A note on solutions of an equation modelling arterial deformation

The derivation of exact solutions for a partial differential equation modelling arterial deformation in large arteries is considered. Amongst other results, we show that, for any values of the parameters appearing in the equation, solutions in terms of the first Painlevé transcendent can be obtained. This is in spite of the non-integrability of the equation. We also establish a connection, via an approximation of the equation under study by the Korteweg-de Vries equation, with the second Painlevé equation. Our results thus serve to further demonstrate the wide applicability and importance of the Painlevé equations.     

A note on the Painlevé analysis of a (2 + 1) dimensional Camassa–Holm equation

We investigate the Painlevé analysis for a (2 + 1) dimensional Camassa–Holm equation. Our results show that it admits only weak Painlevé expansions. This then confirms the limitations of the Painlevé test as a test for complete integrability when applied to non-semilinear partial differential equations.     

On the discrete form of the Fokas–Ablowitz equation

The Fokas–Ablowitz equation, quadratic in the second derivative, is obtained from Painlevé VI by a Miura transformation. We present here two different discretisations in terms of difference- and multiplicative, q-, equations.     

Special functions as solutions to discrete Painlevé equations

We present results on special solutions of discrete Painlevé equations. These solutions exist only when one constraint among the parameters of the equation is satisfied and are obtained through the solutions of linear second-order (discrete) equations. These linear equations define the discrete analogues of special functions.     

The Painlevé integrability of two parameterized nonlinear evolution equations using symbolic computation

An algorithm is presented to prove the Painlevé integrability of parameterized nonlinear evolution equations such that one can filter out Painlevé integrable models from nonlinear equations with general forms. Then two well known nonlinear models with physical interests illustrate the effectiveness of this algorithm. Some new results are reported for the first time.     

Tanh-travelling wave solutions, truncated Painlevé expansion and reduction of Bullough–Dodd equation to a quadrature in magnetohydrodynamic equilibrium

The equations of magnetohydrodynamic (MHD) equilibria for a plasma in gravitational field are investigated. For equilibria with one ignorable spatial coordinate, the MHD equations are reduced to a single nonlinear elliptic equation for the magnetic potential , known as the Grad–Shafranov equation. Specifying the arbitrary functions in this equation, the Bullough–Dodd equation can be obtained. The truncated Painlevé expansion and reduction of the partial differential equation to a quadrature problem (RQ method) are described and applied to obtain the travelling wave solutions of the Bullough–Dodd equation for the case of isothermal magnetostatic atmosphere, in which the current density J is proportional to the exponentially of the magnetic flux and moreover falls off exponentially with distance vertical to the base, with an “e-folding” distance equal to the gravitational scale height.     

New Darboux transformation for Hirota–Satsuma coupled KdV system

A new Darboux transformation (DT) is presented for the Hirota–Satsuma coupled KdV system. It is shown that this DT can be constructed by means of two methods: Painlevé analysis and reduction of a binary DT. By iteration of the DT, the Grammian type solutions are found for the coupled KdV system.     

Exact solutions of the one-dimensional generalized modified complex Ginzburg–Landau equation

The one-dimensional (1D) generalized modified complex Ginzburg–Landau (MCGL) equation for the traveling wave systems is analytically studied. Exact solutions of this equation are obtained using a method which combines the Painlevé test for integrability in the formalism of Weiss–Tabor–Carnevale and Hirota technique of bilinearization. We show that pulses, fronts, periodic unbounded waves, sources, sinks and solution as collision between two fronts are the important coherent structures that organize much of the dynamical properties of these traveling wave systems. The degeneracies of the 1D generalized MCGL equation are examined as well as several of their solutions. These degeneracies include two important equations: the 1D generalized modified Schrödinger equation and the 1D generalized real modified Ginzburg–Landau equation. We obtain that the one parameter family of traveling localized source solutions called “Nozaki–Bekki holes” become a subfamily of the dark soliton solutions in the 1D generalized modified Schrödinger limit.     

Integrability and singularity structure of coupled nonlinear Schrödinger equations

Multimode propagation of electromagnetic waves in optical fibre is often described by coupled nonlinear Schrödinger (NLS) equations. To understand the integrability properties of such coupled NLS systems, we extend the Painlevé singularity structure analysis of two coupled systems to three coupled systems and identify four integrable sets of parameters. We bilinearize these cases to obtain soliton solutions. The results are extended to N-coupled systems, completing the earlier analysis of Sahadevan, Tamizhmani and Lakshmanan.     

Solitary wave solution for inhomogeneous nonlinear Schrödinger system with loss/gain

This paper deals with the propagation of solitons in real fibres, governed by the system of inhomogeneous nonlinear Schrödinger (INLS) equations. The Painlevé singularity structure analysis is utilized to check for the integrability of the system and from the analysis, the system is found to admit soliton-type lossless wave propagation. The system is transformed to its homogeneous counterpart using a suitable variable transformation and the soliton solutions are obtained through Bäcklund transformation after constructing the explicit Lax pair for the system. The one-soliton solutions are plotted for different choices of inhomogeneity parameters and the evolutionary characteristics of the solutions are analyzed.     

