Table Of ContentNonlinear Equations and Operator Algebras
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science. Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGV, Moscow, U.S.S.R.
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R.
S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R.
M. C. POL YVANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Vladimir A. Marchenko
Low Temperature Institute, Academy of Sciences,
Kharkov, Ukrainian SSR
Nonlinear Equations
and Operator Algebras
r
Translated by V Rublinetsky
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster / Tokyo
Library of Congress Cataloging in Publication Data
Marchenko, V. A. (Vladimir Aleksandrovich), 1922-
[NeIineinye uravnenila i operatornye algebry. English)
Nonlinear equations and operator algebras / by Vladimir A. Marchenko; translated
by V. I. Rublinetsky.
p. cm. - (Mathematics and its applications (Soviet series»
Translation of: NeIineinye uravneniJa i operatornye algebry.
Includes bibliographies and index.
ISBN-13 :978-94-0 10 -7800-9 e-ISBN-13 :978-94-009-2887-9
DOl: 10 .1 007/978-94-009-2887-9
1. Differential equations, Nonlinear. 2. Operator algebras. I. Title.
II. Series: Mathematics and its applications (D. Reidel Publishing Company.
Soviet series)
QA372.M36513 1987
515.3'5~c 19 87-28906
CIP
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Original Russian edition puhlished hy Naukol'a Dumka
All Rights Reserved
© 1988 by D. Reidel Publishing Company, Dordrecht, Holland
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No part of the material protected by this copyright notice may be reproduced or
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SERIES EDITOR'S PREFACE
Approach your problems from the right end It isn't that they can't see the solution. It is
and begin with the answers. Then one day, that they can't see the problem.
perhaps you will find the final question.
G.K. Chesterton. The Scandal of Father
'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'.
van Gulik's The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and textbooks on
increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields
does not grow only by putting forth new branches. It also happens, quite often in fact, that
branches which were thought to be completely disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various sciences has
changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical
economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the
structure of water meet one another in packing and covering theory; quantum fields, crystal defects
and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering;
and prediction and electrical engineering can use Stein spaces. And in addition to this there are
such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable
systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the
existing classification schemes. They draw upon widely different sections of mathematics. This pro
gramme, Mathematics and Its Applications, is devoted to new emerging (sub)disciplines and to such
(new) interrelations as exempla gratia:
- a central concept which plays an important role in several different mathematical and/or
scientific specialized areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have and have had on
the development of another.
The Mathematics and Its Applications programme tries to make available a careful selection of
books which fit the philosophy outlined above. With such books, which are stimulating rather than
definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better
communication among the practitioners in diversified fields.
Because of the wealth of scholarly research being undertaken in the Soviet Union. Eastern
Europe, and Japan, it was decided to devote special attention to the work emanating from these
particular regions. Thus it was decided to start three regional series under the umbrella of the main
MIA programme.
It is probably by now largely superfluous to write about the importance of the so-called soliton
equations such as the Korteweg-de Vries equation, the cubic nonlinear SchrOdinger equation, and so
many others. They are important in terms of direct applications, in terms of model equations for
physical theories (such as quantum field theory) and in terms of the interrelations with various parts
of mathematics such as representation theory and algebraic geometry.
One of the astonishing, and so far not understood, aspects of these equations is the variety of
different ways which exist to obtain their solutions, and the relations between these various
approaches.
vi SERIES EDITOR'S PREFACE
This book presents one more way of finding solutions of clearly very considerable power and
about which it is still largely an open question just how powerful it will turn out to be. It starts with
r,....r- r,
the not unknown observation that the fractional linear transformation I takes the solutions
of a linear set of equations to solutions of the nonlinear KdV equation. The next step consists of
the couple of beautiful and far reaching ideas that first the same thing can be done for suitable (non
commutative) operator algebras to give solutions of an operator KdV equation and that these solu
tions can then be projected to yield solutions of the original KdV, including solutions not accessible
by the inverse scattering transform method and the algebraic-geometric Jacobian variety method.
