54 research outputs found

### Comment on: "New exact solutions for the Kawahara equation using Exp-function method"

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, J. Com.
Appl. Math. 233 (2009) 97--102] are analyzed. It is shown that all solutions do
not satisfy the Kawahara equation and consequently all nontrivial solutions by
Assas are wrong.Comment: 4 pages, This comment was sent in september 2009 to the Journal of
Computational and Applied Mathematics. It was the long story with this
comment. However recently I received the decision from the journal that the
paper being commented on will be retracted

### Special polynomials associated with the K2 hierarchy

New special polynomials associated with the rational solutions of analogue to
the Painleve hierarchies are introduced. The Hirota relations for these special
polynomials are found. Differential - difference hierarchies for finding
special polynomials are presented. These formulae allow us to search the
special polynomials associated with the hierarchy studied. The ordinary
differential hierarchy for the Yablonskii - Vorob'ev polynomials is given.Comment: 24 pages, 5 figure

### Self-similar solutions of the Burgers hierarchy

Self-similar solutions of the equations for the Burgers hierarchy are
presented

### Remarks on rational solutions for the Korteveg - de Vries hierarchy

Differential equations for the special polynomials associated with the
rational solutions of the second Painleve hierarchy are introduced. It is shown
rational solutions of the Korteveg - de Vries hierarchy can be found taking the
Yablonskii - Vorob'ev polynomials into account. Special polynomials associated
with rational solutions of the Korteveg - de Vries hierarchy are presented.Comment: 8 page

### Logistic function as solution of many nonlinear differential equations

The logistic function is shown to be solution of the Riccati equation, some
second-order nonlinear ordinary differential equations and many third-order
nonlinear ordinary differential equations. The list of the differential
equations having solution in the form of the logistic function is presented.
The simple method of finding exact solutions of nonlinear partial differential
equations (PDEs) is introduced. The essence of the method is based on
comparison of nonlinear differential equations obtained from PDEs with standard
differential equations having solution in the form of the logistic function.
The wide application of the logistic function for finding exact solutions of
nonlinear differential equations is demonstrated.Comment: 19 page

### Approximate Solutions of Nonlinear Heat Equation for Given Flow

The one-dimensional problem of the nonlinear heat equation is considered. We
assume that the heat flow in the origin of coordinates is the power function of
time and the initial temperature is zero. Approximate solutions of the problem
are given. Convergence of approximate solutions is discussed.Comment: 7 pages, 1 figur

### Explicit form of the Yablonskii - Vorob'ev polynomials

Special polynomials associated with rational solutions of the second
Painlev\'{e} equation and other members of its hierarchy are discussed. New
approach, which allows one to construct each polynomial is presented. The
structure of the polynomials is established. Formulas of their coefficients are
found. Correlations between the roots of every polynomial are obtained.Comment: 21 page

### Newton polygons for finding exact solutions

A method for finding exact solutions of nonlinear differential equations is
presented. Our method is based on the application of the Newton polygons
corresponding to nonlinear differential equations. It allows one to express
exact solutions of the equation studied through solutions of another equation
using properties of the basic equation itself. The ideas of power geometry are
used and developed. Our approach has a pictorial rendition, which is is
illustrative and effective. The method can be also applied for finding
transformations between solutions of the differential equations. To demonstrate
the method application exact solutions of several equations are found. These
equations are: the Korteveg - de Vries - Burgers equation, the generalized
Kuramoto - Sivashinsky equation, the fourth - order nonlinear evolution
equation, the fifth - order Korteveg - de Vries equation, the modified Korteveg
- de Vries equation of the fifth order and nonlinear evolution equation of the
sixth order for the turbulence description. Some new exact solutions of
nonlinear evolution equations are given.Comment: 24 pages, 10 figure

### Power expansions for the self-similar solutions of the modified Savada-Kotera equation

The fourth-order ordinary differential equation, defining new transcendents,
is studied. The self-similar solutions of the Kaup-Kupershmidt and
Savada-Kotera equations are shown to be found taking its solutions into
account. Equation studied belongs to the class of fourth-order analogues of the
Painlev\'{e} equations. All the power and non-power asymptotic forms and
expansions near points $z=0$, $z=\infty$ and near arbitrary point $z=z_0$ are
found by means of power geometry methods. The exponential additions to the
solutions of studied equation are also determined.Comment: 31 pages, 9 figure

### Power and non-power expansions of the solutions for the fourth-order analogue to the second Painlev\'{e} equation

Fourth - order analogue to the second Painlev\'{e} equation is studied. This
equation has its origin in the modified Korteveg - de Vries equation of the
fifth order when we look for its self - similar solution. All power and non -
power expansions of the solutions for the fouth - order analogue to the second
Painlev\'{e} equation near points $z=0$ and $z=\infty$ are found by means of
the power geometry method. The exponential additions to solutions of the
equation studied are determined. Comparison of the expansions found with those
of the six Painlev\'{e} equations confirm the conjecture that the fourth -
order analogue to the second Painlev\'{e} equation defines new transcendental
functions.Comment: 34 pages, 8 figures; submitted to Chaos,Solitons & Fractal

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