On the Painlevé integrability, periodic wave solutions and soliton solutions of generalized coupled higher-order nonlinear Schrödinger equations

It is proven that generalized coupled higher-order nonlinear Schrödinger equations possess the Painlevé property for two particular choices of parameters, using the Weiss–Tabor–Carnevale method and Kruskal’s simplification. Abundant families of periodic wave solutions are obtained by using the Jacobi elliptic function expansion method with the assistance of symbolic manipulation system, Maple. It is also shown that these solutions exactly degenerate to bright soliton, dark soliton and mixed dark and bright soliton solutions with physical interests.     

Bäcklund transformations for higher order Painlevé equations

We present a new generalized algorithm which allows the construction of Bäcklund transformations (BTs) for higher order ordinary differential equations (ODEs). This algorithm is based on the idea of seeking transformations that preserve the Painlevé property, and is applied here to ODEs of various orders in order to recover, amongst others, their auto-BTs. Of the ODEs considered here, one is seen to be of particular interest because it allows us to show that auto-BTs can be obtained in various ways, i.e. not only by using the severest of the possible restrictions of our algorithm.     

Right-hand solutions of the differential equations of dynamics for mechanical systems with sliding friction

As a further development of Painlevé's theory [1], the existence, continuability and uniqueness of righ-hand solutions of the differential equations of dynamics, and, under certain additional conditions, of the equations of motion of holonomic mechanical systems with sliding friction [2] are considered. In classical mechanics, acceleration is essentially defined as the right-hand derivative of velocity (see [3, 4]). Hence the most meaningful definition of the “solution of a differential equation” in problems of the dynamics of mechanical systems with sliding friction is that using the concept of right derivative [5].     

Heat transfer and turbulent diffusion in one-dimensional hydrodynamics without pressure

A one-dimensional transient non-linear problem of continuum mechanics is considered, the possibility of an accurate analytic solution of which is later based on a general local analysis of singular solutions known as the Painlevé test. For one-dimensional non-linear hydrodynamic models without pressure, with the transfer of a passive impunity, which generalizes the well-known Burgers' model, it is shown that it is possible to reduce the problem to linear problems when the kinetic coefficients (viscosity and thermal conductivity) are equal. Using examples of their accurate solutions, the high sensitivity of the structure of shock waves with impurity fronts to the satisfaction of the law of conservation of impurity in the models is demonstrated. When it is satisfied, each steady propagating shock wave with a viscous structure of the velocity field is accompanied by an impurity soliton. When several such shock waves merge (the accurately solved problem), concentration of the impurity in one overall soliton occurs. It is shown that, when the action of time-dependent Gaussian random forces is taken into account, an additional diffusive spreading of the perturbations, with a time-dependent diffusion coefficient, is superimposed on the linearized viscous behaviour of the main models.     

Modified equation for adaptive monotone difference schemes and its convergent analysis

A modified parabolic equation for adaptive monotone difference schemes based on equal-arclength mesh, applied to the linear convection equation, is derived and its convergence analysis shows that solutions of the modified equation approach a discontinuous (piecewise smooth) solution of the linear convection equation at order one rate in the -norm. It is well known that solutions of the monotone schemes with uniform meshes and their modified equation approach the same discontinuous solution at a half-order rate in the -norm. Therefore, the convergence analysis for the modified equation provided in this work demonstrates theoretically that the monotone schemes with adaptive grids can improve the solution accuracy. Numerical experiments also confirm the theoretical conclusions.

     

The Toda molecule equation and the -algorithm

One of the well-known convergence acceleration methods, the -algorithm is investigated from the viewpoint of the Toda molecule equation. It is shown that the error caused by the algorithm is evaluated by means of solutions for the equation. The acceleration algorithm based on the discrete Toda molecule equation is also presented.

     

Degenerate two-phase incompressible flow III. Sharp error estimates

Summary. This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we consider a finite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin finite element method. A fully discrete approximation is analyzed. Sharp error estimates in energy norms are obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the degenerate equation. Also, the analysis does not impose any restriction on the nature of degeneracy. Finally, it respects the minimal regularity on the solution of the differential system. Received March 9, 1998 / Revised version received July 17, 2000 / Published online May 30, 2001     

On the slowness of phase boundary motion in one space dimension

We study the limiting behavior of the solution of with a Neumann boundary condition or an appropriate Dirichlet condition. The analysis is based on “energy methods”. We assume that the initial data has a “transition layer structure”, i.e., u^? ≈ ±+M 1 except near finitely many transition points. We show that, in the limit as ? → 0, the solution maintains its transition layer structure, and the transition points move slower than any power of ?.     

Strange nonchaotic attractors in the externally modulated Rayleigh–Bénard system

The amplitude equation associated with an externally modulated Rayleigh–Bénard system of binary mixtures near the codimension-two point is considered. