Shock waves in dispersive hydrodynamics with non-convex dispersion

Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose-Einstein condensates. As in the classical theory of hyperbolic equations where a non-convex flux leads to non-classical solution structures, a non-convex linear dispersion relation provides an intriguing dispersive hydrodynamic analogue. Here, the fifth order Korteweg-de Vries (KdV) equation, also known as the Kawahara equation, a classical model for shallow water waves, is shown to be a universal model of Eulerian hydrodynamics with higher order dispersive effects. Utilizing asymptotic methods and numerical computations, this work classifies the long-time behavior of solutions for step-like initial data. For convex dispersion, the result is a dispersive shock wave (DSW), qualitatively and quantitatively bearing close resemblance to the KdV DSW. For non-convex dispersion, three distinct dynamic regimes are observed. For small jumps, a perturbed KdV DSW with positive polarity and orientation is generated, accompanied by small amplitude radiation from an embedded solitary wave leading edge, termed a radiating DSW or RDSW. For moderate jumps, a crossover regime is observed with waves propagating forward and backward from the sharp transition region. For jumps exceeding a critical threshold, a new type of DSW is observed we term a translating DSW or TDSW. The TDSW consists of a traveling wave that connects a partial, non-monotonic, negative solitary wave at the trailing edge to an interior nonlinear periodic wave. Its speed, a generalized Rankine-Hugoniot jump condition, is determined by the far-field structure of the traveling wave. The TDSW is resolved at the leading edge by a harmonic wavepacket moving with the linear group velocity. The non-classical TDSW exhibits features common to both dissipative and dispersive shock waves.Comment: 20 pages, 16 figure

A linearly implicit conservative difference scheme for the generalized Rosenau-Kawahara-RLW equation

This paper concerns the numerical study for the generalized Rosenau-Kawahara-RLW equation obtained by coupling the generalized Rosenau-RLW equation and the generalized Rosenau-Kawahara equation. We first derive the energy conservation law of the equation, and then develop a three-level linearly implicit difference scheme for solving the equation. We prove that the proposed scheme is energy-conserved, unconditionally stable and second-order accurate both in time and space variables. Finally, numerical experiments are carried out to confirm the energy conservation, the convergence rates of the scheme and effectiveness for long-time simulation.Comment: accepted in Applied Mathmatics and Computation

Meromorphic solutions of nonlinear ordinary differential equations

Exact solutions of some popular nonlinear ordinary differential equations are analyzed taking their Laurent series into account. Using the Laurent series for solutions of nonlinear ordinary differential equations we discuss the nature of many methods for finding exact solutions. We show that most of these methods are conceptually identical to one another and they allow us to have only the same solutions of nonlinear ordinary differential equations

A note on the stability for Kawahara-KdV type equations

In this paper we establish the nonlinear stability of solitary traveling-wave solutions for the Kawahara-KdV equation $$u_t+uu_x+u_{xxx}-\gamma_1 u_{xxxxx}=0,$$ and the modified Kawahara-KdV equation $$u_t+3u^2u_x+u_{xxx}-\gamma_2 u_{xxxxx}=0,$$ where $\gamma_i\in\mathbb{R}$ is a positive number when $i=1,2$. The main approach used to determine the stability of solitary traveling-waves will be the theory developed by AlbertComment: 8 pages, no figure

Error Estimate for a Fully Discrete Spectral Scheme for Korteweg-de Vries-Kawahara Equation

We are concerned with the convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (in short Kawahara equation), which is a transport equation perturbed by dispersive terms of 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the standard Fourier- Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L2-error bound of spectral accuracy in space and of second-order accuracy in time.Comment: 15 page

Convergence of numerical schemes for the Korteweg-de Vries-Kawahara equation

We are concerned with the convergence of a numerical scheme for the initial-boundary value problem associated to the Korteweg-de Vries- Kawahara equation (in short Kawahara equation), which is a transport equation perturbed by dispersive terms of 3rd and 5th order. This equation appears in several uid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in uid dynamics. We prove here the convergence of both semi-discrete as well as fully-discrete finite difference schemes for the Kawahara equation. Finally, the convergence is illustratred by several examples.Comment: 20 Page

Solitary wave solutions of several nonlinear PDEs modeling shallow water waves

We apply the version of the method of simplest equation called modified method of simplest equation for obtaining exact traveling wave solutions of a class of equations that contain as particular case a nonlinear PDE that models shallow water waves in viscous fluid (Topper-Kawahara equation). As simplest equation we use a version of the Riccati equation. We obtain two exact traveling wave solutions of equations from the studied class of equations and discuss the question of imposing boundary conditions on one of these solutions.Comment: 14 pages, no figure

Lattice Boltzmann Model for High-Order Nonlinear Partial Differential Equations

A general lattice Boltzmann (LB) model is proposed for solving nonlinear partial differential equations with the form $\partial_t \phi+\sum_{k=1}^{m} \alpha_k \partial_x^k \Pi_k (\phi)=0$, where $\alpha_k$ are constant coefficients, and $\Pi_k (\phi)$ are the known differential functions of $\phi$, $1\leq k\leq m \leq 6$. The model can be applied to the common nonlinear evolutionary equations, such as (m)KdV equation, KdV-Burgers equation, K($m,n$) equation, Kuramoto-Sivashinsky equation, and Kawahara equation, etc. Unlike the existing LB models, the correct constraints on moments of equilibrium distribution function in the proposed model are given by choosing suitable \emph{auxiliary-moments}, and how to exactly recover the macroscopic equations through Chapman-Enskog expansion is discussed in this paper. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.Comment: 18 pages, 4 figure

Traveling Wave Solutions to Fifth- and Seventh-order Korteweg-de Vries Equations: Sech and Cn Solutions

In this paper we review the physical relevance of a Korteweg-de Vries (KdV) equation with higher-order dispersion terms which is used in the applied sciences and engineering. We also present exact traveling wave solutions to this generalized KdV equation using an elliptic function method which can be readily applied to any scalar evolution or wave equation with polynomial terms involving only odd derivatives. We show that the generalized KdV equation still supports hump-shaped solitary waves as well as cnoidal wave solutions provided that the coefficients satisfy specific algebraic constraints. Analytical solutions in closed form serve as benchmarks for numerical solvers or comparison with experimental data. They often correspond to homoclinic orbits in the phase space and serve as separatrices of stable and unstable regions. Some of the solutions presented in this paper correct, complement, and illustrate results previously reported in the literature, while others are novel.Comment: 12 pages, 4 figures, minor text modifications, updated bibliograph

Well-posedness and controllability of Kawahara equation in weighted Sobolev spaces

We consider the Kawahara equation, a fifth order Korteweg-de Vries type equation, posed on a bounded interval. The first result of the article is related to the well-posedness in weighted Sobolev spaces, which one was shown using a general version of the Lax--Milgram Theorem. With respect to the control problems, we will prove two results. First, if the control region is a neighborhood of the right endpoint, an exact controllability result in weighted Sobolev spaces is established. Lastly, we show that the Kawahara equation is controllable by regions on $L^2$ Sobolev space, the so-called regional controllability, that is, the state function is exact controlled on the left part of the complement of the control region and null controlled on the right part of the complement of the control region.Comment: 22 pages. To appear in Nonlinear Analysis. arXiv admin note: text overlap with arXiv:1401.683