On the wave length of smooth periodic traveling waves of the Camassa-Holm equation

This paper is concerned with the wave length $\lambda$ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height $a$ (or "peak-to-peak amplitude"). Our main result establishes monotonicity properties of the map $a\longmapsto \lambda(a)$, i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated to the equation, which distinguish between the two possible qualitative behaviours of $\lambda(a)$, namely monotonicity and unimodality. The key point is to relate $\lambda(a)$ to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.Comment: 14 pages, 5 figure

On asymptotically equivalent shallow water wave equations

The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it family} of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama-Helmholtz transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de Vries equation, and a generalization of the Camassa-Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to 5th order. The travelling wave solutions of the CH equation are shown to agree to 5th order with the exact solution

Some new exact solutions of (4+1)-dimensional Davey–Stewartson-Kadomtsev–Petviashvili equation

Exact solutions of nonlinear equations have got formidable attraction of researchers because these solutions demonstrate the physical behaviour of a model. In this paper, we focus on extracting some new exact solutions of a (4+1)-dimensional Davey–Stewartson-Kadomtsev–Petviashvili (DSKP) equation. To find new travelling wave solutions of the DSKP equation, we use ()-expansion technique. The obtained solutions are in the form of the exponential and trigonometric functions. We obtain different kinds of waves solutions for specific values of parameters. We simulate the achieved solutions in 3D and 2D plots.The authors are grateful to the Basque Government, Spain for its support through Grant IT1555-22 and to MCIN/AEI 269.10.13039/5011 00011033 for Grant PID2021-1235430B-C21/C22. All authors approved the version of the manuscript to be published

Smooth and Peaked Solitons of the CH equation

The relations between smooth and peaked soliton solutions are reviewed for the Camassa-Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann-Hilbert problem. The momentum map from the action-angle scattering variables $T^*({\mathbb{T}^N})$ to the flow momentum ($\mathfrak{X}^*$) provides the Eulerian representation of the $N$-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The dispersionless limit of the CH equation and its resulting peakon solutions are examined by using an asymptotic expansion in the dispersion parameter. The peakon solutions of the dispersionless CH equation in one dimension are shown to generalize in higher dimensions to peakon wave-front solutions of the EPDiff equation whose associated momentum is supported on smoothly embedded subspaces. The Eulerian representations of the singular solutions of both CH and EPDiff are given by the (cotangent-lift) momentum maps arising from the left action of the diffeomorphisms on smoothly embedded subspaces.Comment: First version -- comments welcome! Submitted to JPhys

An infinite branching hierarchy of time-periodic solutions of the Benjamin-Ono equation

We present a new representation of solutions of the Benjamin-Ono equation that are periodic in space and time. Up to an additive constant and a Galilean transformation, each of these solutions is a previously known, multi-periodic solution; however, the new representation unifies the subset of such solutions with a fixed spatial period and a continuously varying temporal period into a single network of smooth manifolds connected together by an infinite hierarchy of bifurcations. Our representation explicitly describes the evolution of the Fourier modes of the solution as well as the particle trajectories in a meromorphic representation of these solutions; therefore, we have also solved the problem of finding periodic solutions of the ordinary differential equation governing these particles, including a description of a bifurcation mechanism for adding or removing particles without destroying periodicity. We illustrate the types of bifurcation that occur with several examples, including degenerate bifurcations not predicted by linearization about traveling waves.Comment: 27 pages, 6 figure

Dimension-breaking for Traveling Waves in Interfacial Flows

Fluid flow models in two spatial dimensions with a one-dimensional interface are known to support overturned traveling solutions. Computational methods of solving the two-dimensional problem are well developed, even in the case of overturned waves. The three-dimensional problem is harder for three prominent reasons. First, some formulations of the two-dimensional problem do not extend to three-dimensions. The technique of conformal mapping is a prime example, as it is very efficient in two dimensions but does not have a three-dimensional equivalent. Second, some three-dimensional models, such as the Transformed Field Expansion method, do not allow for overturned waves. Third, computational time can increase by more than an order of magnitude. For example, the Birkhoff-Rott integral has a cost of O(N2) in two-dimensions but O(N4M2) in three-dimensions, where N is the number of discretized points in the lateral directions and M is the number of truncated summation terms. This study seeks to bridge the gap between efficient two-dimensional numerical solvers and more computationally expensive three-dimensional solvers. The dissertation does so by developing a dimension-breaking continuation method, which is not limited to solving interfacial wave models. The method involves three steps: first, conduct N-dimensional continuation to large amplitude; second, extend the solution trivially to a (N+1)-dimensional solution and solve the linearization; and third, use the linearization to begin (N+1)-dimensional continuation. This method is successfully applied to Kadomtsev-Petviashvili and Akers-Milewski interfacial models and then in a reduced Vortex Sheet interfacial formulation. In doing so, accurate search directions are calculated for use in higher-dimension quasi-Newton solvers

Stability of periodic traveling flexural‐gravity waves in two dimensions

In this work, we solve the Euler’s equations for periodic waves travelling under a sheet of ice using a reformulation introduced in [1]. These waves are referred to as flexural-gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic travelling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier-Floquet-Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyse high frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice