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What Is Integrability?

دسته بندی: مکانیک: پویایی و هرج و مرج غیرخطی
ویرایش: reprint 
نویسندگان: Vladimir E. Zakharov (ed.)  
سری:  
ISBN (شابک) : 3642887058, 9783642887055 
ناشر: Springer 
سال نشر: 1991,2012 
تعداد صفحات: 338 
زبان: English 
فرمت فایل : DJVU (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود) 
حجم فایل: 3 مگابایت

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The idea of devoting a complete book to this topic was born at one of the Workshops on Nonlinear and Turbulent Processes in Physics taking place regularly in Kiev. With the exception of E. D. Siggia and N. Ercolani, all authors of this volume were participants at the third of these workshops. All of them were acquainted with each other and with each other's work. Yet it seemed to be somewhat of a discovery that all of them were and are trying to understand the same problem - the problem of integrability of dynamical systems, primarily Hamiltonian ones with an infinite number of degrees of freedom. No doubt that they (or to be more exact, we) were led to this by the logical process of scientific evolution which often leads to independent, almost simultaneous discoveries. Integrable, or, more accurately, exactly solvable equations are essential to theoretical and mathematical physics. One could say that they constitute the "mathematical nucleus" of theoretical physics whose goal is to describe real classical or quantum systems. For example, the kinetic gas theory may be considered to be a theory of a system which is trivially integrable: the system of classical noninteracting particles. One of the main tasks of quantum electrodynamics is the development of a theory of an integrable perturbed quantum system, namely, noninteracting electromagnetic and electron-positron fields. Another well-known example is that in solid-state physics where linear equations describe a system of free oscillators representing atoms connected to each other by linear elastic forces. On the other hand, nonlinear forces yield nonlinear equations for this system. Nonlinear integrable systems were discovered as early as the 18th century. At that time only a few were known and with no real understanding of their characteristics and solutions. Now, however, it is correct to state that it is impossible to overestimate their importance in the development of all areas of science. Among their applications is the integrable problem arising for the motion of a particle in a central field, associated with atomic and nuclear physics. The problem of a particle moving in the fields of two Coulomb centers is fundamental to celestial mechanics and molecular physics. Also in molecular and nuclear physics the integrability of the Euler problem for the motion of a heavy rigid body is used. The development of the theory of gyroscopes would have been impossible without the Lagrange solution of a symmetric top in a gravitational field. Only one of the classical nonlinear integrable systems, namely, the Kovalewsky top, has not yet found direct physical applications. But within mathematics this problem the multiscale expansion method, the results, as derived for chosen models, are found to be true on a more general level. This can be shown easily if the original system possesses a Hamiltonian structure. Several examples of this type may be found in the paper by V. E. Zakharov and E. I. Schulman which is primarily devoted to a quite different question: how can we determine whether a given system is integrable or not? This problem has recently become more and more urgent, and is therefore thoroughly addressed in this volume. There are essentially three approaches to solving it, all discussed here. They originate from classical work initiated in the previous century. The approach used in the paper by E. D. Siggia and N. Ercolani and in the contribution by H. Flaschka, A. C. Newell and M. Tabor is essentially based on the classic paper by S. Kovalewskaya discussing the integrability of a top in a gravitational field. Kovalewskaya observed that the majority of known integrable systems is integrated in terms of elliptic and, consequently, meromorphic functions and thus cannot have any movable critical points. This particular condition of the nonexistence of movable critical points led subsequently to the integrable equation for the Kovalewsky top. Kovalewskaya's idea was pursued further by Painleve. This method of verifying the integrability of equations through an analysis of the arrangement of critical points of their solutions in the complex plane is called the Painleve test. In the contribution by Flaschka, Newell, and Tabor the Painleve test is used on partial differential equations and is proved to be a powerful tool. It allows not only to verify the integrability of systems but also, in the case of a positive answer, it helps to find their Lax representation as a compatibility condition (imposed on an overdetermined linear system), symmetries, and Hirota's bilinear form. One of the highlights of the third workshop in Kiev was the demonstration (by A. C. Newell) of the power of the Painleve test as applied to the integrable system found by A. V. Mikhailov and A. B. Shabat. It is worth noting that in spite of all the advances of the Painleve test there is no reliable assurance for systems not satisfying this test to be definitely nonintegrable. It should also be added that further research is required to provide an even more solid mathematical foundation for this quite useful and successful method. The next paper in the volume is from A. V. Mikhailov, V. V. Sokolov, and A. B. Shabat. They develop a symmetry approach originating from the famous Sophus Lie. The question posed is under which conditions does a class of partial differential equations admit a nontrivial group of local symmetry transformations (depending on a finite number of derivatives). In the cases under consideration the authors succeed in constructing a complete classification of systems possessing symmetries. They also prove that when a few symmetries exist it follows that there are actually an infinite number of them. It should be noted that in this paper not only Hamiltonian but also dissipative systems are considered which cannot be integrable in the classical sense but may be C-integrable, i.e., they may be reduced to linear systems by changing variables. The paper by V. E. Zakharov and E. I. Schulman is based on Poincar6's works. Rather than choosing some differential equations and transforming them to their Fourier representation where differential and pseudo-differential operators differ only in coefficient functions, a Hamiltonian translationally invariant system is taken as the starting point. The question posed is whether at least one additional invariant motion for this system exists. It is shown that the existence of such an integral implies rather important conclusions, discussed thoroughly in the paper. They are formulated as restrictions on the perturbation series in the vicinity of linearized (and trivially integrable) systems. In particular, the existence of an additional invariant of motion implies the existence of an infinite number of invariants. This result agrees with the paper by Mikhailov, Sokolov, and Shabat. An extremely important result of this report is to make clear that the existence of an infinite set of invariants of motion does not always mean integrability in Liouville's sense. The set of integrals may be incomplete. Effective criteria for identifying such cases are presented. The contribution by A. P. Veselov is devoted to systems with discrete time and thus has significant applications in physics. In this paper the particular concept of integrability of systems of this type is defined. The contribution by V. A. Marchenko devoted to the solution of the Cauchy problem of the KdV equation (with nondecaying boundary conditions at infinity) lies to some degree outside the general scope of this volume. It has been incorporated here, however, because it seems to me that the inclusion of a classic paper of modern mathematical physics can only increase the value and beauty of any presentation of associated problems. Moscow, August 1990 V.E. Zakharov