Thus this book adds one more fascinating and promising chunk of theory and technique to the
fascinating literature on integrable systems, and on the (much related) Riemann problem.
The unreasonable effectiveness of mathemat As long as algebra and geometry proceeded
ics in science ... along separate paths. their advance was slow
and their applications limited.
Eugene Wigner But when these sciences joined company
they drew from each other fresh vitality and
Well, if you know of a better 'ole, go to it. thenceforward marched on at a rapid pace
towards perfection.
Bruce Bairnsfather
Joseph Louis Lagrange.
What is now proved was once only ima
gined.
William Blake
Bussum, November 1987 Michiel Hazewinkel
CONTENTS
Preface ix
Introduction xi
Chapter 1. General Scheme 1
§1 Generalized Derivation and Logarithmic Derivatives 1
§2 Examples of Nonlinear Equations 6
§3 Projection Operation 18
Chapter 2. Realization of the General Scheme in
Matrix Rings and N-Soliton Solutions 22
§1 Wronsky Matrices 22
§2 Conditions of Invertibility of Some Wronsky Matrices 25
§3 N-Soliton Solutions of Nonlinear Equations 34
§4 Singular Solutions of Nonlinear Equations 50
Chapter 3. Realization of the General Scheme in
Operator Algebras 53
§1 Extenstion of Algebra COO (B(Ho) 55
§2 Solving Linear Equations in Algebra COO (B(H)) 59
§3 Additional Equations 66
§4 Choice of Parameters 70
§5 Properties of Logarithmic Derivatives with Respect to
Conjunction Operation 88
r
§6 Invertibility Conditions for Operators 104
Chapter 4. Classes of Solutions to Nonlinear
Equations 121
§1 Examples of Solutions to Nonlinear Equations 122
§2 Connection with Inverse Problems of Spectral Analysis 137
§3 KP Equation 149
Appendix 153
References 156
PREFACE
In 1967, Gardner, Green, Kruskal, and Miura [30] discovered a method for inte
grating the Korteweg-de Vries equation with the aid of the inverse problem of
scattering theory. Further development of this method by Lax [15]' Zakharov,
and Shabat [7], [8], and other mathematicians and physicists made it possible
to find solutions of a number of nonlinear equations important in physics, and
to understand the connection of this method with the theory of Hamiltonian
systems with an infinite number of degrees of freedom (Gardner [29]' Zakharov,
and Faddeev [6]). In addition, various modifications of the method were con
structed (Zakharov and Shabat [8], Zakharov and Mikhailov [5]). In 1974-75, a
method of finding periodic and almost periodic solutions of these equations was
suggested (Novikov [21], Lax [31], Marchenko [16]).
Already in 1961, Akhiezer [1] discovered the connection between inverse
problems for some Sturm-Liouville operators with a finite gap spectrum and the
problem of inverting Jacobi-Abel integrals. Developing Akhiezer's idea, Its and
Matveew [12], Dubrovin and Novikov [2], McKean and Van Moerbeke [33] found
explicit formulas for finite zone solutions of the Korteweg-de Vries equation.
Soon similar results were obtained for a number of other nonlinear equations
[10], [11], [13]. The authors substantially used algebraic geometry methods which
were later developed by Krichever [14].
The inverse problem method and the methods connected with algebraic
geometry were substantially presented in the monograph [4] by Zakharov, Man
akov, Novikov, and Pitayevsky (the book contains a comprehensive bibliography,
including important surveys and collections of articles).
Intensive consideration was also given to other approaches that enable find
ing integrable equations and their solutions (the Backlund transformation, the
Hirota method [26], the Adler scheme [28], the Matveew method based on the
Darboux-Krum transformation [32], the Fokas-Ablowitz method [27], and oth
ers).
The present book describes the method of integrating nonlinear equations
suggested by the author in 1979. Some applications of this method were studied
in articles [17], [22]-[24], and in the thesis by Tarapova [25]. Chapters 1 and 2
are partially devoted to these results.
