4,950 research outputs found
Traveling wave solutions of nonlinear partial differential equations
We propose a simple algebraic method for generating classes of traveling wave
solutions for a variety of partial differential equations of current interest
in nonlinear science. This procedure applies equally well to equations which
may or may not be integrable. We illustrate the method with two distinct
classes of models, one with solutions including compactons in a class of models
inspired by the Rosenau-Hyman, Rosenau-Pikovsky and Rosenau-Hyman-Staley
equations, and the other with solutions including peakons in a system which
generalizes the Camassa-Holm, Degasperis-Procesi and Dullin-Gotwald-Holm
equations. In both cases, we obtain new classes of solutions not studied
before.Comment: 5 pages, 2 figures; version to be published in Applied Mathematics
Letter
Special solutions to a compact equation for deep-water gravity waves
Recently, Dyachenko & Zakharov (2011) have derived a compact form of the well
known Zakharov integro-differential equation for the third order Hamiltonian
dynamics of a potential flow of an incompressible, infinitely deep fluid with a
free surface. Special traveling wave solutions of this compact equation are
numerically constructed using the Petviashvili method. Their stability
properties are also investigated. Further, unstable traveling waves with
wedge-type singularities, viz. peakons, are numerically discovered. To gain
insights into the properties of singular traveling waves, we consider the
academic case of a perturbed version of the compact equation, for which
analytical peakons with exponential shape are derived. Finally, by means of an
accurate Fourier-type spectral scheme it is found that smooth solitary waves
appear to collide elastically, suggesting the integrability of the Zakharov
equation.Comment: 17 pages, 14 figures, 41 references. Other author's papers can be
downloaded at http://www.lama.univ-savoie.fr/~dutykh
Wave Structures and Nonlinear Balances in a Family of 1+1 Evolutionary PDEs
We study the following family of evolutionary 1+1 PDEs that describe the
balance between convection and stretching for small viscosity in the dynamics
of 1D nonlinear waves in fluids: m_t + \underbrace{um_x \}
_{(-2mm)\hbox{convection}(-2mm)} + \underbrace{b u_xm \}
_{(-2mm)\hbox{stretching}(-2mm)} = \underbrace{\nu m_{xx}\
}_{(-2mm)\hbox{viscosity}}, \quad\hbox{with}\quad u=g*m . Here
denotes We study exchanges of
stability in the dynamics of solitons, peakons, ramps/cliffs, leftons,
stationary solutions and other solitary wave solutions associated with this
equation under changes in the nonlinear balance parameter .Comment: 69 pages, 26 figure
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
Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
We consider two physically and mathematically distinct regularization
mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the
combination of diffusion and dispersion are known to give rise to monotonic and
oscillatory traveling waves that approximate shock waves. The zero-diffusion
limits of these traveling waves are dynamically expanding dispersive shock
waves (DSWs). A richer set of wave solutions can be found when the flux is
non-convex. This review compares the structure of solutions of Riemann problems
for a conservation law with non-convex, cubic flux regularized by two different
mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation;
and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation.
In the first case, the possible dynamics involve two qualitatively different
types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the
second case, in addition to RWs, there are traveling wave solutions
approximating both classical (Lax) and non-classical (undercompressive) shock
waves. Despite the singular nature of the zero-diffusion limit and rather
differing analytical approaches employed in the descriptions of dispersive and
diffusive-dispersive regularization, the resulting comparison of the two cases
reveals a number of striking parallels. In contrast to the case of convex flux,
the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is
identified as an undercompressive DSW. Other prominent features, such as
shock-rarefactions, also find their purely dispersive counterparts involving
special contact DSWs, which exhibit features analogous to contact
discontinuities. This review describes an important link between two major
areas of applied mathematics, hyperbolic conservation laws and nonlinear
dispersive waves.Comment: Revision from v2; 57 pages, 19 figure
Nonintegrable Schrodinger Discrete Breathers
In an extensive numerical investigation of nonintegrable translational motion
of discrete breathers in nonlinear Schrodinger lattices, we have used a
regularized Newton algorithm to continue these solutions from the limit of the
integrable Ablowitz-Ladik lattice. These solutions are shown to be a
superposition of a localized moving core and an excited extended state
(background) to which the localized moving pulse is spatially asymptotic. The
background is a linear combination of small amplitude nonlinear resonant plane
waves and it plays an essential role in the energy balance governing the
translational motion of the localized core. Perturbative collective variable
theory predictions are critically analyzed in the light of the numerical
results.Comment: 42 pages, 28 figures. to be published in CHAOS (December 2004
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