Do All Integrable Equations Satisfy Integrability Criteria?
© B. Grammaticos et al. 2008
Received: 17 December 2007
Accepted: 16 May 2008
Published: 20 May 2008
At the price of sacrificing all suspense, we can already announce that the answer to the question of the title is "no." It is indeed our belief that one may find counterexamples to all integrability conjectures, unless one constrains the definition of integrability to the point that the integrability criterion becomes tautological. This review is devoted to a critical analysis of the situation.
The study of integrable systems is intimately related to integrability criteria . This is due to the fact that integrable systems are very rare but also very interesting (which explains the intense activity in this domain). The usefulness of integrability criteria is that they allow the proposal of conjectures which serve for integrability prediction. The integrability detectors thus elaborated to find their full usefulness in the nonconstructive approach to integrability. The opposite, constructive, approach consists in deriving integrable systems starting from the solution, or what is admittedly more customary, from an overdetermined linear system (the Lax pair), the nonlinear integrable equation resulting from the compatibility of the linear one. The nonconstructive approach starts from a nonlinear system resulting from some, often physical, usually realistic, model. It is then of the utmost importance to know whether this system is integrable since integrability conditions the long-term behaviour of its solutions. A reliable integrability detector is a most valuable tool.
In the domain of continuous systems, the use of complex analysis has made possible the development of specific and efficient tools for integrability prediction, and actual integration of systems expressed as (ordinary or partial) differential equations. According to Poincaré , to integrate a differential equation is to find for the general solution an expression, possibly multivalued, in terms of a finite number of functions. The word "finite" indicates that integrability is related to a global rather than a local knowledge of the solution. However, this definition is not very useful unless one defines more precisely what is meant by "function." By extending the solution of a given ordinary differential equation (ODE) into the complex domain, one has the possibility, instead of asking for a global solution for an ODE, to look for solutions locally and obtain a more global result by analytic continuation. If we wish to define a function, we must find a way to treat branch points, that is, points around which two (at least) determinations are exchanged. This can be done through various uniformization procedures provided the branch points are fixed. Linear ODEs are such that all the singularities of their solutions are fixed and are thus considered integrable. In the case of nonlinear ODEs, the situation is not so simple due to the fact that the singular points in this case may depend on the initial conditions: they are movable. The genius of Painlevé [3, 4] was to decide to look for those of the nonlinear ODEs the solutions of which were free from movable branch points. The success of this approach is well known: the Painlevé transcendents have been discovered in that way and their importance in mathematical physics is ever growing. The Painlevé property, that is absence of movable branch points, has been since used with great success in the detection of integrability [5–8].
We must stress one important point here. The Painlevé property as introduced by Painlevé is not just a predictor of integrability but practically a definition of integrability. As such it becomes a tautology rather than a criterion. It is thus crucial to make the distinction between the Painlevé property and the algorithm for its investigation. The latter can only search for the absence of Painlevé property within certain assumptions. The search can thus lead to a conclusion the validity of which is questionable: if we find that the system passes what is usually referred to as the Painlevé test (in one of its several variants), this does not necessarily mean that the system possesses the Painlevé property. Kruskal  has stressed this important point on various occasions that, at least as far as its usual practical application is concerned, the Painlevé test may not be sufficient for integrability. The situation becomes further complicated if we consider systems that are integrable through quadratures or linearisation. If we extend the notion of integrability in order to include such systems, it turns out that the connection to the Painlevé property breaks down. As we have shown in , the integrable character of linearisable systems is not associated to the Painlevé property. (As a matter of fact, no linearisability detector appears to exist to date, to the authors knowledge.)