While writing the book, I was much aided by I. V. Ostrovskii, especially in
questions concerning the Carleson and Muckenhoupt theorems. I wish to thank
him heartily for his friendly help.
Kharkov, 1986
ix
INTRODUCTION
This book describes a method for integrating nonlinear differential equations.
The method is based upon the following observations on one-soliton solutions of
the equation
(1)
which is equivalent, as can be easily seen, to the Korteweg-de Vries (KdV)
equation Vt - 6vv", + v"''''''' = 0 (v = -2u",).
1. Propagating waves u = u(x - a2t) are the simplest solutions of Eq. (1).
Such solutions are called one-soliton and can be easily found, since this case is
reduced to solving an ordinary differential equation integrable in quadratures.
Once the appropriate values for integration constants are chosen, the solutions
obtained acquire the form
(2) r = ea",-4a3t + e( -ax-4a3t) m,
where a and m are arbitrary numerical parameters.
Note that the function r = r(x, t) satisfies the following linear differential
equations with constant coefficients,
(3) r +4r",,,,,,, =0,
t
that determine r uniquely to within a numerical factor. The equation rt +
4r xxx = 0 is the linear part of Eq. (1), while its general solution can be rep
resented as a superposition of waves propagating with different velocities. The
second equation, r xx = a2r, separates among the waves the one which prop
agates with the fixed velocity a2, i.e., it selects one-soliton solutions from the
set of solutions of the first equation. Thus, we have arrived at the following
statement:
One-soliton solutions of the nonlinear Eq. (1) are logarithmic derivatives
(u = r-1 r "') of one-soliton solutions (r) of its linear part (the irrelevant numer
ical coefficient equal to 4 can be removed by rescaling).
2. Surely, one can give the direct proof of the above statement, checking
immediately that the logarithmic derivative u = r-1 r x of any function satis
fying Eq. (3) is a solution of Eq. (1). To do this, one has to carry out easy
calculations that are, after all, nothing but standard arithmetic operations and
differentiations. In differentiating, one uses only linearity and commutativity, as
xii Introduc tion
well as the Leibniz formula for the product. Commutativity of multiplication is
not used in these calculations, so the above statement is valid also for operator
valued solutions u(x, t) of Eq. (1) whose values belong to an arbitrary operator
algebra, provided that commutative differential operators are defined correctly,
so that these operators at, ax possess the standard algebraic properties
a(u + v) = a(u) + a(v), a(uv) = a(u)v + ua(v).
By analogy with the scalar case, operator functions l' = r-1(ar) will be
r
called logarithmic derivatives of operator functions with respect to differenti
a.
ations
The above considerations show that in passing from Eq. (1) to the equation
of similar form,
(4) tit + 6u; + uxxx = 0,
in operator-valure-d1 fru nctions u = u(x, t), onre carn( confirm that the logarithmic
derivatives l' = x of operatorfunctions = x, t) that satisfy the equations
r 4r 0,
(5) t + xxx = .r..... . xx = a.. ..2r-,
a
where stand for arbitrary constant operators, solve Eq. (4).
Eq. (2) is also valid in the operator case if constant numbers a and mare
a m:
substituted by constant operators and
(6)
r
Operrator f4urnctions then can naturally be called operator solitro-n1s rof the
equation t + xxx = 0, while their logarithmic derivatives l' = x are
called operator solitons of Eq. (4).
3. Note now that if an operator function u = U(x, t) satisfies Eq. (4), then
functions u(x, t) + c, where c are arbitrary constant operators, also satisfy these
equations. In other words, Eq. (4) admits an additive transformation group
u -+ u + c consisting of all constant operators c.
Every solution u(x, t) of Eq. (4) can be decomposed into a sum of two terms
u(x, t) = u(x, t)P + u(x, t) (I - P),
where P = p2 is an arbitrarily chosen constant projection operator onto a one
dimensional space.
ii,
If in the above decomposition the second term is a constant operator
(7) u(x,'t)(I - P) = ii,
then the first term u(x, t)P will also satisfy Eq. (4),
