25 research outputs found
Abelian versus non-Abelian Baecklund Charts: some remarks
Connections via Baecklund transformations among different non-linear
evolution equations are investigated aiming to compare corresponding Abelian
and non Abelian results. Specifically, links, via Baecklund transformations,
connecting Burgers and KdV-type hierarchies of nonlinear evolution equations
are studied. Crucial differences as well as notable similarities between
Baecklund charts in the case of the Burgers - heat equation, on one side and
KdV -type equations are considered. The Baecklund charts constructed in [16]
and [17], respectively, to connect Burgers and KdV-type hierarchies of operator
nonlinear evolution equations show that the structures, in the non-commutative
cases, are richer than the corresponding commutative ones.Comment: 18 page
New exact solutionsand numerical approximations of the generalized kdv equation
This paper is devoted to create new exact and numerical solutions of the generalized
Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element
method based on cubic B-splines over finite elements. Propagation of single solitary
wave is investigated to show the efficiency and applicability of the proposed methods.
The performance of the numerical algorithm is proved by computing L2 and L∞ error
norms. Also, three invariants I1, I2, and I3 have been calculated to determine the
conservation properties of the presented algorithm. The obtained numerical solutions
are compared with some earlier studies for similar parameters. This comparison
clearly shows that the obtained results are better than some earlier results and
they are found to be in good agreement with exact solutions. Additionally, a linear
stability analysis based on Von Neumann’s theory is surveyed and indicated that
our method is unconditionally stable
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