this simply means that and the memory of the initial condition disappears from the iteration. What does the confinement mean in this case? Clearly, the mapping must recover the lost degree of freedom and the only way to do this is by the appearance of an indeterminate form , in the subsequent iterations. As Kruskal points out, the way to treat this difficulty is to use an argument of continuity with respect to the initial conditions and to introduce a small parameter . For an explicit example, refer to . While singularity confinement has been instrumental in discovering a host of integrable discrete systems [13, 14], it turned out that the confinement property was not sufficient in order to guarantee integrability . We will not go into detailed explanations here. It suffices to say that for discrete systems to be integrable, a proper local singularity structure is not enough. The growth properties of the solutions at infinity enter into play and the best way to qualify this is through the Nevanlinna approach . To put it in a nutshell, for a discrete system to be integrable the requirement is that the Nevanlinna order of the solution be finite (which guarantees not too fast a growth). Just as in the continuous case, the algorithm for the calculation of the Nevanlinna order is not precise enough. The conditions obtained are sufficient but usually largely nonnecessary . In this case, implementing the singularity confinement requirement on top of the "Nevanlinna" constraints allows one to reduce substantially the available parameter space. Linearisable discrete systems are a class of their own. As we have shown in , linearisability does not require confined singularities although the solutions must still have finite Nevanlinna order. The finiteness of the Nevanlinna order is to be understood as a condition for mappings with constant coefficients. The case of nonautonomous systems presents additional difficulties. An algorithm which calculates the growth was proposed by Hietarinta and Viallet  and is commonly referred to as the algebraic entropy technique.
Finally, there exist systems which are generalised cellular automata and are obtained from discrete systems following the ultradiscretisation procedure . We remind here that the latter consists into introducing an ansatz (where is the solution of the discrete system, which should obviously be positive definite) and obtain for an equation by going to the limit . The essential identity that allows to derive easily the ultradiscrete forms is . The ultradiscretisation procedure preserves any integrable character of the initial system. The question of the existence of the ultradiscrete analogue of integrability-related properties, like the singularity confinement, has been already addressed by Joshi and Lafortune . They proposed a singularity analysis approach which is perceived as the ultradiscrete equivalent of singularity confinement. In , we have critically examined this approach and have shown that, just as in the discrete case, there exist integrable ultradiscrete systems with unconfined singularities but also nonintegrable systems with confined singularities.
In what follows, we will present a review of these findings, illustrating all situations through concrete examples.
2. Continuous Systems
There are movable singularities where near some singular point , one has , with , arbitrary. Taking the cube of the variables is not sufficient to regularise them, however. Indeed, a detailed analysis of complex-time singularities shows that their expansions contain all powers of . The fact that some multivaluedness was compatible with integrability led to the introduction of the notion of "weak Painlevé" property . However, it was soon realised  that (2.1) was a member of a vaster family of integrable Hamiltonian systems associated to the potential where . Since the two functions and are free, one can easily show that the singularities of the solutions of the equations of motion can be arbitrary. The Hamiltonians of this family are integrable through quadratures and, in fact, the associated Hamilton-Jacobi equations are separable. This leads to the conclusion that this type of integrability is not necessarily related to the Painlevé property. (As a matter of fact, the same conclusion could have been reached if we had simply considered one-dimensional Hamiltonian systems). One may justifiably argue that in the case of Hamiltonian systems the term integrability is to be understood as Liouville integrability which is not the one we refer to in relation to the Painlevé property. Still, Liouville integrability, and the dynamical symmetries to which it is associated, may be of utmost importance for physical applications and a systematic method for the detection would have been most welcome.
(In , we have shown that this form is the term linear in in the right-hand side of (2.3) can be put to zero without loss of generality). The precise form of the coupling introduced in (2.4) is due to integrability requirements. In fact, the application of singularity analysis shows that the Gambier system cannot be integrable unless the coefficient of the term in (2.4) is an integer . This is not the only integrability requirement. Depending on the value of , one can find constraints on the , , (where the latter is traditionally taken to be or , without loss of generality) which are necessary for integrability. On the other hand, the integration of the two Riccati equations in cascade can always be performed through reduction to linear second-order equations, even when the Painlevé constraints are not satisfied.
