3,581 research outputs found
Wronskians, Generalized Wronskians and Solutions to the Korteweg-de Vries Equation
A bridge going from Wronskian solutions to generalized Wronskian solutions of
the Korteweg-de Vries equation is built. It is then shown that generalized
Wronskian solutions can be viewed as Wronskian solutions. The idea is used to
generate positons, negatons and their interaction solutions to the Korteweg-de
Vries equation. Moreover, general positons and negatons are constructed through
the Wronskian formulation. A few new exact solutions to the KdV equation are
explicitly presented as examples of Wronskian solutions.Comment: 11 pages, 6 figures, to be published in Chaos, Solitons & Fractal
Explicit solutions to the Korteweg-de Vries equation on the half line
Certain explicit solutions to the Korteweg-de Vries equation in the first
quadrant of the -plane are presented. Such solutions involve algebraic
combinations of truly elementary functions, and their initial values correspond
to rational reflection coefficients in the associated Schr\"odinger equation.
In the reflectionless case such solutions reduce to pure -soliton solutions.
An illustrative example is provided.Comment: 17 pages, no figure
A multiple exp-function method for nonlinear differential equations and its application
A multiple exp-function method to exact multiple wave solutions of nonlinear
partial differential equations is proposed. The method is oriented towards ease
of use and capability of computer algebra systems, and provides a direct and
systematical solution procedure which generalizes Hirota's perturbation scheme.
With help of Maple, an application of the approach to the dimensional
potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and
2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton
type solutions. Two cases with specific values of the involved parameters are
plotted for each of 2-wave and 3-wave solutions.Comment: 12 pages, 16 figure
Solitary Waves and Compactons in a class of Generalized Korteweg-DeVries Equations
We study the class of generalized Korteweg-DeVries equations derivable from
the Lagrangian: L(l,p) = \int \left( \frac{1}{2} \vp_{x} \vp_{t} - {
{(\vp_{x})^{l}} \over {l(l-1)}} + \alpha(\vp_{x})^{p} (\vp_{xx})^{2} \right)
dx, where the usual fields of the generalized KdV equation are
defined by u(x,t) = \vp_{x}(x,t). This class contains compactons, which are
solitary waves with compact support, and when , these solutions have the
feature that their width is independent of the amplitude. We consider the
Hamiltonian structure and integrability properties of this class of KdV
equations. We show that many of the properties of the solitary waves and
compactons are easily obtained using a variational method based on the
principle of least action. Using a class of trial variational functions of the
form we
find soliton-like solutions for all , moving with fixed shape and constant
velocity, . We show that the velocity, mass, and energy of the variational
travelling wave solutions are related by , where , independent of .\newline \newline PACS numbers: 03.40.Kf,
47.20.Ky, Nb, 52.35.SbComment: 16 pages. LaTeX. Figures available upon request (Postscript or hard
copy
Seven common errors in finding exact solutions of nonlinear differential equations
We analyze the common errors of the recent papers in which the solitary wave
solutions of nonlinear differential equations are presented. Seven common
errors are formulated and classified. These errors are illustrated by using
multiple examples of the common errors from the recent publications. We show
that many popular methods in finding of the exact solutions are equivalent each
other. We demonstrate that some authors look for the solitary wave solutions of
nonlinear ordinary differential equations and do not take into account the well
- known general solutions of these equations. We illustrate several cases when
authors present some functions for describing solutions but do not use
arbitrary constants. As this fact takes place the redundant solutions of
differential equations are found. A few examples of incorrect solutions by some
authors are presented. Several other errors in finding the exact solutions of
nonlinear differential equations are also discussed.Comment: 42 page
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations
In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of
the focusing nonlinear Schr\"odinger (NLS), focusing modified Korteweg-de Vries
(mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and
breather solutions, they demonstrate the lack of local well-posedness for these
equations below their respective endpoint regularities. In this paper, we study
the defocusing analogues of these equations, namely defocusing NLS, defocusing
mKdV, and real KdV, all in one spatial dimension, for which suitable soliton
and breather solutions are unavailable. We construct for each of these
equations classes of modified scattering solutions, which exist globally in
time, and are asymptotic to solutions of the corresponding linear equations up
to explicit phase shifts. These solutions are used to demonstrate lack of local
well-posedness in certain Sobolev spaces,in the sense that the dependence of
solutions upon initial data fails to be uniformly continuous. In particular, we
show that the mKdV flow is not uniformly continuous in the topology,
despite the existence of global weak solutions at this regularity.
Finally, we investigate the KdV equation at the endpoint regularity
, and construct solutions for both the real and complex KdV
equations. The construction provides a nontrivial time interval and a
locally Lipschitz continuous map taking the initial data in to a
distributional solution H^{-3/4})$ which is uniquely
defined for all smooth data. The proof uses a generalized Miura transform to
transfer the existing endpoint regularity theory for mKdV to KdV.Comment: minor edit
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