An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and the CH equation as limiting cases. It arises as the compatibility condition for a second order isospectral eigenvalue problem and a first order equation for the evolution of its eigenfunctions. This integrable equation is shown to be a shallow water wave equation derived by asymptotic expansion at one order higher approximation than KdV. We compare its traveling wave solutions to KdV solitons.Comment: 4 pages, no figure

Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave

In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. In the current work, the focus is on the numerical approximation of traveling-wave solutions of such equations. We describe our efforts to write a dedicated Python code which is able to compute traveling-wave solutions of nonlinear dispersive equations of the general form \begin{equation*} u_t + [f(u)]_{x} + \mathcal{L} u_x = 0, \end{equation*} where $\mathcal{L}$ is a self-adjoint operator, and $f$ is a real-valued function with $f(0) = 0$. The SpectraVVave code uses a continuation method coupled with a spectral projection to compute approximations of steady symmetric solutions of this equation. The code is used in a number of situations to gain an understanding of traveling-wave solutions. The first case is the Whitham equation, where numerical evidence points to the conclusion that the main bifurcation branch features three distinct points of interest, namely a turning point, a point of stability inversion, and a terminal point which corresponds to a cusped wave. The second case is the so-called modified Benjamin-Ono equation where the interaction of two solitary waves is investigated. It is found that is possible for two solitary waves to interact in such a way that the smaller wave is annihilated. The third case concerns the Benjamin equation which features two competing dispersive operators. In this case, it is found that bifurcation curves of periodic traveling-wave solutions may cross and connect high up on the branch in the nonlinear regime

Numerical study of a Whitham equation exhibiting both breaking waves and continuous solutions

We consider a Whitham equation as an alternative for the Korteweg–de Vries (KdV) equation in which the third derivative is replaced by the integral of a kernel, i.e., ηxxx in the KdV equation is replaced by ∫∞−∞Kν(x−ξ)ηξ(ξ,t)dξ. The kernel Kν(x) satisfies the conditions limν→∞Kν(x) = δ″(x), where δ(x) is the Dirac delta function and limν→0Kν(x) = 0. The questions studied here, by means of numerical examples, are whether adjustment of the parameter ν produces both continuous solutions and shocks of the kernel equation and how well they represent KdV solutions and solutions of the underlying hyperbolic system. A typical example is for resonant forced oscillations in a closed shallow water tank governed by the kernel equation, which are compared with those governed by a partial differential equation. The continuous solutions of the kernel equation associated with frequency dispersion in the KdV equations limit to the shocks of the shallow water equations as ν → 0. Two experimental problems are solved in a single equation. As another example, suitable adjustment of ν in the kernel equation produces solutions reminiscent of a hydraulic and undular bore

On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, eg. traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical 1D dissipative and dispersive shock problem shows significant differences in shock structure and shock front speed. Numerical results associated with the three dimensional experiment show that three and two dimensional approximations are in excellent agreement and one dimensional approximations are in good qualitative agreement. Using one dimensional DSW theory it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.Comment: 24 pages, 28 figures, submitted to Phys Rev

Transformation of a shoaling undular bore

We consider the propagation of a shallow-water undular bore over a gentle monotonic bottom slope connecting two regions of constant depth, in the framework of the variable-coefficient Korteweg-de Vries equation. We show that, when the undular bore advances in the direction of decreasing depth, its interaction with the slowly varying topography results, apart from an adiabatic deformation of the bore itself, in the generation of a sequence of isolated solitons - an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore. Using nonlinear modulation theory we construct an asymptotic solution describing the formation and evolution of this solitary wavetrain. Our analytical solution is supported by direct numerical simulations. The presented analysis can be extended to other systems describing the propagation of undular bores (dispersive shock waves) in weakly non-uniform environments

Theory, Experiments, and Simulations of Internal Waves in Deep Water

This dissertation concerns internal waves occurring in density-stratified fluids wherein one layer is much deeper than the other. Such stratifications are characteristic of internal waves in the ocean, but despite the practical importance of such phenomena, the best-known models fail to capture some of the features of large-amplitude waves. In this thesis we derive a model suitable for large-amplitude waves in the deep regime. We are specifically interested in solitary internal waves, which arise as a balance between nonlinear and dispersive terms. The nonlinearity is intrinsic to the fluid dynamics system and, in contrast with well-studied models, we make no assumptions on the maximum amplitude of the waves. We will, however, exploit the fact that the waves are typically long with respect to the upper layer fluid, and average physical quantities over this layer. The dispersive part of the model comes from pressure contribution of the deep lower layer, and arises as an integral operator. On account of the nonlinearity and the nonlocality of the model, the initial value problem is analytically intractable. Thus we produce a numerical method for solving the model equations which is fast, accurate and flexible. The method exploits the physical properties of the model, specifically by using the natural variables which arise through the Hamiltonian formalism. The numerical method is shown to be pseudospectrally accurate in space and fourth-order accurate in time. Finally we compare the model results to laboratory experiments and show that the model does a reasonably good job at capturing features of large-amplitude solitary waves. These experiments are performed on a much larger scale than results available in the literature, as well as the being the first experiments using miscible fluids (fresh and salt water) in the deep regime. Some mathematical results pertaining to variational principles for stratified fluids are contained in the appendices. New results contained herein apply to two-layer models with the fluid interface in contact with the boundary of the fluid domain, and the Hamiltonian principle for the incompressible, variable-density fluid motion.Doctor of Philosoph

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

Numerical and analytical study of undular bores governed by the full water wave equations and bi-directional Whitham-Boussinesq equations

Undular bores, also termed dispersive shock waves, generated by an initial discontinuity in height as governed by two forms of the Boussinesq system of weakly nonlinear shallow water wave theory, the standard formulation and a Hamiltonian formulation, two related Whitham-Boussinesq equations, and the full water wave equations for gravity surface waves are studied and compared. It is found that the Whitham-Boussinesq systems give solutions in excellent agreement with numerical solutions of the full water wave equations for the positions of the leading and trailing edges of the bore up until the onset on modulational instability. The Whitham-Boussinesq systems, which are far simpler than the full water wave equations, can then be used to accurately model surface water wave undular bores. Finally, comparisons with numerical solutions of the full water wave equations show that the Whitham-Boussinesq systems give a slightly lower threshold for the onset of modulational instability in terms of the height of the initial step generating the undular bore