Once the Painlevé property is deemed unnecessary for the linearisation of the Gambier system, it is straightforward to extend the latter to the form
where , and are arbitrary functions of their arguments. Again, system (2.6) does not possess, generically, the Painlevé property while it can be linearised and integrated in cascade.
where is a (possibly inhomogeneous) polynomial of degree two in together with its derivatives, but linear in , and with a constant. We then ask that the derivatives of both equations with respect to the independent variable, that is the resulting third-order equations, be identical up to an overall factor. This is a novel linearisation approach. The explicit integration procedure is the following. We start from (2.8) with given and initial conditions for some value of the independent variable . We use (2.8) to compute at . Having these values, we can use (2.7) to compute the value of . Since the latter is assumed to be a constant, we can integrate the linear equation (2.7) for all values of . Since this solution will satisfy the third-order equation mentioned above, it will also be a solution of (2.8).
which is equation XII in the Chazy classification . Given and the corresponding , (2.10) is integrable by linearisation through (2.9). On the other hand, (2.10) violates the Painlevé property. Solving it for , we find a term proportional to (or, for that matter, to ) which is incompatible with it.
which is related to the KdV equation. Equations of this class quite often possess the "weak" Painlevé property. This is the case for the Dym equation. The expansion around a singularity manifold is . Moreover, there exist equations in the Calogero list of C-integrable PDEs belonging to the class of solvable through hodograph transformations which do not satisfy the Painlevé property at all. An example of such an equation is , where , and are arbitrary functions.
This is obviously a linearisable equation since its solution proceeds through the solution of a linearisable equation and a linear one, in cascade. The solution of this equation does not possess the Painlevé property. Instead of performing a standard Painlevé analysis, let us profit from the fact that the solution of (2.13) can be explicitly constructed. Solving the equation for , we find . Next we integrate for and obtain . A singularity will appear in the expansion of whenever we have such that . We solve for and find (where is the inverse function of ). Expanding around , we have and the integration for can be performed order by order. We find . Thus, since the exponent is arbitrary, the solution does not possess the Painlevé property.
Equation (2.13) may be easily generalised. The principle remains the same. One starts from a linearisable equation in one independent and one dependent variable, say . If, for instance, we take for a higher-order projective equation, we are guaranteed that the solution for will satisfy the Painlevé property. Next we couple this equation to a linear PDE of the form , where , and can be taken as inhomogeneous linear functions of . Eliminating , one obtains an equation for which is linearisable and can be shown to violate the Painlevé property, the exponent of the leading singular term being again an arbitrary function of .
3. Discrete Systems
has a confined singularity pattern but it is not integrable. This can be established by studying the growth properties of the iterates of some initial condition following the approach of . Indeed starting from initial data , we introduce homogeneous variables through and compute the homogeneity degree of the iterates of the mapping in , to which we assign the same degree 1, while is assigned the degree 0. For a generic, nonintegrable mapping, the degree growth of the iterates is exponential. In the case of (3.1) we obtain the following sequence: 0, 1, 2, 4, 8, 14, 24, 40, 66, 108, 176, 286 which, for large enough, obeys the recursion relation . Thus we have indeed an exponential growth of the degree, the asymptotic ratio of two consecutive being , that is, the same as that of the Fibonacci sequence.
For integrable mappings, the growth is just polynomial. Moreover, a detailed analysis of discrete Painlevé equations  and linearisable mappings  has shown that the latter have even slower growth properties (which can be used not only as a detector of integrability but as an indicator of the integration method). In what follows, we will examine the results of the application of the two methods to integrable discrete systems. We will not present in any detail the case of projective mappings . It suffices to say that any singularity appearing in projective systems is confined in one step. Moreover, the study of the degree of the iterates  shows that there is no growth at all: the degree is constant. Thus both criteria are satisfied in this case.
It goes without saying that the degree growth of the iterates of (3.2) is linear as expected from algebraic entropy arguments.
