328 2.2 Special Function Solutions for Symmetric Discrete Painlev´e Equations.. While this school focuses on discrete integrable systems we feel it nec-essary, if only for reasons of co
Trang 1B Grammaticos Y Kosmann-Schwarzbach
T Tamizhmani (Eds.)
Discrete Integrable Systems
1 3
Trang 2Basil Grammaticos
GMPIB, Universit´e Paris VII
Tour 24-14, 5eétage, case 7021
B Grammaticos, Y Kosmann-Schwarzbach, T Tamizhmani (Eds.), Discrete Integrable
Sys-tems, Lect Notes Phys 644 (Springer, Berlin Heidelberg 2004), DOI 10.1007/b94662
Library of Congress Control Number: 2004102969
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Trang 3Lecture Notes in Physics
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Trang 4The Editorial Policy for Edited Volumes
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Trang 5Three Lessons on the Painlev´ e Property
and the Painlev´ e Equations
M D Kruskal, B Grammaticos, T Tamizhmani 1
1 Introduction 1
2 The Painlev´e Property and the Naive Painlev´e Test 2
3 From the Naive to the Poly-Painlev´e Test 7
4 The Painlev´e Property for the Painlev´e Equations 11
Sato Theory and Transformation Groups A Unified Approach to Integrable Systems R Willox, J Satsuma 17
1 The Universal Grassmann Manifold 17
1.1 The KP Equation 18
1.2 Pl¨ucker Relations 20
1.3 The KP Equation as a Dynamical System on a Grassmannian 22
1.4 Generalization to the KP Hierarchy 23
2 Wave Functions,τ-Functions and the Bilinear Identity 24
2.1 Pseudo-differential Operators 24
2.2 The Sato Equation and the Bilinear Identity 25
2.3 τ-Functions and the Bilinear Identity 28
3 Transformation Groups 31
3.1 The Boson-Fermion Correspondence 31
3.2 Transformation Groups andτ-Functions 34
3.3 B¨acklund Transformations for the KP Hierarchy 36
4 Extensions and Reductions 41
4.1 Extensions of the KP Hierarchy 42
4.2 Reductions of the KP Hierarchy 46
Special Solutions of Discrete Integrable Systems Y Ohta 57
1 Introduction 57
2 Determinant and Pfaffian 58
2.1 Definition 58
2.2 Linearity and Alternativity 62
Trang 6XII Table of Contents
2.3 Cofactor and Expansion Formula 71
2.4 Algebraic Identities 72
2.5 Golden Theorem 74
2.6 Differential Formula 76
3 Difference Formulas 77
3.1 Discrete Wronski Pfaffians 77
3.2 Discrete Gram Pfaffians 78
4 Discrete Bilinear Equations 80
4.1 Discrete Wronski Pfaffian 80
4.2 Discrete Gram Pfaffian 80
5 Concluding Remarks 81
Discrete Differential Geometry Integrability as Consistency A I Bobenko 85
1 Introduction 85
2 Origin and Motivation: Differential Geometry 85
3 Equations on Quad-Graphs Integrability as Consistency 88
3.1 Discrete Flat Connections on Graphs 89
3.2 Quad-Graphs 90
3.3 3D-Consistency 92
3.4 Zero-Curvature Representation from the 3D-Consistency 94
4 Classification 96
5 Generalizations: Multidimensional and Non-commutative (Quantum) Cases 100
5.1 Yang-Baxter Maps 100
5.2 Four-Dimensional Consistency of Three-Dimensional Systems 101
5.3 Noncommutative (Quantum) Cases 103
6 Smooth Theory from the Discrete One 105
Discrete Lagrangian Models Yu B Suris 111
1 Introduction 111
2 Poisson Brackets and Hamiltonian Flows 112
3 Symplectic Manifolds 115
4 Poisson Reduction 118
5 Complete Integrability 118
6 Lax Representations 119
7 Lagrangian Mechanics onRN . 121
8 Lagrangian Mechanics onT P and on P × P 123
9 Lagrangian Mechanics on Lie Groups 125
10 Invariant Lagrangians and the Lie–Poisson Bracket 128
10.1 Continuous–Time Case 129
10.2 Discrete–Time Case 131
Trang 711 Lagrangian Reduction and Euler–Poincar´e
Equations on Semidirect Products 134
11.1 Continuous–Time Case 135
11.2 Discrete–Time Case 138
12 Neumann System 141
12.1 Continuous–Time Dynamics 141
12.2 B¨acklund Transformation for the Neumann System 144
12.3 Ragnisco’s Discretization of the Neumann System 147
12.4 Adler’s Discretization of the Neumann System 149
13 Garnier System 150
13.1 Continuous–Time Dynamics 150
13.2 B¨acklund Transformation for the Garnier System 151
13.3 Explicit Discretization of the Garnier System 152
14 Multi–dimensional Euler Top 153
14.1 Continuous–Time Dynamics 153
14.2 Discrete–Time Euler Top 156
15 Rigid Body in a Quadratic Potential 159
15.1 Continuous–Time Dynamics 159
15.2 Discrete–Time Top in a Quadratic Potential 161
16 Multi–dimensional Lagrange Top 164
16.1 Body Frame Formulation 164
16.2 Rest Frame Formulation 166
16.3 Discrete–Time Analogue of the Lagrange Top: Rest Frame Formulation 168
16.4 Discrete–Time Analogue of the Lagrange Top: Moving Frame Formulation 169
17 Rigid Body Motion in an Ideal Fluid: The Clebsch Case 171
17.1 Continuous–Time Dynamics 171
17.2 Discretization of the Clebsch Problem, CaseA = B2 . 173
17.3 Discretization of the Clebsch Problem, CaseA = B 174
18 Systems of the Toda Type 175
18.1 Toda Type System 175
18.2 Relativistic Toda Type System 177
19 Bibliographical Remarks 179
Symmetries of Discrete Systems P Winternitz 185
1 Introduction 185
1.1 Symmetries of Differential Equations 185
1.2 Comments on Symmetries of Difference Equations 191
2 Ordinary Difference Schemes and Their Point Symmetries 192
2.1 Ordinary Difference Schemes 192
2.2 Point Symmetries of Ordinary Difference Schemes 194
2.3 Examples of Symmetry Algebras of O∆S 199
Trang 8XIV Table of Contents
3 Lie Point Symmetries of Partial Difference Schemes 203
3.1 Partial Difference Schemes 203
3.2 Symmetries of Partial Difference Schemes 206
3.3 The Discrete Heat Equation 208
3.4 Lorentz Invariant Difference Schemes 211
4 Symmetries of Discrete Dynamical Systems 213
4.1 General Formalism 213
4.2 One-Dimensional Symmetry Algebras 217
4.3 Abelian Lie Algebras of DimensionN ≥ 2 218
4.4 Some Results on the Structure of Lie Algebras 220
4.5 Nilpotent Non-Abelian Symmetry Algebras 222
4.6 Solvable Symmetry Algebras with Non-Abelian Nilradicals 222
4.7 Solvable Symmetry Algebras with Abelian Nilradicals 224
4.8 Nonsolvable Symmetry Algebras 224
4.9 Final Comments on the Classification 225
5 Generalized Point Symmetries of Linear and Linearizable Systems 225
5.1 Umbral Calculus 225
5.2 Umbral Calculus and Linear Difference Equations 227
5.3 Symmetries of Linear Umbral Equations 232
5.4 The Discrete Heat Equation 234
5.5 The Discrete Burgers Equation and Its Symmetries 235
Discrete Painlev´ e Equations: A Review B Grammaticos, A Ramani 245
1 The (Incomplete) History of Discrete Painlev´e Equations 247
2 Detectors, Predictors, and Prognosticators (of Integrability) 253
3 DiscreteP’s Galore 262
4 Introducing Some Order into the d-P Chaos 268
5 What Makes Discrete Painlev´e Equations Special? 274
6 Putting Some Real Order to the d-P Chaos 282
7 More Nice Results on d-P’s 300
8 Epilogue 317
