Higher-Order Painlevé Equations in the Polynomial Class II: Bureau Symbol P1

In this article, we complete the Painlevé classification of fourth-order differential equations in the polynomial class that was begun in paper I, where the subcase having Bureau symbol P 2 was treated. This article treats the more difficult subcase having Bureau symbol P 1. Some of the calculations involve the use of computer searches to find all cases of integer resonances. Other cases are better handled with the Conte–Fordy–Pickering test for negative resonances. The final list consists of 19 equations denoted F-I, F-II, … , F-XIX, 17 of which have the Painlevé property while 2 (F-II, F-XIX) have Painlevé violations but are nevertheless very interesting from the point of view of Painlevé analysis. The main task of this article is to prove that the 17 Painlevé-type equations and the equivalence classes that they generate provide the complete classification of the fourth-order polynomial class. Equations F-V, F-VI, F-XVII, and F-XVIII define higher-order Painlevé transcendents. Of these, F-VI was new in paper I while the other three are group-invariant reductions of the KdV5, the modified KdV5, and the modified Sawada–Kotera equations, respectively. Seven of the 19 equations involve hyperelliptic functions of genus 2. Partial results on the fourth-order classification problem have been obtained previously by Bureau, Exton, and Martynov, the latter author finding all but four of the relevant reduced equations. Complete solutions are given except in the cases that define the aforementioned higher-order transcendents.     

Chazy Classes IX–XI Of Third-Order Differential Equations

In this article, we study Classes IX–XI of the 13 classes introduced by Chazy (1911) in his classification of third-order differential equations in the polynomial class having the Painlevé property. Classes IX and X are the only Chazy classes that have remained unsolved to this day, and they have been at the top of our "most wanted" list for some time. (There is an incorrect claim in the literature that these classes are unstable.) Here we construct their solutions in terms of hyperelliptic functions of genus 2, which are globally meromorphic. (We also add a parameter to Chazy Class X, overlooked in Chazy's original paper.) The method involves transforming to a more tractable class of fourth- and fifth-order differential equations, which is the subject of an accompanying paper (paper I). Most of the latter equations involve hyperelliptic functions and/or higher-order Painlevé transcendents. In the case of Chazy Class XI, the solution is elementary and well known, but there are interesting open problems associated with its coefficient functions, including the appearance of one of the aforementioned transcendents. In an appendix, we present the full list of Chazy equations (in the third-order polynomial class) and the solutions of those that are not dealt with in the body of this article.     

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

In this article our concern is with the third Painlevé equation
d² y /d x ²= (1/ y )(d y /d x )²− (1/ x )(d y /d x ) + ( αy ²+ β )/ x + γy ³+ δ / y
where α, β, γ, and δ are arbitrary constants. It is well known that this equation admits a variety of types of solution and here we classify and characterize many of these. Depending on the values of the parameters the third Painlevé equation can admit solutions that may be either expressed as the ratio of two polynomials in either x or x ^1/3 or related to certain Bessel functions. It is thought that all exact solutions of (1) can be categorized into one or other of these hierarchies. We show how, given a few initial solutions, it is possible to use the underlying structures of these hierarchies to obtain many other solutions. In addition, we show how this knowledge concerning the continuous third Painlevé equation (1) can be adapted and used to derive exact solutions of a suitable discretized counterpart of (1). Both the continuous and discrete solutions we find are of potential importance as it is known that the third Painlevé equation has a large number of physically significant applications.     

Toward the Classification of Scalar Nonpolynomial Evolution Equations: Polynomiality in Top Three Derivatives

We prove that arbitrary (nonpolynomial) scalar evolution equations of order    m  ≥ 7  , that are integrable in the sense of admitting the canonical conserved densities   ρ⁽¹⁾, ρ⁽²⁾  , and   ρ⁽³⁾   introduced in [ 1 ], are polynomial in the derivatives    u _{m −  i}    for  i  = 0, 1, 2. We also introduce a grading in the algebra of polynomials in     u _k     with     k  ≥  m  − 2    over the ring of functions in     x ,  t ,  u , … ,  u _{m −3}    and show that integrable equations are scale homogeneous with respect to this grading .     

Binomial-Type Ordinary Differential Equations of the Third Order

The complete Painlevé classification of the binomial ordinary differential equations of the third order is built. Eight classes of equations with Painlevé property are obtained. All of these equations are solved in terms of elementary functions and known Painlevé transcendents.     

