Construction of nonlinear differential equations for description of propagation pulses in optical fiber

A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of 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 interpretation, which is illustrative and effective. The method can be also applied for finding transformations between solutions of 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 fifth-order modified Korteveg–de Vries equation and the sixth-order nonlinear evolution equation describing turbulent processes. Some new exact solutions of nonlinear evolution equations are given.

Polygons of differential equations for finding exact solutions

We obtain all asymptotic expansions of solutions of the sixth Painlevé equation near all three singular points x = 0, x = 1, and x = ∞ for all values of four complex parameters of this equation. The expansions are obtained for solutions of five types: power, power-logarithmic, complicated, semiexotic, and exotic. They form 117 families. These expansions may contain complex powers of the independent variable x. First we use methods of two-dimensional power algebraic geometry to obtain those asymptotic expansions of all five types near the singular point x = 0 for which the order of the leading term is less than 1. These expansions are called basic expansions. They form 21 families. All other asymptotic equations near three singular points are obtained from basic ones using symmetries of the equation. The majority of these expansions are new. Also, we present examples and compare our results with previously known ones. Introduction In 1884–1885 L. Fuchs [47] and H. Poincaré [70, 71, 72] suggested looking for differential equations whose solutions do not have movable critical points and cannot be expressed in terms of previously known functions. In 1889 S. Kovalevskaya [36] had shown that the absence of movable critical points of solutions allows one to construct solutions analytically. A singular point x = x0 of a function y(x) of a complex variable x is called a critical singular point if the value of the function y(x) changes as x moves along the path surrounding x0. A movable singular point of a solution of a differential equation is a singular point such that its position depends on initial conditions of the problem. For example for the solution y = 1/ √ x− x0, where x0 is an arbitrary constant, the point x = x0 is a movable critical point. By a meromorphic function we mean a function whose only singularities in the finite part of the complex place are poles. In 1887 E. Picard [68] suggested studying the following class of second order equations: (1) y′′ = F (x, y, y′), where the function F is rational in y and y′ and meromorphic in x, and to find, among equations (1), those that have only immovable critical singular points. At the beginning of the 20th century P. Painlevé [65, 66, 67], and his students B. Gambier [49] and R. Garnier [50, 51] solved the problem formulated by Fuchs and Picard. They have found 50 canonical equations of the form (1) whose solutions have no movable critical singular 2010 Mathematics Subject Classification. Primary 34E05; Secondary 34M55. The work was supported by the Russian Foundation of Fundamental Research (Project 08–01–00082) and the Foundation for the Assistance to Russian Science. Editorial Note: The following text incorporates changes and corrections submitted by the authors for the English translation. c ©2010 American Mathematical Society 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2 A. D. BRUNO AND I. V. GORYUCHKINA points. Solutions of 44 of these equations could be expressed in terms of known (elementary or special) functions, and solutions of the remaining six equations determine new special functions, which are now called Painlevé transcendents. The sixth Painlevé equation first appeared in the paper by R. Fuchs [48]. In has the form y′′ = (y′)2 2 ( 1 y + 1 y − 1 + 1 y − x ) − y′ ( 1 x + 1 x− 1 + 1 y − x ) + y(y − 1)(y − x) x2(x− 1)2 [ a+ b x y2 + c x− 1 (y − 1)2 + d x(x− 1) (y − x)2 ] , (2) where a, b, c, d are complex parameters, x and y are complex variables, y′ = dy dx . It has three singular points x = 0, x = 1 and x = ∞, and is usually denoted P6. E. Picard [69] has found solutions of this equation in an explicit form for special values of four parameters: a = b = c = 0, d = 12 . R. Garnier [51] was studying solutions of this equation without any restrictions on parameters. A new wave of interest to Painlevé equations occurred in the 1970s after M. Ablowitz, A. Ramani, and H. Segur [1, 42, 43] discovered that integrable nonlinear partial differential equations are related to Painlevé equations (see also [35, 37]). For example, the sixth Painlevé equation is a reduction of the Ernst equation in general relativity. Nowadays, the followng problems for the Painlevé equations are being studied: asymptotic behavior of solutions near singular points, local and global properties of solutions, rational and algebraic solutions, discretization, applications of Painlevé equations (mainly in physics). In the present paper we study asymptotic expansions of solutions of the sixth Painlevé equation at the singular points x = 0, 1, ∞. Expansions in nonsingular points were described in [54, § 46], and, using power geometry, in [13, 24]. Similar studies were performed by many authors. S. Shimomura [73]–[76], M. Jimbo [61], H. Kimura [62], K. Okamoto [64] proved, using a variety of methods, existence and convergence of twoparameter families of expansions for solutions of the sixth Painlevé equation. In the book [54, § 46] by I. V. Gromak, I. Laine, and S. Shimomura the authors describe asymptotic expansions of solutions in integer powers of the independent variable. For some special values of parameters of the sixth Painlevé equation, B. Dubrovin and M. Mazzocco [46, 63], and also D. Guzzetti [55] obtained several initial terms of nonpower and exotic expansions. A comparison of these results with ours is presented at the end of the Introduction and in Section 2 of Chapter 4. The study of asymptotic extensions and asymptotic properties of solutions of Painlevé equations near singular points consists of the following three main steps. Step 1. To find formal solutions in the form of asymptotic expansions

/pdf/asymptotic-expansions-of-solutions-of-the-sixth-painleve-1d4830sz4w.pdf

Asymptotic expansions of solutions of the sixth Painlevé equation

Asymptotic approaches in mechanics: New parameters and procedures

Solitary and periodic waves of the hierarchy for propagation pulse in optical fiber

An ordinary differential equation of quite general form is considered. It is shown how to find the following near a finite or infinite value of the independent variable by using algorithms of power geometry: (i) all power-law asymptotic expressions for solutions of the equation; (ii) all power-logarithmic expansions of solutions with power-law asymptotics; (iii) all non-power-law (exponential or logarithmic) asymptotic expressions for solutions of the equation; (iv) certain exponentially small additional terms for a power-logarithmic expansion of a solution that correspond to exponentially close solutions. Along with the theory and algorithms, examples are presented of calculations of the above objects for one and the same equation. The main attention is paid to explanations of algorithms for these calculations.

https://iopscience.iop.org/article/10.1070/RM2004v059n03ABEH000736/pdf

Asymptotic behaviour and expansions of solutions of an ordinary differential equation

The prime application of the ideas and algorithms of power geometry is in the study of parameter-free partial differential equations. To each differential monomial we assign a point in : the vector exponent of this monomial. To a differential equation corresponds its support, which is the set of vector exponents of the monomials in the equation. The forms of self-similar solutions of an equation can be calculated from the support using the methods of linear algebra. The equations of a combustion process, with or without sources, are used as examples. For a quasi-homogeneous ordinary differential equation, this approach enables one to reduce the order and to simplify some boundary-value problems. Next, generalizations are made to systems of differential equations. Moreover, we suggest a classification of levels of complexity for problems in power geometry. This classification contains four levels and is based on the complexity of the geometric objects corresponding to a given problem (in the space of exponents). We give a comparative survey of these objects and of the methods based on them for studying solutions of systems of algebraic equations, ordinary differential equations, and partial differential equations. We list some publications in which the methods of power geometry have been effectively applied.

A D Bruno

Papers

Asymptotic behaviour and expansions of solutions of an ordinary differential equation

Self-similar solutions and power geometry