H. Flaschka and A. Newell. Monodromy and spectrum preserving deformations, I. Comm. Math. Phys., 76(1):67-116, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Define Xco to be a random variable with the distribution function Fl(X) As indicated in Introduction, we need new classes of distribution functions to describe the transitions from XGSE to XGOE and from Xcu to Xco2. First we consider the Riemann Hilbert problem (RHP) for the Painlev II equation [20, 29]. Let F be the real line 1, oriented from cx to cx, and let m( x) be the solution of the following RHP: ra(z; x) is analytic in z C F, ra (z; x) ra (z;x) e2i( za :rz) for z F, 2.15) ra(z;x) J o( asz . Here ra (z;x) resp, ra ) is the limit of ra(z;x) as z z from the left (resp. ....

....z F, 2.15) ra(z;x) J o( asz . Here ra (z;x) resp, ra ) is the limit of ra(z;x) as z z from the left (resp. right) of the contour r: ra(z; x) lim,0 ra(z = ie; x) Relation (2. 15) corresponds to the RHP for the PII equation with the special 10 monodromy data p q 1, r 0 (see [20, 29], also [22, 19] In particular if the solution is expanded at m(z;x) I ] ml(x) o(zl ) as z cx) 2.16) z 2i(m1(x) 12 2i(m1(x) 21 u(x) 2.17) 2i(m1(x) 22 2i(1(x) 11 v(x) 2.18) where u(x) and v(x) are defined in (2.1) 2.5) Now we define two one parameter ....

H. Flaschka and A. Newell. Monodromy and spectrum preserving deformations, I. Comm. Math. Phys., 76(1):67-116, 1980.

....functions. Therefore F (x) is indeed a distribution function. As indicated in Introduction, we need new classes of distribution functions to describe the phase transitions from GSE to GOE and from GUE to GOE 2 . First we consider the Riemann Hilbert problem (RHP) for the Painlev e II equation [16, 23]. Let be the real line R, oriented from 1 to 1. Let m( x) be the solution of the following RHP : 8 : m(z; x) is analytic in z 2 C n , m (z; x) m (z; x) 0 1 e 2i( 4 3 z 3 xz) e 2i( 4 3 z 3 xz) 0 1 A for z 2 , m(z; x) I O 1 z as z 1. ....

....= I O 1 z as z 1. 2.15) Here m (z; x) resp, m ) is the limit of m(z 0 ; x) as z 0 z from the left (resp. right) of the contour : m (z; x) lim #0 m(z i ; x) Relation (2. 15) corresponds to the RHP for the PII equation with the special monodromy data p = q = 1; r = 0 (see [16, 23], also [18, 15] In particular if the solution is expanded at z = 1, m(z; x) I m 1 (x) z O 1 z 2 ; as z 1; 2.16) we have 2i(m 1 (x) 12 = 2i(m 1 (x) 21 = u(x) 2.17) 2i(m 1 (x) 22 = 2i(m 1 (x) 11 = v(x) 2.18) where u(x) and v(x) are de ned in (2.1) 2.5) Therefore the ....

H. Flaschka and A. Newell. Monodromy and spectrum preserving deformations, I. Comm. Math. Phys., 76(1):67-116, 1980.

....etc. Rather, for the convenience of the reader who may not be familiar with Riemann Hilbert theory, we will use the above calculations and computations as a guide, and verify all the steps directly as they arise. LONGEST INCREASING SUBSEQUENCE 11 We now consider the RHP for the PII equation ([FN], JMU] see also [IN] FZ] DZ2] We will consider two equivalent versions of the RHP for PII. These two RHP s will be used in the later sections for the construction of parametrices for the solution of (1.26) Let P II denote the oriented contour consisting of 6 rays in Figure 3. Thus ....

....6= z 2 P II 0 ; m P II 0 (z) I as z 1; 2.13) in the sense of (2.2) Let m P II 1 (x) denote the residue at 1 of m P II 0 (z) given by m P II 0 (z; x) I m P II 1 (x) z O( 1 z 2 ) as z 1. Then u(x) 2im P II 1;12 (x) 2im P II 1;21 (x) 2. 14) solves PII (see [FN], JMU] u xx = 2u 3 xu; x 2 R; where m P II 1;12 (x) resp. m P II 1;21 (x) denotes the (12) entry (resp, 21) entry) of m P II 1 (x) It is easy to see that m P II 1 (x) and hence u(x) in (2.14) is independent of the choice of 0 2 (0; 3) A solution of the RHP ( P II 0 ....

H.Flaschka and A.Newell, Monodromy and spectrum preserving deformations, I, Comm. Math. Phy., 76, no.1, 67-116, (1980).

....solutions contain ample analytic information and in many cases (for instance generic linear or nonlinear differential systems [16] the actual solutions of the problem can be reconstructed from their transseries by modern resummation methods. Unlike the isomonodromic deformation approach (see [2, 10, 5, 9, 4]) the method introduced in this paper is suitable to be used in nonintegrable systems as well. 1.1 Notes on asymptotics beyond all orders for P1 Proofs of the various technical statements in this section can be found in [16] in the more general context of analytic rank one differential systems. ....

Flaschka, H. and Newell, A. C. Monodromy- and spectrum-preserving deformations. I Comm. Math. Phys. 76: 65--116, 1980

....which are now known as the Painlev e equations (P I P VI ) The rst two of these are P I d 2 y dz 2 = 6y 2 z; 1) 2 P II d 2 y dz 2 = 2y 3 zy ; 2) where is a constant. The six Painlev e equations are integrable (solvable) via associated (linear) isomonodromy problems [14, 23]. There are many ways in which one could discretize these equations (Euler s method, for example) Most of these will no longer possess the many special properties that the di erential equations (1 2) possess; e.g. related isomonodromy problems and solutions that are asymptotically well behaved. ....

H. Flaschka and A.C. Newell. Monodromy- and spectrum preserving deformations. i. Commun. Math. Phys., 76:65-116, 1980.

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Flaschka, H., Monodromy- and Spectrum-Preserving Deformations I, Commun. math. Phys., 76, 65-116 (1980)

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