3,353 research outputs found
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
Elliptic solutions and solitary waves of a higher order KdV--BBM long wave equation
We provide conditions for existence of hyperbolic, unbounded periodic and
elliptic solutions in terms of Weierstrass functions of both third and
fifth-order KdV--BBM (Korteweg-de Vries--Benjamin, Bona \& Mahony) regularized
long wave equation. An analysis for the initial value problem is developed
together with a local and global well-posedness theory for the third-order
KdV--BBM equation. Traveling wave reduction is used together with zero boundary
conditions to yield solitons and periodic unbounded solutions, while for
nonzero boundary conditions we find solutions in terms of Weierstrass elliptic
functions. For the fifth-order KdV--BBM equation we show that a parameter
, for which the equation has a Hamiltonian, represents a
restriction for which there are constraint curves that never intersect a region
of unbounded solitary waves, which in turn shows that only dark or bright
solitons and no unbounded solutions exist. Motivated by the lack of a
Hamiltonian structure for we develop bounds, and
we show for the non Hamiltonian system that dark and bright solitons coexist
together with unbounded periodic solutions. For nonzero boundary conditions,
due to the complexity of the nonlinear algebraic system of coefficients of the
elliptic equation we construct Weierstrass solutions for a particular set of
parameters only.Comment: 13 pages, 6 figure
Stability of Compacton Solutions of Fifth-Order Nonlinear Dispersive Equations
We consider fifth-order nonlinear dispersive type equations to
study the effect of nonlinear dispersion. Using simple scaling arguments we
show, how, instead of the conventional solitary waves like solitons, the
interaction of the nonlinear dispersion with nonlinear convection generates
compactons - the compact solitary waves free of exponential tails. This
interaction also generates many other solitary wave structures like cuspons,
peakons, tipons etc. which are otherwise unattainable with linear dispersion.
Various self similar solutions of these higher order nonlinear dispersive
equations are also obtained using similarity transformations. Further, it is
shown that, like the third-order nonlinear equations, the fifth-order
nonlinear dispersive equations also have the same four conserved quantities and
further even any arbitrary odd order nonlinear dispersive type
equations also have the same three (and most likely the four) conserved
quantities. Finally, the stability of the compacton solutions for the
fifth-order nonlinear dispersive equations are studied using linear stability
analysis. From the results of the linear stability analysis it follows that,
unlike solitons, all the allowed compacton solutions are stable, since the
stability conditions are satisfied for arbitrary values of the nonlinear
parameters.Comment: 20 pages, To Appear in J.Phys.A (2000), several modification
Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics
For surface gravity waves propagating in shallow water, we propose a variant
of the fully nonlinear Serre-Green-Naghdi equations involving a free parameter
that can be chosen to improve the dispersion properties. The novelty here
consists in the fact that the new model conserves the energy, contrary to other
modified Serre's equations found in the literature. Numerical comparisons with
the Euler equations show that the new model is substantially more accurate than
the classical Serre equations, specially for long time simulations and for
large amplitudes.Comment: 24 pages, 4 figures, 41 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Dissipative perturbations for the K(n,n) Rosenau-Hyman equation
Compactons are compactly supported solitary waves for nondissipative
evolution equations with nonlinear dispersion. In applications, these model
equations are accompanied by dissipative terms which can be treated as small
perturbations. We apply the method of adiabatic perturbations to compactons
governed by the K(n,n) Rosenau-Hyman equation in the presence of dissipative
terms preserving the "mass" of the compactons. The evolution equations for both
the velocity and the amplitude of the compactons are determined for some linear
and nonlinear dissipative terms: second-, fourth-, and sixth-order in the
former case, and second- and fourth-order in the latter one. The numerical
validation of the method is presented for a fourth-order, linear, dissipative
perturbation which corresponds to a singular perturbation term
Finite depth effects on solitary waves in a floating ice sheet
A theoretical and numerical study of two-dimensional nonlinear flexural-gravity waves propagating at the surface of an ideal fluid of finite depth, covered by a thin ice sheet, is presented. The ice-sheet model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchhoff׳s hypothesis, which yields a conservative and nonlinear expression for the bending force. From a Hamiltonian reformulation of the governing equations, two weakly nonlinear wave models are derived: a 5th-order Korteweg–de Vries equation in the long-wave regime and a cubic nonlinear Schrödinger equation in the modulational regime. Solitary wave solutions of these models and their stability are analysed. In particular, there is a critical depth below which the nonlinear Schrödinger equation is of focusing type and thus admits stable soliton solutions. These weakly nonlinear results are validated by comparison with direct numerical simulations of the full governing equations. It is observed numerically that small- to large-amplitude solitary waves of depression are stable. Overturning waves of depression are also found for low wave speeds and sufficiently large depth. However, solitary waves of elevation seem to be unstable in all cases
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