Special Solutions for Discrete Painlev´ e Equations K M Tamizhmani, T Tamizhmani, B Grammaticos, A Ramani 323
1 What Is a Discrete Painlev´e Equation? 324
2 Finding Special-Function Solutions 328
2.1 The Continuous Painlev´e Equations and Their Special Solutions 328
2.2 Special Function Solutions for Symmetric Discrete Painlev´e Equations 332
2.3 The Case of Asymmetric Discrete Painlev´e Equations 338
Trang 93 Solutions by Direct Linearisation 345
3.1 Continuous Painlev´e Equations 346
3.2 Symmetric Discrete Painlev´e Equations 349
3.3 Asymmetric Discrete Painlev´e Equations 356
3.4 Other Types of Solutions for d-P’s 365
4 From Elementary to Higher-Order Solutions 366
4.1 Auto-B¨acklund and Schlesinger Transformations 366
4.2 The Bilinear Formalism for d-Ps 368
4.3 The Casorati Determinant Solutions 370
5 Bonus Track: Special Solutions of Ultra-discrete Painlev´e Equations 377
Ultradiscrete Systems (Cellular Automata) T Tokihiro 383
1 Introduction 383
2 Box-Ball System 385
3 Ultradiscretization 386
3.1 BBS as an Ultradiscrete Limit of the Discrete KP Equation 386
3.2 BBS as Ultradiscrete Limit of the Discrete Toda Equation 391
4 Generalization of BBS 395
4.1 BBS Scattering Rule and Yang-Baxter Relation 395
4.2 Extensions of BBSs and Non-autonomous Discrete KP Equation 399
5 From Integrable Lattice Model to BBS 405
5.1 Two-Dimensional Integrable Lattice Models andR-Matrices 405
5.2 Crystallization and BBS 408
6 Periodic BBS (PBBS) 412
6.1 Boolean Formulae for PBBS 414
6.2 PBBS and Numerical Algorithm 415
6.3 PBBS as PeriodicA(1)M Crystal Lattice 417
6.4 PBBS asA(1)N−1 Crystal Chains 419
6.5 Fundamental Cycle of PBBS 421
7 Concluding Remarks 423
Time in Science: Reversibility vs Irreversibility Y Pomeau 425
1 Introduction 425
2 On the Phenomenon of Irreversibility in Physical Systems 426
3 Reversibility of Random Signals 429
4 Conclusion and Perspectives 435
Index 437
Trang 10Three Lessons on the Painlev´ e Property
M D Kruskal1, B Grammaticos2, and T Tamizhmani3
1 Department of Mathematics, Rutgers University, New Brunswick, NJ 08903,
USA, kruskal@math.rutgers.edu
2 GMPIB, Universit´e Paris VII, Tour 24-14, 5e´etage, case 7021, 75251 Paris,
France, grammati@paris7.jussieu.fr
3 Department of Mathematics, Kanchi Mamunivar Centre for Postgraduate
Studies, Pondicherry 605008, India, arasi55@yahoo.com
Abstract While this school focuses on discrete integrable systems we feel it
nec-essary, if only for reasons of comparison, to go back to fundamentals and introducethe basic notion of the Painlev´e property for continuous systems together with acritical analysis of what is called the Painlev´e test The extension of the latter towhat is called the poly-Painlev´e test is also introduced Finally we devote a lesson
to the proof that the Painlev´e equations do have the Painlev´e property
M.D Kruskal, B Grammaticos, and T Tamizhmani, Three Lessons on the Painlev´ e Property and the Painlev´ e Equations, Lect Notes Phys.644, 1–15 (2004)
http://www.springerlink.com/ Springer-Verlag Berlin Heidelberg 2004c
Trang 11These may look like more or less random equations, but that is not thecase Apart from some simple transformations they cannot have a form otherthan shown above They are very special.
The equations form a hierarchy Starting from the highest we can, throughappropriate limiting processes, obtain the lower ones (after some rescalingsand changes of variables):
PVI −→ PV −→ PIV
PIII−→ PII −→ PI
Note that PIVand PIIIare at the same level since they can both be obtainedfrom PV What makes these equations really special is the fact that theypossess the Painlev´e property [2]
The Painlev´e property can be loosely defined as the absence of movablebranch points A glance at the Painlev´e equations above reveals the fact
that some of them possess fixed branch points Equation PIII for instance
has t = 0 as (fixed) singular point At such points one can expect bad
be-haviour, branching, of the solutions In order to study this one has to go tothe complex plane of the independent variable This is a most interestingfeature Typically when the six Painlev´e and similar equations arise from
physical applications, the variables are real and t represents physical time,
which is quintessentially real The prototypical example that springs to mind
is the “Kowalevski top” [3] It is surprising that the behaviour of the solution
for complex values of t should be relevant.
Kovalevskaya set out to study the integrability of a physical problem,namely the motion of an ideal frictionless top in a uniform gravitationalfield, spinning around a fixed point in three dimensions, using what today
we call singularity-analysis techniques The equations of motion of a movingCartesian coordinate system based on the principal axes of inertia with theorigin at its fixed point, known as Euler’s equations, are:
Trang 12Three Lessons on the Painlev´e Property and the Painlev´e Equations 3
dβ
dt = γp − αr dγ
dt = αq − βp where (p, q, r) are the components of angular velocity, (α, β, γ) the direction cosines of the force of gravity, (A, B, C) the moments of inertia, (x0, y0, z0) the
centre of mass of the system, M the mass of the top, and g the acceleration of
gravity Complete integrability of the system requires four integrals of motion.Three such integrals are straightforward: the geometric constraint
A fourth integral was known only in three special cases:
i) Spherical: A = B = C with integral px0+ qy0+ rz0= K,
ii) Euler: x0= y0= z0= 0 with integral A2p2+ B2q2+ C2r2= K, and iii) Lagrange: A = B and x0= y0= 0 with integral Cr = K.
In each of these cases the solutions of the equations of motion were given in
terms of elliptic functions and were thus meromorphic in time t Kovalevskaya
set out to investigate the existence of other cases with solutions meromorphic
in t, and found the previously unknown case
with integral
[C(p + iq)2+ M g(x0+ iy0)(α + iβ)][C(p − iq)2+ M g(x0− iy0)(α − iβ)] = K
(2.6) This case has been dubbed the Kowalevski top in her honour.
Using (2.6) Kovalevskaya was able to show that the solution can be pressed as the inverse of a combination of hyperelliptic integrals Such inversesare not meromorphic in general, but it turns out that the symmetric combi-nations of hyperelliptic integrals involved in the solution of the Kowalevskitop do have meromorphic inverses, called hyperelliptic functions
ex-Going back to the question of singularities and the Painlev´e property, werequire that the solutions be free of movable singularities other than poles.(Poles can be viewed as nonsingular values of ∞ on the “complex sphere,”
the compact closure of the complex plane obtained by adjoining the point atinfinity.)
