The Painlevé test for C P N − 1 sigma models

We test the C P N − 1 sigma models for the Painlevé property. While the construction of finite action solutions ensures their meromorphicity, the general case requires testing. The test is performed for the equations in the homogeneous variables, with their first component normalised to one. No constraints are imposed on the dimensionality of the model or the values of the initial exponents. This makes the test nontrivial, as the number of equations and dependent variables are indefinite. The C P N − 1 system proves to have a (4N − 5)-parameter family of solutions whose only movable singularities are poles, while the order of the investigated system is 4N − 4. The remaining degree of freedom, connected with an extra negative resonance, may correspond to a branching movable essential singularity. An example of such a solution is provided.


Introduction
Numerous physical applications of models with effective Lagrangians, in particular the  -P N 1 sigma models [4,5,14,27,29,36,39], make these models an interesting subject of study [15][16][17]. In particular, these models have been shown to play an essential role in several applications to nonlinear phenomena in such areas of physics as quantum field theory, string theory (both bosonic [26,35] and superstrings) [6], statistical physics, e.g. the Ising model [31], gauge field theories, e.g. the reduction of the self-dual Yang-Mills equations to the Ernst model for cylindrical gravitational waves [1,3], phase transitions (e.g. the growth of crystals, deformations of membranes [13,33]), coherent states obtained via the  -P N 1 sigma models [18,23], fluid dynamics e.g. the motion of boundaries between regions of different densities and viscosities [9]. In biochemistry and biology there are applications such as biological membranes and vesicles, for example long protein molecules [14,28,34] and the Canham-Helfrich-Evans membrane models [24,27]. These macroscopic models can be derived from microscopic ones and allow us to explain basic features and equilibrium shapes both for biological membranes and for liquid interfaces [37]. In mathematics, P N models have been shown to play a pertinent role for the systematic description of surfaces immersed in Lie algebras and isomonodromic deformations in connection with surfaces and Painlevé type equations [7,19]. The question of the integrability of the equations governing these models has found an apparently positive answer in the works of Din and Zakrzewski [17]. Moreover, the linear spectral problem is known for them, so (in principle) the initial problem may be solved by the inverse scattering method. However the above results only concern systems with finite action. On the other hand, if we are interested in the dynamics of the systems, we start from the corresponding Euler-Lagrange (EL) equations, which allow for a much larger class of solutions. Singularities are an intrinsic property of nonlinear equations and some of them may make the action infinite. A natural question arises, as to whether the equations remain integrable if we remove the assumption of finite action. In the present paper we will discuss this question and provide a self-contained approach to the subject.
The first approach which we try when testing a system of equations for integrability is usually the Painlevé test in the form introduced in [2], or its generalisation to partial differential equations (PDEs) [38], with possible further refinements (as discussed in [10,11,32], which provide a comprehensive review of the method). Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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In our case, the Painlevé test entails extra difficulties due to the fact that the dimensionality of the  -P N 1 model and the number of equations are arbitrary. Nevertheless, the test can be carried out (see section 3).
In what follows, section 2 contains a short summary of  -P N 1 models and various methods of their description. We conclude that section with selecting the description (system of PDEs) suitable for the Painlevé test. In section 3 we perform the test, obtaining a 'nearly general' local solution in the form of a Laurent series. By 'nearly general' we mean that our solution provides 4N−5 out of the 4(N−1) first integrals in the general solution, i.e. one integral fewer than the order of the system. Section 4 contains a discussion of the missing first integrals. A counterexample, i.e. an example of the non-Painlevé behaviour, is given in the form of a solution which has an essential singular manifold with branching. The manifold depends on four parameters (although not on an arbitrary function), which means that the position of the singularity depends on the initial conditions.
2.  -P N 1 sigma models Sigma models describe complex systems by a simple Lagrangian defined in terms of an effective field which lies in an appropriate space, while the complexity remains in the metrics of the space.
i j represent the field variables in  N , while g ij is the metric tensor. A bar over a symbol denotes its complex conjugate.
The models prove to be rich in interesting properties provided that the metric depends on the fields, i.e. the model is nonlinear. Even simple nonlinear cases, like the  -P N 1 models, have many applications, from two dimensional gravity to biological membranes [8,20,27]. In these models the independent variables ξ 1 , ξ 2 take values in the Riemann sphere or in a 2D Minkowski space, Î z S N , while the differential in (1) is expressed in terms of the z-dependent covariant derivatives D μ by m = ¶ - producing a Lagrangian density of the form where the convention of summation over repeating Greek indices is assumed, z and z † are complex unit vectors in  N , a dagger denotes the Hermitian conjugate, while ∂ and ¶ are the derivatives with respect to ξ=ξ 1 +iξ 2 and x x x =i 1 2 respectively. The normalisation of z requires that The EL equations corresponding to the Lagrangian are simple, but they are not suitable for testing the Painlevé property: due to the normalisation (4), a pole of z has to correspond to a zero of † z , at least for real ξ μ . For the same reason, we do not analyse even simpler equations satisfied by the rank-1 projectors The necessary freedom is achieved if we use the homogeneous unnormalised field variables f, such that N 1 2 0 1 whose dynamics are governed by the unconstrained EL equations The way in which these vector functions are constructed makes them elements of a Grassmannian space Gr  ( ) 1, N [39] and suggests that equations (8) are invariant under multiplication of fby any scalar function (which may easily be checked by direct calculation). This property leaves too much freedom for the shape of possible singularities. However if we normalise the homogeneous variables in such a way that the first component f 0 is equal to 1, we eventually obtain a system of equations suitable for the Kovalevsky-Gambier analysis, commonly known as the Painlevé test. The equations in terms of the affine variables w=(w 1 , K, w N−1 ), such that where the complex conjugates of (10a) have been written separately as (10b) because the complex conjugation will no longer link the variables w i withw i when we extend the independent variables analytically to the double complex plane  2 (as it is done in the Painlevé test). Therefore, in what follows, we put 'complex conjugation' in quotation marks while naming the symmetry which turns the unbarred quantities into the barred ones and vice versa.
Equations (10a) (10b) will be the subject of further analysis. They constitute a system of 2(N−1) secondorder PDEs, which requires 4(N−1) first integrals to build the general solution.