Next we turn now to the case of the Gambier mapping . The latter is, in perfect analogy to the continuous case, a system of two (discrete) Riccati equations in cascade:
where and are all functions of the independent discrete variable . In , it was shown that system (3.4a) and (3.4b) is not confining unless the coefficients entering in the equation satisfy certain conditions. On the other hand, the same argument presented in the continuous case can be transposed here: the integration of the two Riccati equations in cascade can always be performed through reduction to linear second-order mappings. The study of the degree growth of the iterates of (3.4a) and (3.4b) was performed in  where it was found that the growth is always linear, independently of the conditions we referred to above.
This result leads naturally to the following generalisation of the discrete Gambier system, the singularities of which are, in general, not confined:
where are polynomials in the coefficients of which may depend on the independent variable . The study of the degree growth of the iterates of (3.5a) and (3.5b) is straightforward. We find that the degree growth of is linear. Again, system (3.5a) and (3.5b) can be integrated in cascade. On the other hand, (3.5a) and (3.5b) cannot be written as a three-point mapping for . Indeed, if we eliminate between (3.5a), (3.5b) and the upshift of the latter, we obtain an equation relating and which is polynomial in all three variables, generically not linear in . This does not define a mapping but rather a correspondence which in general leads to exponential proliferation of the number of images and preimages. This correspondence is not integrable but this is not in contradiction with the integrability of (3.5a) and (3.5b). The two systems are not equivalent.
where is globally polynomial of degree two in all the 's but not more than linear separately in each of and . Writing that the left-hand side of (3.6) is the same as that of its upshift, we get an equation relating , , , and . For appropriate choices of , this four-point equation can be identical (up to unimportant factors) to the four-point equation obtained from (3.7) by writing . The integration method is quite similar to that described in the continuous case. Given , and starting with at some , one gets from (3.7). Implementing (3.6), this fixes the value of . From now on, one integrates the linear equation (3.6) for all . Since the four-point equation is always satisfied, this means that computed at any has a constant value, which is just , so (3.7) is satisfied.
where and . Mapping (3.8) is generically nonconfining unless is a constant.
is both chaotic and integrable (since it is linearisable). The value of the algebraic entropy is . The same would apply to all cases : they all have a positive algebraic entropy and are also linearisable. The main difference of this mapping compared to the previous example is that while (3.12) is rational the transformation leading to the linear equation (3.10) is transcendental.
While this equation is linearisable, it does not have confined singularities. Indeed, if at some lattice position we have (which is perfectly possible given the appropriate initial conditions), iterating (3.15) we find that for all . On the other hand, since (3.15) is linearisable, we expect the growth of the sequence of its iterates to be linear. This turns out to be indeed the case. Taking initial conditions , , (with arbitrary functions of their argument) and computing the global homogeneous degree in , we find that for .
Generalising (3.15) is quite straightforward. It suffices to start from a linearisable equation for of higher order (of which several examples do exist). Next, we couple to a linear equation of the form where are first degree in , and using the first equation we eliminate . We surmise that the equation for will in general have unconfined singularities. However, this has to be examined on a per-case basis since there does not seem to exist a general argument for the singularity structure of the final equation.
4. Ultradiscrete Systems
Before proceeding to the analysis of ultradiscrete systems, it is interesting to spend a few lines again on their discrete counterparts focusing on the notion of singularity. Given a mapping of the form , we are in the presence of a singularity whenever , that is "loses" its dependence on . When this is due to a particular choice of initial conditions, we are referring to this singularity as a movable one. Movable singularities may be bad, for integrability, because they may lead, after a few mapping iterations, to an indeterminate form ( ) or propagate indefinitely. In the former case, provided we can lift the indeterminacy while recovering the lost degree of freedom (using an argument of continuity with respect to the initial conditions), we are talking about a confined singularity. The typical singularity pattern in the case of confined singularities is the following: the solution is regular for all values of the index up to some value , then a singularity appears and propagates up to whereupon it disappears and the solution is again regular for all values of the index larger than . In some cases, we are in presence of the reciprocal situation. The solution is singular for all values of , becomes regular between and , and is again singular for . This singularity is called weakly confined by Takenawa  and is considered to be compatible with integrability. At the limit where there exists no interval where the solution may be regular, and the solution is singular throughout, we are in the presence of what we call a "fixed" singularity (which again does not hinder integrability).