Trang 13Fixed singularities do not pose a major problem Linear equations canhave only the singularities of their coefficients and thus these singularitiesare fixed The case of fixed singularities of nonlinear equations can also be
dealt with Consider for example the t = 0 branch point of PIII The change
of variable t = e z removes the fixed singularity by moving it to ∞ (without
creating any new singularity in the finite plane) The same or somethingsimilar can be done for all the Painlev´e equations Thus we can rationaliseignoring fixed singularities
The simplest singularities are poles Consider the equation x = x2, an
extremely simple nonlinear equation Its solution is x = −1/(t − t0), with apole of residue−1 at the point t0 So no problem arises in this case (But what
about essential singularities? Consider the function x = ae 1/(t −t0 ), which
satisfies the equation (x /x − x 2/x2)2+ 4x 3/x3 = 0 This function has nobranching but its movable singularity is an essential one, not a pole.) Painlev´ehimself decreed that any movable singularities should be no worse than poles,i.e no movable branch points or essential singularities should be present.Next we can ask for a method to investigate whether there are movablesingularities other than poles, the “Painlev´e test” There exists a standardpractice for the investigation of the Painlev´e property which we call the naivePainlev´e test [4] It is not really satisfactory but we can consider it as a usefulworking procedure We present an example like PIbut generalised somewhatto
where f (t) is an analytic function of its argument in some region If the
solutions are not singlevalued then the equation does not possess the Painlev´e
property We use the test to find a condition (on f ) for the equation to have all
its solutions singlevalued We look for branched solutions in a straightforward
way Assume x ∼ a(t − t0)p which is branched unless p is an integer We look
for something like a Laurent series with a leading term (or even Taylor series,depending on the exponents) and write
x ∼ a0(t − t0)p0+ a1(t − t0)p1+· · · with p0< p1< · · ·
Looking for branching in such an expansion can be done algorithmically This
is an asymptotic series; we do not care (in the present context) whether itconverges We do not say that this is a solution, only that it is asymptotic to asolution Sincep0< p1< · · · , the first term is dominant as t → t0 If thereare two codominant leading terms (two terms with the samep0at dominantorder), then even the leading behaviour is bad; however this situation does
not arise in practice If two terms have complex conjugate p i’s at orders otherthan the dominant one, this violates the condition for asymptoticity, but stillthe formalism goes through
We substitute the series into the differential equation and differentiateterm by term (though this is generally not allowed for asymptotic series, it
is all right here because we are operating at a formal level) and find
Trang 14Three Lessons on the Painlev´e Property and the Painlev´e Equations 5
p0− 2 : 2p0: 0and two must be equal and dominate the third for a balance (or all threemay be equal) There are three ways to equate a pair of these exponents:
First way 2p0 = 0 Then p0 = 0 and the ignored exponent p0− 2 is
−2, which dominates (has real part less than) the two assumed dominant
exponents So this is not a possible case
Second way p0− 2 = 0 Then p0= 2 and the ignored exponent 2p0 is 4,which is, satisfactorily, dominated by the two assumed dominant exponents
However, p0 is a integer so no branched behaviour has appeared A propertreatment would develop the series with this leading behaviour to see whetherbranching occurs at higher order, but we do not pursue that issue here
Third way p0− 2 = 2p0 Then p0 =−2 so the two balanced exponents
are −4, which is, satisfactorily, less than the other exponent 0 There is no
branching to dominant order, but now we will test higher order terms Weassume an expansion in integer powers and determine the coefficients one byone (A more general procedure is to generate the successive terms recursivelyand see whether branching such as fractional powers or logarithms arise, as we
will demonstrate almost immediately.) Assume x =∞
n= −2 a n (t − t0)n ,
sub-stitute into the equation, and obtain a recursion relation for the coefficients
If at some stage the coefficient of a n vanishes, this is called “resonance” Inthe case of (2.7) we find a recurrence relation of the form
(n + 3)(n − 4)a n = F n (a n −1 , a n −2 , · · · , a0) (2.8) where F n is a definite polynomial function of its arguments The resonances
are at n = −3 and 4 The one at −3 is outside the range of meaningful values
of n but was to be expected: a formal resonance at p0− 1 is always present (unless p0 = 0), because infinitesimal perturbation of the free constant t0
in the leading term gives the derivative with respect to t0 and thereby the
formally dominant power p0−1; this is called the “universal resonance” From (2.8) we see that a4 drops out and so is not determined Since the only free
parameter in the solution we had till now was t0, it is natural for the second
order equation to have another, here a4 (Of course there is no guarantee thatwhat we find within the assumptions we made, in particular on the dominantbalance, will be a general solution.) Since the left side of (2.8) vanishes, theright side must also vanish if a power series is to work There is no guaranteefor this If the right side is not zero the test fails: the equation does nothave the Painlev´e property (More properly, the test succeeds: it succeeds inshowing that the equation fails to have the property.) However it turns outthat for PI this condition is indeed satisfied But what about the generalisedequation (2.7)?
Trang 15We will now present the more general way to set up the recursion togenerate a series that is not prejudiced against the actual appearance ofterms exhibiting branching when it occurs What we do, analogous to whatPicard did to solve an ordinary differential equation near an ordinary point,
is to integrate the equation formally, obtain an integral equation to view
as a recursion relation, and iterate it If we look at the dominant terms
of the equation near the singularity (for “the third way” above) we have
x = 6x2+· · · , and these terms we can integrate explicitly after multiplying
No confusion should result from the convenient impropriety of using t for
both the variable of integration and the upper limit of integration The lower
limit of choice would have been t0 but since x behaves dominantly like a
double pole there the integrand would not be integrable, so we choose some
arbitrary other point t1 instead
A second integration of the dominant terms is now possible For this we
take the square root and multiply by the integrating factor x −3/2:
(2.10) we find immediately that the dominant behaviour of x is (t − t0)−2 ,
the double pole as expected Starting from this we can iterate (2.10) (raised
to the−2 power) and obtain an expansion for x with leading term The only
term that might create a problem is
xf (t) dt which, because of the double
pole leading term in the expansion of x, might have a residue and contribute
a logarithm In order to investigate this we expand in the neighbourhood of
t0: f (t) = f (t0) + f (t0)(t − t0) +· · · The term f (t0)(t − t0) times thedouble pole, when integrated, gives rise to a logarithm This multivaluedness
is incompatible with the Painlev´e property Thus f (t0) must vanish if we
are to have the Painlev´e property Since t0 is an arbitrary point this means
that f (t) = 0 and f must be linear (We can take f (t) = at + b but it is
then straightforward to transform it to just f (t) = t.) So the only equation
of the form (2.7) that has the Painlev´e property is P
Trang 16Three Lessons on the Painlev´e Property and the Painlev´e Equations 7
The technique of integrating dominant terms and generating expansionscan be used to analyse the remaining Painlev´e equations PII and PIII have
simple poles (x = ∞), but for the latter x = 0 is also singular Thus here we
must consider not only poles but also zeros and ensure that these are purezeros without logarithms appearing The Painlev´e equations are very special
in the sense that they do indeed satisfy the Painlev´e property What is lessclear is why they appear so often in applications
What we presented above is the essence of the naive Painlev´e test out assuming anything we can seek dominant balances and for each one gen-erate a series for the solution, finding the possible logarithms (and fractional