فهرست مطالب

Why Are Certain Nonlinear PDEs Both Widely Applicable 
and Integrable? 1 
By F. Calogero 
Summary 1 
Introduction 1 
1. The Main Ideas in an Illustrative Context 4 
2. Survey of Model Equations 19 
3. C-Integrable Equations 33 
4. Envoi 56 
Addendum 57 
References 61 
Painleve Property and Integrability 63 
By N. Ercolani and E.D. Siggia 
1. Background 63 
1.1 Motivation 63 
1.2 History 64 
2. Integrability 64 
3. Riccati Example 65 
4. Balances 66 
5. Elliptic Example 67 
6. Augmented Manifold 68 
7. Argument for Integrability 69 
8. Separability 70 
References 72 
Integrability 73 
By H. Flaschka, A.C. Newell and M. Tabor 
1. Integrability 73 
2. Introduction to the Method 80 
2.1 The WTC Method for Partial Differential Equations 81 
2.2 The WTC Method for Ordinary Differential Equations .. 84 
2.3 The Nature of φ 86 
2.4 Truncated Versus Non-truncated Expansions 89 
3. The Integrable Henon-Heiles System: A New Result 90 
3.1 The Lax Pair 90 

3.2 The Algebraic Curve and Integration 
of the Equations of Motion 92 
3.3 The Role of the Rational Solutions 
in the Painleve Expansions 95 
4. A Mikhailov and Shabat Example 97 
5. Some Comments on the KdV Hierarchy 98 
6. Connection with Symmetries and Algebraic Structure 99 
7. Integrating the Nonintegrable 106 
References 113 
The Symmetry Approach to Classification 
of Integrable Equations 115 
By АУ. Mikhailov, AB. Shabat and V.V. SoMov 
Introduction 115 
1. Basic Definitions and Notations 116 
1.1 Classical and Higher Symmetries 116 
1.2 Local Conservation Laws 121 
1.3 PDEs and Infinite-Dimensional Dynamical Systems 123 
1.4 Transformations 124 
2. The Burgers Type Equations 129 
2.1 Classification in the Scalar Case 129 
2.2 Systems of Burgers Type Equations 135 
2.3 Lie Symmetries and Differential Substitutions 142 
3. Canonical Conservation Laws 146 
3.1 Formal Symmetries 146 
3.2 The Case of a Vector Equation 152 
3.3 Integrability Conditions 158 
4. Integrable Equations 161 
4.1 Scalar Third Order Equations 161 
4.2 Scalar Fifth Order Equations 170 
4.3 Schrodinger Type Equations 173 
Historical Remarks 182 
References 183 
Integrability of Nonlinear Systems and Perturbation Theory 185 
By V.E. Zakharov and Ε J. Schulman 
1. Introduction 185 
2. General Theory 187 
2.1 The Formal Classical Scattering Matrix in the Soliton- 
less Sector of Rapidly Decreasing Initial Conditions 187 
2.2 Infinite-Dimensional Generalization of Poincare's 
Theorem. Definition of Degenerative Dispersion Laws .. 193 
2.3 Properties of Degenerative Dispersion Laws 197 

2.4 Properties of Singular Elements of a Classical 
Scattering Matrix. Properties of Asymptotic States 205 
2.5 The Integrals of Motion 209 
2.6 The Integrability Problem in the Periodic Case. 
Action-Angle Variables 213 
3. Applications to Particular Systems 222 
3.1 The Derivation of Universal Models 222 
3.2 Kadomtsev-Petviashvili and Veselov-Novikov 
Equations 227 
3.3 Davey-Stewartson-Type Equations. 
The Universality of the Davey-Stewartson Equation 
in the Scope of Solvable Models 230 
3.4 Applications to One-Dimensional Equations 232 
Appendix I 236 
Proofs of the Local Theorems (of Uniqueness 
and Others from Sect.2.3) 236 
Appendix Π 244 
Proof of the Global Theorem 
for Degenerative Dispersion Laws 244 
Conclusion 247 
References 249 
What b an Integrable Mapping? 251 
By A.P. Veselov 
Introduction 251 
1. Integrable Polynomial and Rational Mappings 252 
1.1 Polynomial Mapping of C: What Is Its Integrability? ... 252 
1.2 Commuting Polynomial Mappings of CN 
and Simple Lie Algebras 254 
1.3 Commuting Rational Mappings of CPn 257 
1.4 Commuting Cremona Mappings of C2 258 
1.5 Euler-Chasles Correspondences 
and the Yang-Baxter Equation 260 
2. Integrable Lagrangean Mappings with Discrete Time 261 
2.1 Hamiltonian Theory 261 
2.2 Heisenberg Chain with Classical Spins 
and the Discrete Analog of the С Neumann System 263 
2.3 The Billiard in Quadrics 264 
2.4 The Discrete Analog of the Dynamics of the Top 266 
2.5 Connection with the Spectral Theory 
of the Difference Operators: A Discrete Analogue 
of the Moser-Trubowitz Isomorphism 267 
Appendix A 269 
Appendix В 270 
References 270