Searcharxiv⌕ Search

arXiv subjects

Nikolai A. Kudryashov

Publications and source records attributed to Nikolai A. Kudryashov.

35 records · Page 2Linked to original sources
arXiv2007

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.

nlin.SI↗
arXiv2006

Relations for zeros of special polynomials associated to the Painleve equations

A method for finding relations for the roots of polynomials is presented. Our approach allows us to get a number of relations for the zeros of the classical polynomials and for the roots of special polynomials associated with rational solutions of the Painleve equations. We apply the method to obtain the relations for the zeros of several polynomials. They are: the Laguerre polynomials, the Yablonskii - Vorob'ev polynomials, the Umemura polynomials, the Ohyama polynomials, the generalized Okamoto polynomials, and the generalized Hermite polynomials. All the relations found can be considered as analogues of generalized Stieltjes relations.

nlin.SI↗
arXiv2006

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 Painlevé 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.

nlin.SI↗
arXiv2006

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.

nlin.SI↗
arXiv2006

Special polynomials associated with the fourth order analogue to the Painlev'e equations

Rational solutions of the fourth order analogue to the Painlev'e equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulas for their coefficients and degrees are derived. It is shown that special solutions of the Fordy - Gibbons, the Caudrey - Dodd - Gibbon and the Kaup - Kupershmidt equations can be expressed through solutions of the equation studied.

nlin.SI↗
arXiv2006

The Yablonskii - Vorob'ev polynomials for the second Painlev'e hierarchy

Special polynomials associated with rational solutions of the second Painlev'e equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of the polynomials is established. Formulaes for their coefficients are found. The degree of every polynomial is obtained. The main achievement of the method lies in the fact that it enables one to construct the family of polynomials corresponding to any member of the second Painlev'e hierarchy. Our approach can be applied for deriving the polynomials related to rational or algebraic solutions of other nonlinear differential equations.

nlin.SI↗
arXiv2006

Mathematical model for blood flow autoregulation by endothelium-derived relaxing factor

The fluid shear stress is an important regulator of the cardiovascular system via the endothelium-derived relaxing factor (EDRF) that is Nitric Oxide. This mechanism involves biochemical reactions in an arterial wall. The autoregulation process is managed by the vascular tonus and gives the negative feedback for the shear stress changing. A new mathematical model for the autoregulation of a blood flow through arteria under the constant transmural pressure is presented. Endothelium-derived relaxing factor Nitric Oxide, the multi-layer structure of an arterial wall, and kinetic-diffusion processes are taken into consideration. The limit case of the thin-wall artery is analytically studied. The stability condition for a stationary point of the linearized system is given. The exact stationary solutions of the origin system are found. The numerical simulation for the autoregulation system is presented. It is shown the arteria adaptation to an initial radial perturbation and the transition of the system to new equilibrium state in response on the blood flow changing.

physics.bio-ph↗
arXiv2006

Explicit form of the Yablonskii - Vorob'ev polynomials

Special polynomials associated with rational solutions of the second Painlevé 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.

nlin.SI↗
arXiv2006

Polygons for finding exact solutions of nonlinear differential equations

New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for finding one -- parameter exact solutions of the generalized Korteveg -- de Vries -- Burgers equation, the generalized Kuramoto - Sivashinsky equation, and the fifth -- order nonlinear evolution equation. All these nonlinear equations contain the term $u^mu_x$. New exact solitary waves are found.

nlin.SI↗
arXiv2005

Power and non-power expansions of the solutions for the fourth-order analogue to the second Painlevé equation

Fourth - order analogue to the second Painlevé 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 Painlevé 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 Painlevé equations confirm the conjecture that the fourth - order analogue to the second Painlevé equation defines new transcendental functions.

nlin.SI↗
arXiv2005

Power solution expansions of the analogue tothe first Painleve equation

The fourth-order analog to the first Painlevé equation is studied. All power expansions for solutions of this equation near points $z=0$ and $z=\infty$ are found. The exponential additions to the expansion of solution near $z=\infty$ are computed. The obtained results confirm the hypothesis that the fourth-order analog of the first Painlevé equation determines new transcendental functions. By means of the methods of power geometry the basis of the plane lattice is also calculated.

nlin.SI↗
arXiv2005

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.

math-ph↗
arXiv2005

Power expansions for solution of the fourth-order analog to the first Painlevé equation

One of the fourth-order analog to the first Painlevé equation is studied. All power expansions for solutions of this equation near points $z=0$ and $z=\infty$ are found by means of the power geometry method. The exponential additions to the expansion of solution near $z=\infty$ are computed. The obtained results confirm the hypothesis that the fourth-order analog of the first Painlevé equation determines new transcendental functions.

nlin.SI↗
arXiv2004

Latent periodicity of serine-threonine and tyrosine protein kinases and another protein families

We identified latent periodicity in catalytic domains of approximately 85% of serine/threonine and tyrosine protein kinases. Similar results were obtained for other 22 protein domains. We also designed the method of noise decomposition, which is aimed to distinguish between different periodicity types of the same period length. The method is to be used in conjunction with the cyclic profile alignment, and this combination is able to reveal structure-related or function-related patterns of latent periodicity. Possible origins of the periodic structure of protein kinase active sites are discussed. Summarizing, we presume that latent periodicity is the common property of many catalytic protein domains.

q-bio.BM↗
arXiv2003

Exact solutions for one of the extensive chaos model

Sampling equation method is presented to look for exact solutions of nonlinear differential equations. Application of this approach to one of the extensive chaos model is considered. Exact solutions of this model in travelling wave are given. Nonlinear evolution equation for the considered extensive chaos model is shown to have solitary and periodical waves.

nlin.SI↗
arXiv2003

Information decomposition of symbolic sequences

We developed a non-parametric method of Information Decomposition (ID) of a content of any symbolical sequence. The method is based on the calculation of Shannon mutual information between analyzed and artificial symbolical sequences, and allows the revealing of latent periodicity in any symbolical sequence. We show the stability of the ID method in the case of a large number of random letter changes in an analyzed symbolic sequence. We demonstrate the possibilities of the method, analyzing both poems, and DNA and protein sequences. In DNA and protein sequences we show the existence of many DNA and amino acid sequences with different types and lengths of latent periodicity. The possible origin of latent periodicity for different symbolical sequences is discussed.

math.SP↗