AUTHOR INDEX (Volume 16)

It is shown that the deformed Nonlinear Schrödinger (NLS), Hirota and AKNS equations with (1 + 1) dimension admit infinitely many generalized (nonpoint) symmetries and polynomial conserved quantities, master symmetries and recursion operator ensuring their complete integrability. Also shown that each of them admits infinitely many nonlocal symmetries. The nature of the deformed equation whether bi-Hamiltonian or not is briefly analyzed.     

Auto-Bäcklund transformations for a matrix partial differential equation

We derive auto-Bäcklund transformations, analogous to those of the matrix second Painlevé equation, for a matrix partial differential equation. We also then use these auto-Bäcklund transformations to derive matrix equations involving shifts in a discrete variable, a process analogous to the use of the auto-Bäcklund transformations of the matrix second Painlevé equation to derive a discrete matrix first Painlevé equation. The equations thus derived then include amongst other examples a semidiscrete matrix equation which can be considered to be an extension of this discrete matrix first Painlevé equation. The application of this technique to the auto-Bäcklund transformations of the scalar case of our partial differential equation has not been considered before, and so the results obtained here in this scalar case are also new. Other equations obtained here using this technique include a scalar semidiscrete equation which arises in the case of the second Painlevé equation, and which does not seem to have been thus derived previously.     

A Generalization of the Sine-Gordon Equation to 2 + 1 Dimensions

Abstract

The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation with just one branch, that is the AKNS equation in 2 + 1 dimensions. The solutions of the former split as linear superposition of two solutions of the second, related by a B¨acklund-gauge transformation. Solutions of both equations are obtained by means of an algorithmic procedure derived from these transformations.     

Bäcklund transformation of matrix equations and a discrete matrix first Painlevé equation

We show that the known auto-Bäcklund transformation for the matrix second Painlevé equation can be generalized to a much wider class of equations. This auto-Bäcklund transformation is an involution and so cannot be used on its own to generate an infinite sequence of different solutions, although for particular equations a second auto-Bäcklund transformation allows this to be done. We also give a Bäcklund transformation for this general class of matrix equations. For the matrix second Painlevé equation we also give a coalescence limit, and a construction of special integrals and of a discrete matrix first Painlevé equation.     

Painlevé integrability and multi-dromion solutions of the 2+1 dimensional AKNS system

The Painlevé integrability of the 2+1 dimensional AKNS system is proved. Using the standard truncated Painlevé expansion which corresponds to a special B?cklund transformation, some special types of the localized excitations like the solitoff solutions, multi-dromion solutions and multi-ring soliton solutions are obtained. Received 31 January 2001 and Received in final form 15 May 2001     

Bäcklund transformations for certain rational solutions of Painlevé VI

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial KP Grassmannian. In this way we obtain transformations that act on the root lattice of A⁵. We also show that this A⁵ root lattice can be related to the F₄⁽¹⁾ root lattice. We thus obtain Bäcklund transformations that relate Painlevé VI tau functions, parametrized by the elements of this F₄⁽¹⁾ root lattice.     

New integrable (2+1)- and (3+1)-dimensional sinh-Gordon equations with constant and time-dependent coefficients

In this work, we mainly address two new integrable (2+1)- and (3+1)-dimensional sinh-Gordon equations, which naturally appear in surface theory and fluid dynamics. The first equation includes constant coefficients, while the other is characterized with time-dependent coefficients. It is of further value to investigate the integrability of each model. This study puts forward a Painlevé test to reveal the Painlevé integrability. We show that the first equation passes the Painlevé test to confirm its integrability. However, the compatibility conditions of the second model with time-dependent coefficients provides the relation between these coefficients to ensure its integrability. We show that the dispersion relations of the two equations are distinct, whereas the phase shifts are identical. We apply the simplified Hirota's method where four sets of multiple soliton are derived for these equations.     

Parameterless discrete Painlevé equations and their Miura relations

We present a study of discrete Painlevé equations which do not have any parameter, apart from those that can be removed by the appropriate scaling. We find four basic equations of this type as well as several more related to the basic ones by Miura transformations, which we derive explicitly. We obtain also the continuous limits of the basic parameterless equations and show that two of them are the discrete analogues of both the continuous Painlevé I and the zero-parameter Painlevé III.     

