Symmetries and hamiltonians of Ince’s XXXVIII and XLIX equations

Abstract

We discuss symmetries of Hamiltonians of I38 and I49 equations that appear on Ince's list of fifty second-order differential equations with Painlevé property. This study is informed by structure of Weyl symmetries of Painlevé PIII and mixed Painlevé PIII−V equations and provides insights into differences between the symmetries of Painlevé equations and symmetries of solvable equations on Ince's list.

On special limits of the Mixed Painlevé PIII–VModel

“…Comparison of actions of s 2 and π 0 on parameters β, γ, δ, κ in equations ( 33) and (34) indeed reveals identical behavior (up to the sign) of those two transformations, which in the discussion below (22) was recognized as a reason for why the geometric interpretation of W [s 0 , s 2 , π 0 , π 2 , π 2 ] as an extended affine Weyl group did not extend to the case of symmetry of I 12 equation. The remaining questions of how to complete Hamiltonian structures seen in this paper in such a way as to obtain full equations I 38 , I 49 and what are the symmetries governing I 38 , I 49 models will be addressed in a paper in preparation [3].…”

Coalescence, Deformation and Bäcklund Symmetries of Painlevé IV and II Equations

“…Note that in order to define the action of authomorphisms π i , ρ i , i = 0, 1, 2 they need to be formulated on an enlarged parameter space that includes σ that transforms nontrivially under these authomorphisms, see tables (2.9), (2.10). The presence of σ affords us also an opportunity to include in the formalism the solvable Painlevé equations (classified by Gambier) that appear on Ince's list [12] (see also [2]). In particular, equations I 30 , I 8 given in equations (3.7),(3.9) were obtained here in the σ → 0 limit.…”

“…Customarily, people set C = 0 and σ = 1. Recall that setting σ = 0 reduces P IV to Ince's XXX equation (see also [2]). The integration constant C can be absorbed by redefining f i 's : f i → g i so that i g i = σz and the system is obviously still invariant under Bäcklund symmetries s i , i = 0, 1, 2 and π.…”

“…More recently, in reference [3] we introduced the hybrid P III−V model that was obtained as reduction of a class of integrable models known as multi-boson systems [6,7] that generalize the AKNS hierarchy [5]. The P III−V model reduces to P III , P V and I 12 , I 38 and I 49 equations from Ince's list [12,2] for special limits of its parameters while for remaining finite values of its parameters preserves enough symmetry under remaining Bäcklund transformations of the extended affine Weyl symmetry group to satisfy Painlevé property [3].…”

Coalescence, deformation and Bäcklund symmetries of Painlevé IV and II equations

“…More recently, in reference [3] we introduced the hybrid P III-V model that was obtained as reduction of a class of integrable models known as multi-boson systems [6,7] that generalize the AKNS hierarchy [5]. The P III-V model reduces to P III , P V and I 12 , I 38 and I 49 equations from Ince's list [2,12] for special limits of its parameters while for remaining finite values of its parameters preserves enough symmetry under remaining Bäcklund transformations of the extended affine Weyl symmetry group to satisfy Painlevé property [3].…”