Painlevé Classification of Binomial Type Ordinary Differential Equations of the Arbitrary Order

The complete Painlevé classification of the binomial ordinary differential equations of the arbitrary order n ≥ 4 is built. Six classes of equations with Painlevé property are obtained. All of these equations are solved in terms of elementary functions and known Painlevé transcendents.     

Conditional Lie Bäcklund Symmetries and Sign-Invariants to Quasi-Linear Diffusion Equations

Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term u _t =[ u ^m ( u _x ) ⁿ ] _x + P ( u ) u _x + Q ( u ) , where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bäcklund conditional symmetry with characteristic η= u _xx + H ( u ) u ² _x + G ( u )( u _x )^{2− n} + F ( u ) u ^{1− n} _x and the Hamilton–Jacobi sign-invariant J = u _t + A ( u ) u ^{n +1} _x + B ( u ) u _x + C ( u ) which preserves both signs, ≥0 and ≤0, on the solution manifold. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems.     

Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight

We study polynomials that are orthogonal with respect to the modified Laguerre weight   z ^{− n +ν} e ^{− Nz} ( z − 1)^{2 b}   , in the limit where   n , N →∞  with   N / n → 1  and ν is a fixed number in     . With the effect of the factor (   z − 1)^{2 b}   , the local parametrix near the critical point z = 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.     

On Discrete Painlevé Equations Associated with the Lattice KdV Systems and the Painlevé VI Equation

A new integrable nonautonomous nonlinear ordinary difference equation is presented that can be considered to be a discrete analogue of the Painlevé V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial differential equations of Korteweg–de Vries (KdV) type. The new equation, which is referred to as generalized discrete Painlevé equation (GDP), contains various "discrete Painlevé equations" as subcases for special values/limits of the parameters, some of which have already been given in the literature. The general solution of the GDP can be expressed in terms of Painlevé VI (PVI) transcendents. In fact, continuous PVI emerges as the equation obeyed by the solutions of the discrete equation in terms of the lattice parameters rather than the lattice variables that label the lattice sites. We show that the bilinear form of PVI is embedded naturally in the lattice systems leading to the GDP. Further results include the establishment of Bäcklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.     

The Second Painlevé Equation in the Large-parameter Limit I: Local Asymptotic Analysis

In this article, we find all possible asymptotic behaviors of the solutions of the second Painlevé equation y "=2 y ³+ xy +α as the parameter α→∞ in the local region x ≪α^2/3. We prove that these are asymptotic behaviors by finding explicit error bounds. Moreover, we show that they are connected and complete in the sense that they correspond to all possible values of initial data given at a point in the local region.     

Justification of a Perturbation Approximation of the Klein–Gordon Equation

Consider the nonlinear wave equation
u_tt − γ ² u_xx + f(u) = 0
with the initial conditions
u ( x ,0) = εφ ( x ), u _t( x ,0) = εψ ( x ),
where f ( u ) is either of the form f ( u )= c ² u −σ u ^{2 s +1}, s =1, 2,…, or an odd smooth function with f '(0)>0 and | f '( u )|≤ C ₀².The initial data φ( x )∈ C ² and ψ( x )∈ C ¹ are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u ( x ,  t ; ɛ), and prove its boundedness in x ∈ R and t >0 for all sufficiently small ɛ>0. Furthermore, we show that the error between the solution u ( x ,  t ; ɛ) and the leading term approximation obtained by the multiple scale method is of the order ɛ³ uniformly for x ∈ R and 0≤ t ≤ T /ɛ², as long as ɛ is sufficiently small, T being an arbitrary positive number.     

On solutions of third and fourth-order time dependent Riccati equations and the generalized Chazy system

We introduce a new transformation (nonlocal) to find the general solutions of some equations belonging to the third and fourth-order time dependent Riccati class of equations. These are in turn related to the Chazy polynomial class and the time dependent F-XVI Bureau symbol PI equations respectively.     

Large Time Asymptotic Behaviors for Periodic Solutions of Generalized Burgers Equations With Spherical Symmetry or Linear Damping

Using the method of balancing arguments, large time asymptotic behaviors for the periodic solutions of generalized Burgers equations   u_t  +  u ³ u_x  +  ju /2 t  =δ/2 u_xx   and   u_t  +  u ³ u_x  +λ u  =δ/2 u_xx   subject to the periodic initial condition     and the vanishing boundary conditions   u (0,  t ) =  u ( l ,  t ) = 0,   t  ≥ 0   or    t ₀,  where   A ,  A ₁, δ, λ,  l ,  t ₀, ∈ R ⁺  and   j  = 1, 2  , are obtained.     

