We have finally obtained for each of the 6 Painlevés an expression of z, w, w′ that behaves as 1/(z − Z0) + O(1) at each kind of movable singular point. This expression is polynomial in w′ (at most quadratic), and rational in w and z. After it is integrated and exponentiated it yields a function that has a simple zero at each of the singular points.

The derivation of gene-transport equations is re-examined. Fisher's assumptions for a sexually reproducing species lead to a Huxley reaction-diffusion equation, with cubic logistic source term for the gene frequency of a mutant advantageous recessive gene. Fisher's equation more accurately represents the spread of an advantaged mutant strain within an asexual species. When the total population density is not uniform, these reaction-diffusion equations take on an additional non-uniform convection term. Cubic source terms of the Huxley or Fitzhugh-Nagumo type allow special nonclassical symmetries. A new exact solution, not of the travelling wave type, and with zero gradient boundary condition, is constructed.

We analyse the various integrability criteria which have been proposed for discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of discrete Painlevé equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the discrete Painlevé equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of discrete integrability based on singularity confinement and low-growth considerations is also proposed.

The art of asymptotology is a powerful tool in applied mathematics and theoretical physics, but can lead to erroneous conclusions if misapplied. A seemingly paradoxical case is presented in which a local analysis of an exactly solvable problem appears to find solutions to an eigenvalue problem over a continuous range of the eigenvalue, whereas the spectrum is known to be discrete. The resolution of the paradox involves the Stokes phenomenon. The example illustrates two of Kruskal's Principles of Asymptotology.

The probability that an interval I is free of eigenvalues in a matrix ensemble with unitary symmetry is given by a Fredholm determinant. When the weight function in the matrix ensemble is a classical weight function, and the interval I includes an endpoint of the support, Tracy and Widom have given a formalism which gives coupled differential equations for the required probability and some auxiliary quantities. We summarize and extend earlier work by expressing the probability and some of the auxiliary quantities in terms of Painlevé transcendents.

The Maxwell-Dirac equations model an electron in an electromagnetic field. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equation, resulting in a nonlinear system of partial differential equations (PDE's). Well-behaved solutions, within reasonable Sobolev spaces, have been shown to exist globally as recently as 1997 [12]. Exact solutions have not been found—except in some simple cases.

We have shown analytically in [6, 18] that any spherical solution surrounds a Coulomb field and any cylindrical solution surrounds a central charged wire; and in [3] and [19] that in any stationary case, the surrounding electron field must be equal and opposite to the central (external) field. Here we extend the numerical solutions in [6] to a family of orbits all of which are well-behaved numerical solutions satisfying the analytic results in [6] and [11]. These solutions die off exponentially with increasing distance from the central axis of symmetry. The results in [18] can be extended in the same way. A third case is included, with dependence on z only yielding a related fourth-order ordinary differential equation (ODE) [3].

Daniel et al. [6] analysed the singularity structure of the continuum limit of the one-dimensional anisotropic Heisenberg spin chain in a transverse field and determined the conditions under which the system is nonintegrable and exhibits chaos. We investigate the governing differential equations for symmetries and find the associated first integrals. Our results complement the results of Daniel et al.

The classical dynamics of non-relativistic particles are described by the Schrödinger wave equation, perturbed by quantum potential nonlinearity. Quantization of this dispersionless equation, implemented by deformation of the potential strength, recovers the standard Schrödinger equation. In addition, the classically forbidden region corresponds to the Planck constant analytically continued to pure imaginary values. We apply the same procedure to the NLS and DNLS equations, constructing first the corresponding dispersionless limits and then adding quantum deformations. All these deformations admit the Lax representation as well as the Hirota bilinear form. In the classically forbidden region we find soliton resonances and black hole phenomena. For deformed DNLS the chiral solitons with single event horizon and resonance dynamics are constructed.

We give an intrinsic construction of a coupled nonlinear system consisting of two first-order partial differential equations in two dependent and two independent variables which is determined by a hyperbolic structure on the complex special linear group regarded as a real Lie group G. Despite the fact that the system is not Darboux semi-integrable at first order, the construction of a family of solutions depending.upon two arbitrary functions, each of one variable, is reduced to a system of ordinary differential equations on the 1-jets. The ordinary differential equations in question are of Lie type and associated with G.

Solitary wave interaction is examined using an extended Benjamin-Bona-Mahony (eBBM) equation. This equation includes higher-order nonlinear and dispersive effects and is is asymptotically equivalent to the extended Korteweg-de Vries (eKdV) equation. The eBBM formulation is preferable to the eKdV equation for the numerical modelling of solitary wave collisions, due to the stability of its finite-difference scheme. In particular, it allows the interaction of steep waves to be modelled, which due to numerical instability, is not possible using the eKdV equation.

Numerical simulations of a number of solitary wave collisions of varying nonlinearity are performed for two special cases corresponding to surface water waves. The mass and energy of the dispersive wavetrain generated by the inelastic collision is tabulated and used to show that the change in solitary wave amplitude after interaction is of O(α4), verifying previously obtained theoretical predictions.

We study the propagation of electromagnetic waves (EMWs) in both isotropic and anisotropic ferromagnetic material media. As the EMW propagates through linear charge-free isotropic and anisotropic ferromagnetic media, it is found that the magnetic field and the magnetic induction components of the EMW and the magnetization excitations of the medium are in the form of solitons. However, the electromagnetic soliton gets damped and decelerates in the case of a charged medium. In the case of a charge-free nonlinear ferromagnetic medium we obtain results similar to those for the linear case.

We discuss Ablowitz-Segur's connection problem for the second Painlevé equation from the viewpoint of WKB analysis of Painlevé transcendents with a large parameter. The formula they first discovered is rederived from a suitable combination of connection formulas for the first Painlevé equation.

A hierarchy of bilinear Lotka-Volterra equations with a unified structure is proposed. The bilinear Bäcklund transformation for this hierarchy and the corresponding canonical Lax pair are obtained. Furthermore, the nonlinear superposition formula is proved rigorously.

In contrast to mono-constrained flows with N degrees of freedom, binary constrained flows of soliton equations, admitting 2 × 2 Lax matrices, have 2N degrees of freedom. Currently existing methods only enable Lax matrices to yield the first N pairs of canonical separated variables. An approach for constructing the second N pairs of canonical separated variables with N additional separated equations is introduced. The Jacobi inversion problems for binary constrained flows are then established. Finally, the separability of binary constrained flows together with the factorization of soliton equations by the spatial and temporal binary constrained flows leads to the Jacobi inversion problems for soliton equations.

It is shown that an integrable class of helicoidal surfaces in Euclidean space E3 is governed by the Painlevé V equation with four arbitrary parameters. A connection with sphere congruences is exploited to construct in a purely geometric manner an associated Bäcklund transformation.

Using the Bäcklund transformations for the solutions of fourth-order differential equations of the P2 and K2 hierarchies, corresponding discrete equations are found.

Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schrödinger operator and hence construct a Darboux transformation, an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discuss the specialisation to the case of a graph with trivial compact part, that is, with diagonal matrix potential.

In this paper we consider a projective connection as defined by the nth-order Adler-Gelfand-Dikii (AGD) operator on the circle. It is well-known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar nth-order differential (AGD) operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for n ≤ 4) under the action of Vect(S1). The solutions of the AGD operator define an immersion R → RPn−1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ(n), for n ≤ 4.