Joshi and Lafortune  have transposed these notions to the ultradiscrete case and proposed an analogue to the singularity confinement property. In the ultradiscrete systems, the nonlinearity is mediated by terms involving the operator. Typically one is in presence of terms like . When, depending on the initial conditions, the value of crosses zero, the result of the operation becomes discontinuous: when is slightly smaller than 0 the result is zero, while for the result is . It is this discontinuity that plays the role of the singularity since it leads nonanalyticity. Typically, if we put , a term propagates with the iterations of the mapping and perpetuates the discontinuity unless by some coincidence it disappears. This disappearance is the equivalent of the singularity confinement for ultradiscrete systems. Joshi and Lafortune have introduced an algorithmic method for testing the confinement property for ultradiscrete systems, linked it to integrability, and reproduced results on ultradiscrete Painlevé equations by initially deautonomising ultradiscrete mappings.
In this case, we are in presence of a weakly confined solution: a regular part around is surrounded by unconfined singularities both for large positive and large negative 's. Thus, the ultradiscrete mapping (4.2) has confined singularities and is not integrable.
We remark readily that while for negative indices the solution is regular, a singularity, mediated by , appears for positive 's and is never confined.
While some progress towards a Nevanlinna-like theory for ultradiscrete systems has been recently accomplished , it is clear that more work is necessary before we can be sure where we stand concerning ultradiscrete integrability.
In the light of the examples analysed in the previous section, we can now present our ideas and beliefs on integrability and its detection. A perfect criterion of integrability would in theory be both necessary and sufficient. However, no real-life integrability criterion meets these stringent requirements. This has also to do with the loose definition of "integrability" which in some cases is used in lieu of "linearisability" or even "solvability."
In the continuous domain, the "Painlevé" criterion is of unmatched success. Systems integrable through spectral methods have the Painlevé property and its rigorous detection constitutes a reliable integrability predictor. (It goes without saying that the practical implementations of singularity analysis , known as the "Painlevé test," offer no such guarantees: they may miss some critical singularities to say nothing of the complexity of the task which sometimes leads to incomplete applications of the test and erroneous conclusions.) If one extends the notion of integrability so as to include systems integrable through linearisation or quadratures, then the relation to the Painlevé property is lost and the Painlevé criterion is violated by these systems.
In the case of discrete systems, the situation is more complicated. Integrability is not conditioned by the local singularity behaviour which explains why the singularity confinement test is not sufficient and one must consider the growth properties of the solution. While the algebraic entropy techniques furnish a successful test, the ideal discrete analogue of the Painlevé property is still lacking. Perhaps something along the lines of the Sakai approach  for the derivation of the discrete Painlevé equations could provide the answer. Just as in the continuous case, systems which are integrable through spectral method possess the singularity confinement property while the linearisable systems do not. The main difference here is that the algebraic entropy techniques constitute a reliable linearisability detector (with a few precautions as to the class of transformations allowed).
Finally, the ultradiscrete case is even less well understood. As we have shown here, there exist ultradiscrete systems with confined singularities (in the sense of Joshi and Lafortune) and which are nonintegrable. Conversely, some systems integrable through linearisation do not have confined singularities, in perfect parallel to the discrete situation. What is more worrisome here is that there exists at least one example  of a linearisable system which possesses an explicit invariant and still has unconfined singularities. In the ultradiscrete case, the notion of integrability through spectral methods is not yet well established. Thus we cannot make any claims concerning the confinement properties of such systems. Another difficulty has to do with the linearisable character which is assessed through the relation of the ultradiscrete system to its discrete parent and not in an intrinsic way. Moreover, the study of the growth properties of ultradiscrete systems (in particular through the equivalent of Nevanlinna theory) is at its initial phase. We hope that all these open questions will find some answers in the next few years.
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