Rational ODEs with Movable Algebraic Singularities

A class of second-order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite-length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations   P_II – P   _VI   are poles.     

Painlevé Classification of a Class of Differential Equations of the Second Order and Second Degree

In this paper we construct all Painlevé-type differential equations of the form (d²y/dx²)² = F(x,y,dy/dx), where F is rational in y and y′=dy/dx, locally analytic in x, and not a perfect square. No further simplifying assumptions are made, but it is found that the absence of a term linear in y″ in the class of equations under investigation forces F to be a polynomial in y and y′. We find exactly six distinct classes of second-degree Painlevé equations, denoted SD-I,??,SD-VI, some of which further subdivide into canonical subcases. Only the first three classes (or at least equations transformable to the first three classes) and part of the sixth have appeared previously in the literature, especially the work of Chazy and Bureau. The fourth and fifth classes are new. The unified treatment of SD-I, which we call the “master Painlevé equation,” is new. Complete solutions are given in terms of the classical Painlevé transcendents, elliptic functions, or solutions of linear equations. In an appendix, it is shown that a class of second-degree equations generalizing the Appell equation can always be reduced to a second-order linear equation.     

Weakly Nonhydrostatic Effects in Compositionally-Driven Gravity Flows

In the study of compositionally-driven gravity currents it is customary to adopt the hydrostatic assumption for the pressure field which, in turn, leads to a depth-independent horizontal velocity field and significant simpilifications to the governing equations. The hydrostatic assumption is reasonable in, say, the case of a two-layer flow when the depth variations of the lower layer are small when considered as a function of space and time. However, for larger deflections of the interface (such as those caused by bottom topography) the flow will deviate in its behavior from the low aspect ratio, slowly varying purely hydrostatic flow because of the presence of vertical accelerations. In this paper we present an approach to capture the contribution of interface curvature to nonhydrostatic effects in fully time-dependent flows in two-fluid systems. Our approach involves expanding the relevant dependent variables in the form of an asymptotic expansion   f = f ⁽⁰⁾+δ² f ⁽¹⁾+ o (δ²)  , where  0 < δ≪ 1  is the aspect ratio of the flow, and obtaining the first-order correction to hydrostatic theory. Numerical results and comparisions with the purely hydrostatic theory are included.     

Eigenfunctions of Two-Scale Difference Equations with Rational Symbol

This paper deals with two-scale difference equations having a formal power series as symbol. We require that the equation has non-zero distributional solutions which are either compactly supported or integrals of compactly supported distributions with support bounded to the left. Such solutions are called eigenfunctions. As main result we determine the necessary and sufficient condition for the existence of eigenfunctions that the symbol must be a rational function with a special structure. We show that the eigenfunctions can be expressed by means of a finite sum of shifted eigenfunctions belonging to the case with a polynomial symbol (characteristic polynomial), which is well investigated.     

Horizontal Growth of Harmonic Functions

We show that positive harmonic functions in the upper halfplane grow at most quadratically in horizontal bands. This bound is sharp in a sense to be specified, which, at least implies that there are examples growing as fast as any power under 2. These results are extended to positive harmonic functions in a half-space of R ^{n +1}, with points represented by ( x , y ), where x ∈R ⁿ , and y ∈R, the sharp maximum rate of growth being now ¦ x ¦ ^{n +1}. The case of Poisson integrals of functions in L_p ( dx /(1+(¦ x ¦)² )^{( n +1)/2}) is also taken up; the bound condition is then O (¦ x ¦^{( n +1)/ p} ).     

Unconditional Nonlinear Stability in Temperature-Dependent Viscosity Flow in a Porous Medium

The equations of flow in porous media attributable to Forchheimer are considered. In particular, the problem of thermal convection in such a medium is addressed when the viscosity varies with temperature. It is shown that nonlinear stability may be achieved naturally for all initial data by working with L ³ or L ⁴ norms. It is also shown that L ² theory is not sufficient for such unconditional stability. Previous work has established nonlinear stability for vanishingly small initial data thresholds, but we believe this is the first analysis that addresses the important physical problem of unconditional stability. It is shown how to extend the nonlinear analysis for a viscosity linear in temperature to the cases when the viscosity may be quadratic or when penetrative convection is allowed in the layer.