With-or complex powers) naturally The main difficulty is in finding all possibledominant behaviours Some equations have a dominant behaviour that is notpower-like We have seen in the example above an equation with an essentialsingularity for which the naive Painlev´e test would not find anything trou-blesome While the solution to that equation was singlevalued, it is straight-forward to generate similar examples with branching Thus, starting from
the branched function x = ae (t −t0 )−1/2
we obtain the differential equation
A lot of mysteries remain While many problems (like the one of valevskaya) are set in real time, one still has to look for branching in thecomplex plane It is not clear why one has to look outside the real line
Ko-If one thinks of a simple one-dimensional system in Newtonian mechanics,
x = F (x) with smooth F, it is always possible to integrate it over real time
but the equation is, in general, not integrable in the complex plane, nor evenanalytically extendable there Another question is, “Why does an equationthat passes the Painlev´e test behave nicely numerically?” Still, the numericalstudy of an equation and the detection of chaotic behaviour is an indicationthat the Painlev´e property is probably absent One should think deeply aboutthese mysteries and try to explain them
As we have seen the application of the naive Painlev´e test makes possible thedetection of multivaluedness related to logarithms But what about fractionalpowers? We illustrate such an analysis with the differential equation
Trang 17We apply the naive test by assuming that the dominant behaviour is x ∼ aτ p
where τ = t − t0 and τ << 1 in the vicinity of the singularity Furthermore
we assume that a = 0 (The case a = 0 seems nonsensical but can be an
indication of the existence of logarithms at dominant order.) Substitutinginto the equation we obtain the possible dominant order terms
ap(p − 1)τ p −2 ∼ −ap2τ p −2 + a5τ 5p+1
2t0aτ
p − α 2a τ
−p
leading to the comparison of powers p −2 : p−2 : 5p : p : −p The principle of
maximum balance [11] requires that two (at least) terms be equal Balancing
p − 2 with −p gives p = 1, which on the face of it gives a simple zero and
so no singularity (though one should pursue its analysis to higher order incase a singularity arises later) However here we concentrate on the balance
p − 2 = 5p which gives p = −1/2: a fractional power appears already in
the leading order! In view of this result we can conclude, correctly, that theequation does not have the Painlev´e property However, computing the series
we find that only half-integer powers appear to all orders Thus if we squarethe solution we may find poles as the only singularities So, while the initialequation does not have the Painlev´e property, there exists a simple change
of variable which transforms it to an equation that does Indeed, multiplying
and putting y = x2we recover the Painlev´e II equation
y = 2y3+ ty + α
the solution of which which has simple poles with leading terms ±1/(t − t0)
as its only singularities
Since at each singular point we have a square root, with a branching intotwo branches, we have potentially an infinite number of branches However,
as we saw, this is not the case for (3.1) How can we determine, given someequation with many (even infinitely many) branch points, that somethinglike what happened here is possible? The answer to this is the poly-Painlev´etest [2, 6] While the naive Painlev´e test studies the solution around just onesingularity, the poly-Painlev´e test considers more than one singularity at atime (hence the name) The idea is that if we start from a first singularityand make a loop around a second one and come back to the first, we mayend up on a different branch of it Thus branching may be detected throughthe “interaction” of singularities
To show how this works in a first-order equation we consider equationswhich are mostly analytic, i.e equations involving functions which are an-alytic except for some special singular points We try to find the simplestnontrivial example Clearly, linear and quadratic (Riccati) equations are toosimple Thus we choose the cubic equation
Trang 18Three Lessons on the Painlev´e Property and the Painlev´e Equations 9
Now, the Painlev´e test looks for any multivaluedness of a solution in theneighbourhood of a (movable) singularity; if any is found the test “fails”(actually the test succeeds, it’s the equation that fails — to be integrable!),and one can go on to the poly-Painlev´e test which looks for “bad” (dense)multivaluedness, generally not in the neighbourhood of a single singular pointbut by following a path winding around several (movable) singular points.Like the Painlev´e test it relies on asymptotic expansions of the solution Thismeans that one must have a small parameter in which to expand
But equation (3.2) does not contain a small parameter, and if it did, such
an “external” parameter wouldn’t suit our purpose We introduce an priate “internal” parameter by transforming variables One way is to look
appro-in an asymptotic region with t large (but not approachappro-ing appro-infinity), a region where t is approximately constant We effect this formally by introducing the change of variable t = N +az where N is a large (complex) number (N >> 1),
a is a parameter, and z is a new variable (to be thought of as taking “finite” values) We must have az much smaller than N (which means that a << N ) and we expand in the small quantity a/N We also rescale x through x = by where b is a parameter (which can be of any size, small or large or even finite).
The equation now becomes
b a
where ∼ N −5/3 Equation (3.3) is autonomous at leading order with a
small nonautonomous perturbation (Here we see an application of anotherasymptotological [11] principle: transform the problem so that you can treat
it by perturbation theory.) The parameter is an internal one, just like the parameter α in the eponymous α-method of Painlev´e
In order to investigate whether (3.3) has the poly-Painlev´e property we
start by inverting the roles of the variables, taking z as independent and y
as dependent Introducing q = 1 + y3we have
Trang 19Next we expand z in powers of , z = z0+ z1+ z2+· · · , and set up the equations for the z i recursively At lowest order we have
1
y3+ 1 =
13
1
loga-defined up to a quantity 2πin, which would introduce a one-dimensional
lat-tice of values of the integration constant Two logarithms would lead to a dimensional lattice, a multivaluedness still acceptable in the poly-Painlev´e
two-spirit In the present case of three logarithms, the integration constant c in the complex plane is defined up to a quantity 2πi(k + mj + nj2)/3, where
k, m, n are arbitrary integers In general such a multivaluedness involving
three integers and arbitrary residues would be dense and thus unacceptable
However, since the three cube roots of unity are related through 1+j +j2= 0,
the multivaluedness of c is not dense Thus at leading order the equation (3.3)
is integrable in the poly-Painlev´e sense
However, to decide the integrability of the full equation (3.2) we mustcontinue with the poly-Painlev´e test to higher orders of (3.3) We are notgoing to give these details here They can be found in the course of two ofthe authors (MDK, BG) together with A Ramani in the 1989 Les Houcheswinter school [2] It turns out that while no bad multivaluedness is introduced
at the next (first) order, the second-order contribution gives an uncertainty(in the value of the integration constant) that accumulates densely as we goaround the singularities Thus no constant of integration can be defined andthe equation is not integrable according to the poly-Painlev´e test
Of course in this problem we studied the behaviour of the solutions onlynear infinity So the question is whether we can apply the results obtainednear infinity in all regions of the complex plane The simple answer to thisquestion is that if an equation violates the poly-Painlev´e criterion in anyregion, then this means that the equation is not integrable However if wefind that the poly-Painlev´e criterion is satisfied in the region we studied then
we cannot conclude that it is so everywhere
Having dealt with Abel’s equation, we return to the case of the
second-order equation (2.7), x = 6x2 + f (t), for which we have found that the
Painlev´e property requires f (t) = t We ask whether some “mild” branching,
compatible with the poly-Painlev´e property, is possible for this equation.Here we shall work around some finite point and introduce the change of
variables t = t0+ δz where δ << 1 We scale x through x = αy and rewrite
the equation as
Trang 20Three Lessons on the Painlev´e Property and the Painlev´e Equations 11
