Nikolai A. Kudryashov

Simplest equation method to look for exact solutions of nonlinear differential equations

Abstract New method is presented to look for exact solutions of nonlinear differential equations. Two basic ideas are at the heart of our approach. One of them is to use the general solutions of the simplest nonlinear differential equations. Another idea is to take into consideration all possible singularities of equation studied. Application of our approach to search exact solutions of nonlinear differential equations is discussed in detail. The method is used to look for exact solutions of the Kuramoto–Sivashinsky equation and the equation for description of nonlinear waves in a convective fluid. New exact solitary and periodic waves of these equations are given.

One method for finding exact solutions of nonlinear differential equations

Abstract One of old methods for finding exact solutions of nonlinear differential equations is considered. Modifications of the method are discussed. Application of the method is illustrated for finding exact solutions of the Fisher equation and nonlinear ordinary differential equation of the seven order. It is shown that the method is one of the most effective approaches for finding exact solutions of nonlinear differential equations. Merits and demerits of the method are discussed.

EXACT SOLITON SOLUTIONS OF THE GENERALIZED EVOLUTION EQUATION OF WAVE DYNAMICS

A Backlund transformation is proposed for the generalized evolution equation of gas dynamics, by means of which exact soliton solutions of this equation are obtained. In recent years, a non-linear fourth-order equation has been used to describe a number of wave processes. In the general case, this takes the form

On types of nonlinear nonintegrable equations with exact solutions

Some nonlinear nonintegrable equations of evolution type have been investigated. Solutions of nonlinear equations frequently used in various fields of physics are expressed in terms of the solutions of the Riccati equation and the equation for the anharmonic oscillator.

Extended simplest equation method for nonlinear differential equations

Abstract The modified simplest equation method to look for the exact solutions of the nonlinear differential equations is presented. Our approach is applied to search for the exact solutions of the Sharma–Tasso–Olver and the Burgers–Huxley equations. The new exact solutions of these equations are obtained.

A note on the G′/G-expansion method

Abstract We demonstrate that the G ′ / G -expansion method which is often used in finding exact solutions of nonlinear differential equation is equivalent to the well-known tanh-method and application of these methods gives the same exact solutions of nonlinear differential equations.

Information decomposition method to analyze symbolical sequences

The information decomposition (ID) method to analyze symbolical sequences is presented. This method allows us to reveal a latent periodicity of any symbolical sequence. The ID method is shown to have advantages in comparison with application of the Fourier transformation, the wavelet transform and the dynamic programming method to look for latent periodicity. Examples of the latent periods for poetic texts, DNA sequences and amino acids are presented. Possible origin of a latent periodicity for different symbolical sequences is discussed.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.

Popular ansatz methods and solitary wave solutions of the Kuramoto-Sivashinsky equation

Some methods to look for exact solutions of nonlinear differential equations are discussed. It is shown that many popular methods are equivalent to each other. Several recent publications with “new” solitary wave solutions for the Kuramoto-Sivashinsky equation are analyzed. We demonstrate that all these solutions coincide with the known ones.

Fourth-order analogies to the Painlevé equations

Using the compatibility condition for the Painleve equations, several new fourth-order ordinary differential equations (ODEs) that are analogies of the Painleve equations are found. The isomonodromic linear problems for these equations are given. Special solutions of the fourth-order ODEs found are discussed. The Painleve test is applied to investigate several fourth-order ODEs.

A note on solutions of the generalized Fisher equation

Abstract The generalized Fisher equation is considered. Possible exact solutions of this equation are found by Q -function method. The velocities of traveling waves are determined and analyzed.