerkin methods. This model, proposed by Whitham [Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974] as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. The proposed numerical method is high order accurate and preserves two invariants, hence producing solutions with satisfying long time behavior. The $L^2$-stability of the scheme for general solutions is a consequence of the energy preserving property. The optimal order of accuracy for polynomial elements of even degree is proven. A series of numerical tests is provided to illustrate both accuracy and capability of the method

An explicit finite difference scheme for the Camassa-Holm equation

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of Camassa-Holm equation.Comment: 45 pages, 6 figure