We balance the terms by taking α/δ2 = α2 or α = δ −2 >> 1, since δ is
assumed to be small The equation can now be written as
We now treat (3.4) by perturbation analysis We start by formally
inte-grating it from the dominant terms as before, first multiplying it by 2 dy/dz:
The integral term is small because of the powers of δ, so the square root can
be expanded as 1 plus powers of that term Integrating the whole equationleads to
So y ∼ (z − z0)−2 +· · · and the leading singularity is a double pole as
expected Next we iteratively construct the solution The problems arise when
we integrate y multiplied by f (t
0)z2, resulting in a logarithm The only way
to avoid having this logarithm is to have f (t
0) = 0, which as before, since
t0 is arbitrary, means that f must be linear In this case the poly-Painlev´etest has uncovered no equations that don’t already satisfy the more stringentnaive Painlev´e test, that is, no instances of (2.7) whose solutions are free ofdense branching other than PI itself, with no branching at all
The Painlev´e equations possess the Painlev´e property, one would say, almost
by definition They were discovered by asking for necessary conditions forthis property to be present But do they really have it? Painlev´e himselfrealized that this had to be shown He did, in fact, produce a proof which
is rather complicated (although it looks essentially correct) [7] MoreoverPainlev´e treated only the PI case, assuming that the remaining equationscan be treated in a similar way (something which is not entirely clear) Asimple proof thus appeared highly desirable
Trang 21In a series of papers [8,9] one of the authors (MDK) together with variouscollaborators has proposed a straightforward proof of the Painlev´e propertythat can be applied to all six equations The latest version of this proof isthat obtained in collaboration with K.M Tamizhmani [10] In what follows
we shall outline this proof in the case of PIII(for x as a function of z), which
is a bit complicated but still tractable:
neigh-not necessarily a singular point of the solution) the solution can be expressed
as a convergent Laurent expansion with leading term We shall examine theseries up to the highest power where an arbitrary constant may enter (“thelast resonance”) We shall not be concerned with the fixed singularity at
z = 0: it suffices to put z = e t to send the fixed singular point to infinitywithout significantly affecting other singularities Infinity is a bad singularityfor the independent variable in all the Painlev´e equations, being a limit point
of poles We are only interested here in singularities in the finite plane
We note that the equation is singular where the dependent variable x = 0
(but not the solution, which has a simple Taylor series around this point) The
other value of the dependent variable where the equation is singular is x = ∞ Any other initial value for x leads, given x , to a solution by the standard
theory of ordinary differential equations Moreover the points 0 and ∞ are reciprocal through the transformation x → 1/x, which leaves the equation
invariant up to some parameter changes Thus the Laurent expansion at apole is essentially like the Taylor expansion at a zero This allows us to confineour study to just one of the two kinds of singularity In order to simplify the
calculations we put a = b = 0 (which turns out not to change anything significant) and rescale the remaining ones to c = 1, d = 1 We have finally
A crucial ingredient of the proof not previously sufficiently exploited is
the localness of the Painlev´e property: if in any given arbitrarily small region
(of the finite plane with the origin removed) an arbitrary solution has no
movable “bad” singularities, then it can have no bad singularities anywhere(in the similarly punctuated finite plane) Use of this localness does awaywith the difficulties encountered in previous proofs where one had to boundintegrals over (finitely) long paths in the complex plane
Consider some region which is a little disk around z1 (which we assume
to be neither a pole nor a zero) with radius (As we have shown in [10]
Trang 22Three Lessons on the Painlev´e Property and the Painlev´e Equations 13
= |z1|/96 suffices for our estimates.) As in the previous lessons we start
by formally integrating our equation so as to be able to iterate; the path ofintegration is to be entirely contained in the little disk We solve it recursively
to obtain an asymptotic series for the solution Near the singular pointsthe important term on the right side (containing the terms not involving
derivatives of u, namely u4−1), is u4when u is large, and −1 when u is small.
In order to integrate (4.1) we need an integrating factor We start by noting
that the left side has the obvious integrating factor 1/(uu ), after multiplying
by which we can write the left side as [ln(u z/u)] or [(u z/u)] /(u z/u), while
the right side becomes (u4− 1)/(uu ) To render the right side integrable we
would like to multiply by u 2 and any function of u alone, while to maintain the integrability of the left side we can multiply by any function of u z/u If
we could do both of these at the same time we would succeed in integratingthe equation exactly, which is more than we can hope for However, here
localness enters effectively: in our little disk z is nearly constant, and we can
treat it as constant up to small corrections
Accordingly, we multiply the latest version of the equation by (u z/u)2
Our aim is to show that the solution is regular everywhere in the little disk
with center z1and radius If u and 1/u are finite along the path of integration
then the integral is small (because length of the integration path is of order
) But what happens when u passes close to 0 or ∞? If z were constant
then the solution of (4.1) would be given in terms of elliptic functions Thelatter have two zeros and two poles in each elementary parallelogram Whenthe parameter (here the integration constant) becomes large, the poles (andthe zeros) of the elliptic functions get packed closely together Thus when
we integrate we may easily pass close to an ∞ (or a zero) It is important
in this case to have a more precise estimate of the integral To this end we
put a little disk around the pole z0 and assume that on its circumferencethe value of|u| becomes large, say A Similarly we can treat the case where
|u| is small, say 1/A, with A large as before The integration path is now a straight line starting at z1, till|u| hits the value A (or 1/A) Then we make a
detour around the circumference of the small disk where|u| > A (or u < 1/A)
and we proceed along the straight line extrapolation of the previous path till
we encounter the next singularity In general the integration path will be a
straight line from z1 to z interspersed with several small detours.
We now make more precise estimates For definiteness we choose to work
with the case of u small, but u large is entirely similar, mut mut To solve the equation by iteration, we note that the contribution of 1/u2is more important
Trang 23than that of u2 The integral of z/u2 may be an important contribution but
since it is taken over a short path it is much smaller than the z2/u2 termoutside the integral in (4.2) The precise bounds can be worked out and the
choice of a small enough guarantees that the integral is indeed subdominant Thus (4.2) becomes (zu /u)2= z2/u2 plus smaller terms or equivalently
u =±(1 + · · · ) More precisely we have
Integrating we find u = ±(z − z0) +· · · where z0 is the point where u = 0.
(This makes the constant of this last integration exactly zero.) We find thus
that u has a simple zero, if we can show that no logarithmic term appears
in the recursively generated expansion (We would have found a pole had we
worked with a u which became large instead of small).
The dangerous term is the integral
Moving the last explicit term to the left side, multiplying by the integrating
factor 1/z, and integrating gives for I the formula
I = − z
u+· · ·
We simultaneously iterate for I, u , and u from this, (4.3), and the
ob-vious u = z
z0u dz treated as three coupled equations, and in this form it
is clear that no logarithm can be generated (This is true only for the
pre-cise z dependence of (4.1): any other dependence would have introduced a
logarithm.)
This completes the proof that the special form of PIII (4.1) has thePainlev´e property
Open problems remain First one has to repeat the proof for the full PIII
without any special choice of the parameters Then the proof should be tended to all the other Painlev´e equations, including their special cases (whereone or more parameters vanish) Still we expect the approach presented above