The Painlevé test
To perform the test, we look for the solution of system (10a), (10b), extended to the double complex plane  x x Î (¯) , 2 , in the form of a Laurent series about a movable non-characteristic singularity manifold Kruskal s simplification , 11 where the function j defining the singularity manifold is a holomorphic function of ξ, while the coefficients of the expansion are analytic in their arguments x x (¯) , . The condition of being non-characteristic excludes the surfaces ξ=0 and x = 0, which in turn eliminates locally holomorphic and locally antiholomorphic functionsw w , , including the solutions of Din and Zakrzewski [15,17]. On the other hand, the selection of non-characteristic singularity manifolds makes possible both the Kruskal simplification [25] and the assumption j x ¢ ¹ ( ) 0. In the series below, we adopt the notation in which a superscript for F is simply an exponent, while a superscript for a dependent variable, e.g. w i n denotes the n-th order coefficient in the Laurent expansion of w i . Additionally, it is convenient to extend the notation to negative n, assuming = < = ¼ -( ) w n i N 0 whenever 0, for all 1, , 1. 12 i n We do not limit the number of dependent variables w i and allow a priori the possibility that the initial exponents of each w i may be different. Thus the Laurent expansion has the form Then we shift the dummy index k in such a way that all terms of the same k become proportional to the same powers of F (the range of k remains unchanged thanks to our convention (12)). Next we require that the coefficients of all powers in F vanish.
Under the assumption that a b > > 0, 0 i i for all i=0, K, N−1, there is no balance of terms with exponents of different form in the lowest order. Therefore, the initial exponents are obtained from the equations satisfied by the coefficients of the lowest-order terms k=0 rather than those satisfied by the exponents of those terms In the lowest-order, we obtain for = ¼ - where the prime denotes the derivative with respect to ξ (we have omitted the ξ-dependence of j and all the w′s).
Equations (14) solves both systems (14). A question arises as to whether this is the only solution in which all a i and β i are positive. We will show that this is indeed the case. The proof below is performed for the system (14a). The proof for (14b) is identical.
Proof. Suppose that our movable singularity manifold intersects the plane 1 are absent from the lhs of the conjugate system (although the systems remain coupled with each other through the right-hand sides (rhs)). This absence means that the matrix of coefficients of the complete linear system is a direct sum of two square matrices and its determinant is a product of their determinants. The Fuchs indices or resonances (we use the second name to avoid misunderstandings in our multi-index notation) are calculated from the requirement that the determinant vanish.
The first matrix has the elements The second component of the direct sum is its 'complex conjugate' (further abbreviated to 'c.c.'). The instances in which the determinant of their direct sum vanishes are listed in table 1.

