32 research outputs found
Smooth and non-smooth traveling wave solutions of some generalized Camassa-Holm equations
In this paper we employ two recent analytical approaches to investigate the
possible classes of traveling wave solutions of some members of a
recently-derived integrable family of generalized Camassa-Holm (GCH) equations.
A recent, novel application of phase-plane analysis is employed to analyze the
singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible
non-smooth peakon and cuspon solutions. One of the considered GCH equations
supports both solitary (peakon) and periodic (cuspon) cusp waves in different
parameter regimes. The second equation does not support singular traveling
waves and the last one supports four-segmented, non-smooth -wave solutions.
Moreover, smooth traveling waves of the three GCH equations are considered.
Here, we use a recent technique to derive convergent multi-infinite series
solutions for the homoclinic orbits of their traveling-wave equations,
corresponding to pulse (kink or shock) solutions respectively of the original
PDEs. We perform many numerical tests in different parameter regime to pinpoint
real saddle equilibrium points of the corresponding GCH equations, as well as
ensure simultaneous convergence and continuity of the multi-infinite series
solutions for the homoclinic orbits anchored by these saddle points. Unlike the
majority of unaccelerated convergent series, high accuracy is attained with
relatively few terms. We also show the traveling wave nature of these pulse and
front solutions to the GCH NLPDEs
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
Recommended from our members
Mini-Workshop: Mathematical Aspects of Nonlinear Wave Propagation in Solid Mechanics
Nonlinear elastodynamics sets a plethora of challenging mathematical problems such as those concerning wave propagation in solids.
Elastic vibrations and acoustic waves have been widely studied because of their applications in nondestructive tests of materials and structures, and, in recent times, several novel aspects of the theory of wave propagation in solids have blossomed thanks to the introduction of metamaterials and new technological devices.
The goal of this workshop was to bring together researchers with different backgrounds to discuss recent advances, and to stimulate future work
Construction of Special Solutions for Nonintegrable Systems
The Painleve test is very useful to construct not only the Laurent series
solutions of systems of nonlinear ordinary differential equations but also the
elliptic and trigonometric ones. The standard methods for constructing the
elliptic solutions consist of two independent steps: transformation of a
nonlinear polynomial differential equation into a nonlinear algebraic system
and a search for solutions of the obtained system. It has been demonstrated by
the example of the generalized Henon-Heiles system that the use of the Laurent
series solutions of the initial differential equation assists to solve the
obtained algebraic system. This procedure has been automatized and generalized
on some type of multivalued solutions. To find solutions of the initial
differential equation in the form of the Laurent or Puiseux series we use the
Painleve test. This test can also assist to solve the inverse problem: to find
the form of a polynomial potential, which corresponds to the required type of
solutions. We consider the five-dimensional gravitational model with a scalar
field to demonstrate this.Comment: LaTeX, 14 pages, the paper has been published in the Journal of
Nonlinear Mathematical Physics (http://www.sm.luth.se/math/JNMP/
Application Extended Tanh Method for Solving Nonlinear Generalized Ito System
In this paper, we generate new exact travelling wave solutions to the nonlinear generalized Ito system based on the extended tanh method with a computerized symbolic computation. It was obtained a various new solutions by applying this method and including solitons, kinks and plane periodic solutions. More importantly, for some equations, we also obtain other new and more general solutions at the same time. It is shown that the extended tanh method is straightforward and concise, and more powerful mathematical tool can be applied to other nonlinear partial differential equations arising in mathematical physics problems. Keywords: Extended tanh method, nonlinear generalized Ito system