ex-to be directly applicable without any fundamental difficulty
Trang 24Three Lessons on the Painlev´e Property and the Painlev´e Equations 15
References
1 E.L Ince, Ordinary Differential Equations, Dover, London, 1956.
2 M.D Kruskal, A Ramani and B Grammaticos, NATO ASI Series C 310,Kluwer 1989, p 321
3 S Kovalevskaya, Acta Math 12 (1889) 177
4 M.J Ablowitz, A Ramani and H Segur, Lett Nuov Cim 23 (1978) 333
5 M.D Kruskal, NATO ASI B278, Plenum 1992, p 187
6 M.D Kruskal and P.A Clarkson, Stud Appl Math 86 (1992) 87
7 P Painlev´e, Acta Math 25 (1902) 1
8 N Joshi and M.D Kruskal, in “Nonlinear evolution equations and dynamicalsystems” (Baia Verde, 1991), World Sci Publishing 1992, p 310
9 N Joshi and M.D Kruskal, Stud Appl Math 93 (1994), no 3, 187
10 M.D Kruskal, K.M Tamizhmani, N Joshi and O Costin, “The Painlev´e erty: a simple proof for Painlev´e equation III”, preprint (2004)
prop-11 M.D Kruskal, Asymptotology, in Mathematical Models in Physical Sciences(University of Notre Dame, 1962), S Drobot and P.A Viebrock, eds., Prentice-Hall 1963, pp 17-48
Trang 25A Unified Approach to Integrable Systems
Ralph Willox1,2 and Junkichi Satsuma1
1 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba,
Meguro-ku, 153-8914 Tokyo, Japan,
{willox, satsuma}@poisson.ms.u-tokyo.ac.jp
2 Theoretical Physics, Free University of Brussels (VUB), Pleinlaan 2,
1050 Brussels, Belgium
Abstract More than 20 years ago, it was discovered that the solutions of the
Kadomtsev-Petviashvili (KP) hierarchy constitute an infinite-dimensional mann manifold and that the Pl¨ucker relations for this Grassmannian take the form
Grass-of Hirota bilinear identities As is explained in this contribution, the resulting fied approach to integrability, commonly known as Sato theory, offers a deep al-gebraic and geometric understanding of integrable systems with infinitely manydegrees of freedom Starting with an elementary introduction to Sato theory, fol-lowed by an expos´e of its interpretation in terms of infinite-dimensional Cliffordalgebras and their representations, the scope of the theory is gradually extended
uni-to include multi-component systems, integrable lattice equations and fully discretesystems Special emphasis is placed on the symmetries of the integrable equationsdescribed by the theory and especially on the Darboux transformations and ele-mentary B¨acklund transformations for these equations Finally, reductions to lowerdimensional systems and eventually to integrable ordinary differential equationsare discussed As an example, the origins of the fourth Painlev´e equation and of itsB¨acklund transformations in the KP hierarchy are explained in detail
1 The Universal Grassmann Manifold
More than 20 years ago, it was discovered by Sato that the solutions ofthe Kadomtsev-Petviashvili (KP) hierarchy constitute an infinite-dimensionalGrassmann manifold (which he called the Universal Grassmann manifold)and that the Pl¨ucker relations for this Grassmannian take the form of Hirotabilinear identities [38, 39] The resulting “unified approach” to integrability,
commonly known as Sato theory [36], offers a deep algebraic and geometric
understanding of integrable systems with infinitely many degrees of freedomand their solutions At the heart of the theory lies the idea that integrablesystems are not isolated but should be thought of as belonging to infinite
families, so-called hierarchies of mutually compatible systems, i.e., systems
governed by an infinite set of evolution parameters in terms of which their(common) solutions can be expressed
R Willox and J Satsuma, Sato Theory and Transformation Groups A Unified Approach to Integrable Systems, Lect Notes Phys.644, 17–55 (2004)
http://www.springerlink.com/ Springer-Verlag Berlin Heidelberg 2004c
Trang 2618 R Willox and J Satsuma
1.1 The KP Equation
One could summarize the original idea of Sato as follows:
Start from an ordinary differential equation and suppose that its solutionssatisfy certain dispersion relations, for a set of supplementary parameters.Then, as conditions on the coefficients of this ordinary differential equation,
we obtain a set of integrable nonlinear partial differential equations
Let us show how this recipe allows one to derive the famous KP equationfrom a particularly simple set of linear dispersion relations Consider thefollowing second-order linear differential equation, denoting derivatives df
then the coefficients a and b will obviously also depend on these new
parame-ters However, this dependence will be of a much more complicated form than
(4) ; a(x; y, t) and b(x; y, t) must satisfy certain nonlinear partial differential
equations which will turn out to be solvable in terms of the KP equation.Let us see what kind of partial differential equations we obtain Denote
by W the second-order differential operator acting on f (x) in (1) (∂ x:=∂x ∂ )
W := ∂2x + a(x; y, t)∂ x + b(x; y, t) , (5)for which, by definition,
W f i (x; y, t) = 0 , (i = 1, 2) (6)
Trang 27By differentiating with respect to y we see that the functions f i also solvethe fourth-order differential equation,
order differential operator B2,
Similarly, from the t derivative of (6), one obtains the following
factor-ization of a sixth-order operator,
of the linear ordinary differential equation (1) – as in (4) – we obtain afar more interesting system of nonlinear partial differential equations (10)and (12) for the parameter-dependence of the coefficients of that ordinarydifferential equation The system (10,12) can be seen to be equivalent to the
KP equation If we use the factorizations (8) and (11) to express the equality
of the cross-derivatives (W y)tand (W t)y, we obtain a compatibility conditionfor the operators B2 and B3,
Trang 2820 R Willox and J Satsuma
(4u t − 12uu x − u 3x)x − 3u 2y = 0 , (14)
in the field u(x, y, t) := ( −a(x; y, t)) x
Furthermore, since solutions a and b to (10) and (12) can be expressed in terms of the function τ (x; y, t) (2), subject to (4),
a(x; y, t) = −τ x (x; y, t)
τ (x; y, t) , b(x; y, t) =
τ 2x (x; y, t) − τ y (x; y, t)
2 τ (x; y, t) , (15)
it follows that the solution u(x, y, t) to the KP equation derived from a(x, y, t)
is also completely determined by this function,
u(x, y, t) = ∂2x log τ (x; y, t) (16)
Exercise 1.1 Show that (13) really yields the KP equation (14).
1.2 Pl¨ ucker Relations
It is the function τ (x; y, t) that will turn out to be the single most important
object in Sato theory In fact, it is directly connected to the notion of aninfinite-dimensional Grassmann manifold To demonstrate this, we start by
expanding the solutions f1(x) and f2(x) of the original differential equation (1) around a common point of analyticity, say, x = 0 for simplicity, (i = 1, 2)
Observe that, due to (3), τ (x; 0, 0) is completely determined by the entries
in this matrix However, due to the linearity of the differential equation (1),
the matrix ζ0itself is only defined up to right-multiplication with an element
of GL(2, C), i.e., a non-singular 2 × 2 matrix The resulting change in τ(x)
being but a mere multiplication with the determinant of that transformation
matrix, these transformations obviously leave a, b and thus also u invariant.
Definition 1.1 Given an n-dimensional vector space V , then the
Grass-mann manifold GM (m; n) (or GrassGrass-mannian for short) is defined as the set
of all m-dimensional linear subspaces of V
Trang 29Alternatively, one may also think of GM (m; n) as the quotient space obtained
by the right-action of the Lie group GL(m, C) on the manifold M(m, n) of all
n × m matrices of rank m, M(m, n)/GL(m, C) In particular, GM(m; n) is m(n −m)-dimensional (see, e.g., [30] or [41] for further details on Grassmann
manifolds)
By extension of these ideas, it can be shown that the set of all ∞ × 2
matrices ζ0, defined up to the right-action of GL(2,C), constitutes an
infinite-dimensional Grassmann manifold, in this case denoted by GM (2; ∞).
It is instructive however to dwell a little longer on the case of a dimensional Grassmannian, the simplest (non-trivial) example of which is
finite-GM (2; 4), i.e., the set W of all 2-dimensional planes passing through the origin of a 4-dimensional vector space V In practice, one needs to introduce
a coordinate system on this Grassmannian If we take v i (i = 1, , 4) to be basis vectors for the vector space V (dim(V ) = 4), we can express a basis
{w1, w2} for a 2-dimensional plane by means of the coordinates ζ ij of the
The 4× 2 matrix (ζ ij)4×2 ∈ M(4, 2) is called a frame of W and we can think
of the Grassmannian GM (2; 4) as the quotient space M (4, 2)/GL(2,C)
Now, using the minor determinants of the frame (ζ ij)4×2, we can define the
following homogeneous coordinates in 5-dimensional projective space (P5),
ξ is invariant under such transformations In this way we see that GM(2; 4)
can also be regarded as a 4-dimensional subvariety ofP5 The homogeneous
coordinates ξ are called the Pl¨ ucker coordinates of this Grassmannian It
is worth observing that a Grassmann manifold GM (m; n) can always be
embedded in the projective spaceP(m n)−1 [30, 41].
It is important to realize however that the Pl¨ucker coordinates are notindependent ; they satisfy (and are in fact fully characterized by) a set of
nonlinear algebraic relations which are called the Pl¨ ucker relations In the case of GM (2; 4) there is only one such relation, which can be obtained from
the Laplace expansion of the (0) determinant,
Trang 3022 R Willox and J Satsuma
actually encodes the KP equation
1.3 The KP Equation as a Dynamical System on a Grassmannian
Let us introduce an evolution with respect to the x-coordinate in GM (2; ∞).