Constructing discrete Painlevé equations: from E8(1) to A1(1) and back

The ‘restoration method’ is a novel method we recently introduced for systematically deriving discrete Painlevé equations. In this method we start from a given Painlevé equation, typically with symmetry, obtain its autonomous limit and construct all possible QRT-canonical forms of mappings that are equivalent to it by homographic transformations. Discrete Painlevé equations are then obtained by deautonomising the various mappings thus obtained. We apply the restoration method to two challenging examples, one of which does not lead to a QRT mapping at the autonomous limit but we verify that even in that case our method is indeed still applicable. For one of the equations we derive we also show how, starting from a form where the independent variable advances one step at a time, we can obtain versions that correspond to multiple-step evolutions.     

Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation

There has been considerable interest in the study on the variable-coefficient nonlinear evolution equations in recent years, since they can describe the real situations in many fields of physical and engineering sciences. In this paper, a generalized variable-coefficient KdV (GvcKdV) equation with the external-force and perturbed/dissipative terms is investigated, which can describe the various real situations, including large-amplitude internal waves, blood vessels, Bose-Einstein condensates, rods and positons. The Painlevé analysis leads to the explicit constraint on the variable coefficients for such a equation to pass the Painlevé test. An auto-B?cklund transformation is provided by use of the truncated Painlevé expansion and symbolic computation. Via the given auto-B?cklund transformation, three families of analytic solutions are obtained, including the solitonic and periodic solutions.     

On Integrability of a (2+1)-Dimensional Perturbed KdV Equation

Abstract

A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlevé; test for integrability well, and its 4×4 Lax pair with two spectral parameters is found. The results show that the Painlevé; classification of coupled KdV equations by A. Karasu should be revised.     

The Painlevé property and a partial differential equation with an essential singularity

It is shown that the equation u²_t = 2uu²_x - (1 + u²)u_xx possesses the Painlevé property for partial differential equations as defined by Weiss, Tabor and Carnevale, yet does not satisfy the necessary conditions of the Painlevé conjecture to be completely integrable since it is reducible, via a similarity reduction, to an ordinary differential equation which has a movable essential singularity. It is further shown that in a more general sense, the equation does not possess the Painlevé property for partial differential equations.     

The Anisotropic Heisenberg spin chain and the nonlinear Schrödinger equation

It is shown that the solutions of the continuous Anisotropic Heisenberg Spin Chain (AHSC) can be obtained from the linear integral equation which was proposed in a previous paper for the solutions of the Isotropic Heisenberg Spin Chain (IHSC) and the Nonlinear Schrödinger equation (NLS). An explicit expression is obtained for the Miura transformation which maps the solutions of the AHSC on solutions of the NLS. In the second part of the paper we investigate the similarity solutions of these partial differential equations which leads to ordinary differential equations of Painlevé type. As an application we discuss some new solutions of Painlevé IV.     

On multivortex solutions of the weierstrass representation

The connection between the complex sine-Gordon equation on the plane associated with a Weierstrass-type system and the possibility of constructing several classes of multivortex solutions is discussed in detail. It is shown that the amplitudes of these vortex solutions represented in polar coordinates satisfy the fifth Painlevé equation. We perform the analysis using the known relations for the Painlevé equations and construct explicit formulas in terms of the Umemura polynomials, which are τ functions for rational solutions to the third Painlevé equation. New classes of multivortex solutions to the Weierstrass system are obtained through the use of this proposed procedure.     

Painlevé Test and Higher Order Differential Equations

Abstract

Starting from the second Painlevé equation, we obtain Painlevé type equations of higher order by using the singular point analysis.     

Note on algebro-geometric solutions to triangular Schlesinger systems

We construct algebro-geometric upper triangular solutions of rank two Schlesinger systems. Using these solutions we derive two families of solutions to the sixth Painlevé equation with parameters (1/8, ?1/8, 1/8, 3=8) expressed in simple forms using periods of differentials on elliptic curves. Similarly for every integer n different from 0 and ?1 we obtain one family of solutions to the sixth Painlevé equation with parameters .     

Painlevé analysis and integrability aspects of some nonlinear partial differential equations

A brief review of the Painlevé singularity structure analysis of some autonomous and nonautonomous nonlinear partial differential equations is discussed. We point out how the Painlevé analysis of solutions of these equations systematically provides the integrability properties of the equation. The Lax pair, Bäcklund transformation and bilinear forms are constructed from the analysis.     

A simple-looking relative of the Novikov,Hirota-Satsuma and Sawada-Kotera equations

We study the simple-looking scalar integrable equation f_xxt 3( f_x f_t 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlevé series, a scaling reduction to a third order ODE and its Painlevé series, and the Hirota form (giving further multisoliton solutions).     

A Riemann–Hilbert approach to Painlevé IV

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ?¹ and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on ?¹ are introduced, inspired by the symmetric form for PIV, studied by M. Noumi and Y. Yamada.