Resonance Multiplicity
Linear dependence of the rows k=0 2 (N−1) Each row identically vanishes. Altogether we have 4N−4 zeros. This number is equal to the total order of the system of PDEs. Hence there are no more resonances.
We now test the compatibility of the resonances by checking whether the rhs of equations (17) have the same linear dependence between rows as their lhs For k = 0, all terms on the rhs contain w with a negative superscript, which according to our convention (12) means that they are equal to zero. Hence the whole rhs is equal to zero, as it should be. Consequently, this leaves room for 2(N−1) arbitrary functions of ξ (first integrals). The above verification of compatibility cannot be performed for negative zeros. One of the two zeros k=−1 corresponds to the arbitrariness of j x ( ). The compatibility of the other zero at k=−1 remains unknown. The verified zeros allow us to introduce a total of 4N−6 arbitrary functions of ξ. These are w i 0 andw i 0 for i=1, K, N−1 and w i 1 andw i 1 for i=2, K, N−1. Together with the arbitrary singularity manifold j (corresponding to one of the two zeros k=−1), they constitute a set of 4N−5 first integrals. There remains the second zero k=−1, which is the cause of the missing (4N−4)-th first integral. This problem will be addressed in the next section.

The question of the double resonance at k=−1
The negative resonances, except for a single k=−1 resonance, correspond to essential singular points. In the  -P N 1 model, they are connected with the coupling between w andw (if not for the coupling we would have two separate systems, each possessing a single resonance k=−1). A singularity connected with the phase may indeed be essential. A question arises: does the essential singularity introduce multivaluedness in the solution or not.
The authors tried to apply the perturbative Painlevé analysis of [12] to the P 1 model. Up to the third order in the perturbation of the Laurent series (13) all the resonances are compatible. However the order at which an incompatibility may occur is difficult to predict. Being unable to prove the Painlevé or non-Painlevé property by any systematic method, we limit ourselves to a counterexample.
An example of a solution (an envelope solitary wave) which has branching at a point dependent on the initial conditions has been derived by Lie group analysis and the corresponding symmetry reduction of the P 1 model in [21,22]. A typical solution of the kind reads To ensure that w andw are complex conjugates of each other when the remaining quantities are real, it is usually assumed that p<−1. However the solution is valid for any p. This solution (as well as several other solutions in the form of elliptic functions) is associated with multileaf surfaces [21,22]. It is singular for For these values of χ, the argument of arctan in (19) becomes infinite, which results in branching (i.e. the multivaluedness of the arctan function). These singularities do not lie on characteristics (x = const and x = const ), which makes them proper for the analysis. The position of the singularities depends on four parameters: , , , 0 , and thus also on the initial conditions, which contradicts the usual understanding of the Painlevé property. However, the authors are aware that a more constructive answer to the question of compatibility at the negative resonance would be provided by a non-Painlevé solution with its position dependent on an arbitrary function rather than a few parameters. We do not have such a solution.
The action integral for the example (19) is not finite, hence it is compatible with the theorem of Din and Zakrzewski [15,17,39]. Neither does it contradict the classical result of [2], because (19) cannot be obtained as a solution of a Gelfand-Levitan-Marchenko equation [30] with a finite integral kernel. The P 1 model is a limit case of  -P N 1 models, where all but one affine coordinates (and all but one 'complex conjugates') tend to zero. Thus the absence of the Painlevé property in the P 1 model infers its absence for all  -P N 1 models.

Conclusion
We have shown that the equations governing the behaviour of  -P N 1 models, without the constraint of finite action, may have solutions with movable singularities in the form of pole manifolds. The order of the poles is equal to one for all dependent variables (the calculation, based on the usual assumption of the Painlevé test, i.e. negative initial exponents, has eliminated poles of other orders). For the  -P N 1 model equations, the Laurent series about a pole manifold is consistent at all 4N−5 nonnegative resonances. In this way, it provides a family of solutions with 4N−5 parameter functions (first integrals) within the domain of convergence of the series. However, branching may still occur at essential singular points. We have provided an example of a solution which is multivalued in the neighbourhood of a sequence of non-characteristic movable singular manifolds. Their position depends on the initial conditions through four parameters. It would be desirable to find a deformation of such solutions turning them into solutions depending on an arbitrary function. The result shows that the  -P N 1 models admit solutions which would branch (multifurcate) at the points at which the action integral diverges. It seems to extend the range of applicability of these models to physical and biological phenomena.
The Painlevé analysis is nontrivial for these models due to the indefinite number of equations and dependent variables.