This can be done by means of the shift matrix (see, e.g., [36] for a detailedaccount)
x3 3! · · ·
Trang 31These conditions are however identical to (4), identifying the functions
h i (x, y, t) with the f i (x; y, t), i = 1, 2, and thus we have succeeded in ducing the parameter dependence (4) into the Grassmannian GM (2; ∞) The
intro-evolution of the function τ (x; y, t) then corresponds to the motion ζ(x; y, t)
of an initial point ζ0 on GM (2; ∞), under the action of the 3-parameter
transformation group exp(xΛ + yΛ2+ tΛ3)
We can now translate Pl¨ucker relation (21) for GM (2; ∞) into an equation for τ (x; y, t) If we introduce the frame (25)
x + 3∂2− 4∂ x ∂ t τ With the help of these expressions, the Pl¨ucker
re-lation (21) can be transformed into a quadratic rere-lation for τ (x; y, t), which
This is nothing but the Hirota bilinear form of the KP equation (14)
Exercise 1.2 Show that the KP equation can be obtained from (28) by
means of the “bilinearizing transformation” (16)
1.4 Generalization to the KP Hierarchy
If one introduces infinitely many evolution parameters, t = (t1, t2, t3, ), the functions τ (t1, t2, t3, ) will correspond to the orbits
where, as compared to the above, the variable t1 plays the rˆole of x and t2
and t3 those of y and t respectively : we shall adhere to this convention from
Trang 3224 R Willox and J Satsuma
Exercise 1.3 Calculate Schur polynomials p0, p1· · · up to p5 explicitly.This generalization to infinitely many evolution parameters, accompanied
by a careful limit GM (m → ∞; ∞), is necessary if one wants to capture all
the evolutions contained in the KP hierarchy For it can be shown that [38]
Theorem 1.1 (Sato 1981) The solution-space of the KP hierarchy is
isomorphic to the infinite-dimensional Grassmannian GM ( ∞/2; ∞) whose Pl¨ ucker relations take the form of Hirota bilinear identities for the equations
in the KP hierarchy The evolution of a KP τ -function is defined by the motion of a point on that Grassmannian, under the action of the Abelian infinite-parameter group (30).
In particular, the τ -function can be expressed in terms of Pl¨ucker coordinates
ξ Y for GM ( ∞/2; ∞) and vice versa,
τ (t) =
Y
ξ Y χ Y (t) or ξ Y = χ Y( ˜∂ t )τ (t) | t=0 , (32)
where ˜∂ t := (t1,12t2,13t3, ) The symbol χ Y (t) denotes the
character-polynomials associated with the irreducible tensor representations of GL(n), classified in terms of Young diagrams, Y The interested reader is referred
to [36, 40] for a more detailed explanation, proofs and for some explicit amples
It is intuitively clear that if one wishes to construct the Universal
Grass-mann Manifold GM ( ∞/2; ∞) by mimicking the construction of GM(2; 4),
performed in the previous sections, one would have to start from a lineardifferential operator of infinite-order and study its deformations in terms ofinfinitely many auxiliary parameters This is exactly the point where so-called
pseudo-differential operators [36, 12] come into play.
where ∂ m := ∂ x m , when m ≥ 0, and ∂ −1 is defined such that ∂∂ −1 = ∂ −1 ∂ = 1.
Definition 2.1 A pseudo-differential operator A(∂) is a linear operator,
Trang 33j ≥0 a j ∂ i denotes the so-called “differential” part of a
pseudo-differential operator and its complement, A(∂) − (A(∂))+, is denoted by
(A(∂)) − A pseudo-differential operator possesses a unique inverse, denoted
simply by A(∂) −1, and its formal adjoint can be calculated from (35) :
a i,kx ∂ i −k . (37)
2.2 The Sato Equation and the Bilinear Identity
Extending (33), we define a pseudo-differential operator,
called the gauge operator, whose coefficients w j depend on infinitely many
parameters, t = (t1≡ x, t2, t3, · · · ), as introduced in Sect 1.4.
In order to generalize the line of thought running through Sect 1.1, we
need to define differential operators, B n (∀n ≥ 1),
Trang 3426 R Willox and J Satsuma
equation actually encodes a “doubly infinite” sequence of partial differential
equations for the w j (t), generalizing (10) and (12).
Furthermore, the gauge operator W can be used to define the operator
which will turn out to be of crucial importance, not least because it underlies
the differential operators B n,
Let us now define so-called wave functions and adjoint wave functions.
Definition 2.2 A wave function (adjoint wave function) Ψ (t, λ) (Ψ ∗ (t, λ))
is defined by the expression
is obtained from the gauge operator W by setting ∂ → λ −1 W ∗−1 (t, λ) can
similarly be obtained from (37).
This leads to the following proposition [38, 11, 12],
Proposition 2.1 (Sato) If a pseudo-differential operator W satisfies the
Sato equation (40), then the operators L and B n , obtained from W by means
of (41) and (39), satisfy ∀n, m,
(B n)t m − (B m)t n = [B m , B n]− (47)
Furthermore, the wave function Ψ (t, λ) and adjoint wave function Ψ ∗ (t, λ)
which can be derived from W satisfy the linear systems,
Trang 35Several remarks are in order First of all it should be clear that (47)
generalizes (13) from Sect 1.1 Secondly, forgetting for a moment the W origins of the operators L and B n and simply parametrizing L as
-L := ∂ + u1(t)∂ −1 + u
2(t)∂ −2+· · · , (50)equations (46) and (42) will yield an infinite system of partial differential
equations in the coefficients u j (t).
Exercise 2.1 Show that at n = 2, 3, the so-called Lax equation (46) yields a
system of equations from which, by elimination of u2, u3, · · · , the KP equation expressed in the field u1(t) is obtained.
Observe that, due to (41), the coefficients u j (t) can easily be connected to
those of the gauge operator W , e.g., u1(t) = −(w1(t)) x , etc More
im-protantly however, observe also that due to Prop 2.1, all the equations
ob-tained for the w j (t) or u j (t) are mutually compatible.
Corollary 2.1 ( [38, 12]).
∂ t m ∂ t n W = ∂ t n ∂ t m W ∀m, n. (51)
In fact, the KP hierarchy is the set of nonlinear (2+1)-dimensional tion equations expressed in u1(t) that can be obtained from (46) and (47) In
evolu-turn, these equations can be thought of as the compatibility conditions of the
linear systems (48) which underly the KP hierarchy The bilinear identity (49)
however, as we shall see in the next section, encodes both the equations of the
KP hierarchy in their Hirota bilinear forms as well as their associated linearformulations (48) A remark regarding the nature of the equations in the sys-
tem (48) is in order here As the “action” (as a differential operator) of ∂ −1 is
not defined on a function, the first equation in each system in (48) should be
thought of as a formal relation linking the coefficients in the (formal) rent expansions on both sides of the equality, defining “∂ −1 exp ξ(t, λ)” to
Lau-be λ −1 exp ξ(t, λ) The second set of equations in each system however only
consists of differential equations, whose compatibility conditions are given
by (47) The compatibility of these differential equations with the formalrelations in (48) is guaranteed by (46)
The converse of Prop.2.1 can be formulated as follows [39, 11, 12]
Proposition 2.2 (Sato) If Ψ (t, λ) and Ψ ∗ (t, λ) of the form
Trang 3628 R Willox and J Satsuma
2.3 τ -Functions and the Bilinear Identity
Just as the coefficients of the second-order operator W in Sect 1.1 could be expressed in terms of a particular (Wronski) determinant τ (x), so can the gauge operator W , and with it the entire KP hierarchy and its associated
linear formulations, be expressed in terms of just a single function τ (t), the
Proposition 2.3 (Sato) There exists a function τ (t), in terms of which the
coefficients of W and V in (52) and (53) can be expressed as
where ξ(t, λ) is as in (44) and the shift ε[λ] on the coordinates t stands for
the infinite sequence
seen by changing to new variables, x and y, as in t = x − y and t = x + y.
Direct calculation of Res in (58) then yields
Trang 37where the symbol ˜D stands for the sequence of “weighted” Hirota
D-operators, ˜D = (D x1, D x2/2, D x3/3, ) The above expression is nothing
but a generating formula for all the Hirota bilinear equations in the KP
hier-archy, e.g., as the coefficients of y3 and y4 in (59) one finds the followingequations,
(4D x1D x3− 3D2
x2− D4
x1)τ · τ = 0 (3D x1D x4− D x2D x31− 2D x2D x3)τ · τ = 0 , (60)
the first of which is the KP equation in bilinear form (28)
Exercise 2.2 Derive (59) from (58).
Exercise 2.3 Compute the coefficient of y n in formula (59) for general n,
in order to find bilinear expressions for all flows x n in the KP hierarchy.Demonstrate (60)
Observe that because of the use of the Hirota bilinear operators, tions (60) are not explicitly in (2 + 1)-dimensional form since the “higherweight” bilinear equation involves a Hirota operator corresponding to the
equa-“lower weight” time variable x3 Elimination of this “lower” time variable is
needed to obtain a genuine (2 + 1)-dimensional equation governing the x4
flow
To show that the bilinear identity also encompasses the linear formulation
of the KP hierarchy, it is convenient to rewrite (58) as an integral identity [11],
for a narrow loopC λ in the complex plane around λ ≈ ∞ Then, introducing
3 points ν i (i = 1, 2, 3) inside C λ, i.e., ∀i |ν i | > |λ|, and identifying t as
n(λ ν)n = ν −λ ν for|ν| > |λ|, i.e., the
par-ticular choice of t made above turns the essential singularity at λ = ∞ in the bilinear identity into simple poles at ν i Use this result to calculate (62)from (61)
Trang 3830 R Willox and J Satsuma
The quadratic expression in the τ -functions (62) is often called the Fay tity for the KP hierarchy, since it is connected to Fay’s tri-secant formula [14]
iden-for theta functions It first appeared in the KP context in [39] where it waspointed out that this identity actually includes all useful information aboutthe KP hierarchy, such as the Hirota forms of the evolution equations orthe underlying linear formulations As we shall see later on, it is also closelyrelated to the discretization procedure for the KP equation
To show that (62) contains all information about the linear system
un-derlying the KP hierarchy, it suffices, e.g., to take the limit ν3→∞ which, at o(ν0) yields,
which, upon expansion in powers of ν −1
1 , which was after all the real meaning
of the shifts introduced in (55), yields the infinite set of linear equations [39],
p n(−˜∂)Ψ = Ψ p n −1(−˜∂)(log τ) x ∀n ≥ 2 (65)Analogously, one can also obtain
p n( ˜∂)Ψ ∗= − Ψ ∗ p
n −1( ˜∂)(log τ ) x ∀n ≥ 2 (66)
These equations are nothing but a recursive formulation of the Shabat (ZS) linear system, or its adjoint form, for the KP hierarchy, i.e.,
Zakharov-of the set Zakharov-of (λ-independent) differential equations in (48) For a combined
approach to the KP hierarchy, its linear system and its symmetries, in thesame vain as the above, see [4]
Exercise 2.5 Calculate the ZS equations at n = 2, 3 from (65), and show
that they are equivalent to those obtained from (48) at the same order, if one
sets u1= (log τ ) 2x and u2= 12[(log τ ) xt2− (log τ) 3x]
Observe that, since the ZS equations (65) do not depend explicitly on
the spectral parameter associated to the wave function Ψ , here ν2, any ear combination of wave functions will solve the same set of equations, andsimilarly for the adjoint case Conversely, it can be shown that [45, 3],
lin-Proposition 2.4 Any solution Φ of the ZS system (65), generally called a
KP eigenfunction, can be expressed as a superposition of wave functions,
Trang 39The relevant formulae for (KP) adjoint eigenfunctions are [45]
Φ ∗ (t) =
C λ
dλ 2πi h
∗ (λ)Ψ ∗ (t, λ) , (69)
where h ∗ (λ) = 1
λ Φ
∗ (t − ε[λ])Ψ(t, λ) (70)
In the above formulas, the loopC λ is taken as in (61)
That (62) contains information on the KP evolution equations themselvescan be seen from the following exercise
Exercise 2.6 Show that subsequent limits, ν2 → ∞, ν1→ ∞, of (63) yield the KP equation in bilinear form (28) at o(ν −1
or free Fermion algebra This description is originally due to Date, Jimbo,
Kashiwara and Miwa (see [11, 19] for a review of results or [21, 22] for aslightly different point of view) We shall see that this description not onlyoffers an interesting perspective on the theory we presented so far, but that italso serves as an extremely convenient starting point for further extensions orgeneralizations of the theory For details of proofs or derivations, the reader
is referred to [30] where an elementary treatment of the case of a finite (free)Fermion algebra can be found
3.1 The Boson-Fermion Correspondence
In terms of the usual anti-commutator, [X, Y ]+:= XY + Y X, we define the
Clifford algebra or free Fermion algebra
Definition 3.1 The algebra over C with generators ψ j and ψ ∗
j that satisfy the anti-commutation relations
F is referred to as the th charge-section of the fermionic Fock space F.
We shall not go into full detail as to how general elements |u ∈ F can be
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constructed Instead, let us define the heighest-weight vectors in this sentation as (∀ ∈ N \ {0}),
repre-| := ψ 1/2 − · · · ψ −1/2 |0 (72)
|− := ψ ∗
1/2 − · · · ψ ∗
for a “cyclic vector”|0 , sometimes called the vacuum state The cyclic vector
|0 is defined in terms of the genuine vacuum state |Ω ,
−j From the above, one immediately obtains that the
highest-weightvectors ∗ ( ∈ Z) are such that (j ∈ Z + 1/2),
ψ j | = 0 if j > − , ψ ∗
j | = 0 if j > , (75)
j = 0 if j < ∗
j = 0 if j < (76)There exists a pairing F ∗ × F → C, the “expectation value”, such that
j and ψ ∗
j carry “charge” +1 and −1
respec-tively, and since the charge-sectors F of the Fock space and its dual areothogonal for this pairing ( = δ , ), it is easily seen that only charge-0combinations of Fermion operators, i.e., combinations with equal amounts of
General vacuum-expectation values are calculated with the help of the well
known Wick theorem, where u i denotes either ψ j i or ψ ∗
j i,
1· · · u r |0 =
0
σ sgn(σ) σ(1) u σ(2) σ(r −1) u σ(r) |0 , (78)depending on whether r ∈ N is odd or even, and where the sumσruns over
all possible permutations of the indices such that σ(1) < σ(2), , σ(r − 1) < σ(r) and σ(1) < σ(3) < · · · < σ(r − 1).
Most importantly however, from the above Fermion operators it is possible
to construct bosonic operators,
H n :=
j ∈Z+1/2
ψ −j ψ j+n ∗ , n ∈ Z \ {0} , (79)
which satisfy the usual commutation relations, [H n , H m]− = nδ n+m,0, and
which can easily be seen to annihilate highest-weight vectors, H n | =
−n = 0 , ∀ ∈ Z, n ≥ 1.
It was discovered by Date, Jimbo, Kashiwara and Miwa that, conversely, it
is also possible to express fermionic operators in terms of bosonic ones [11,30]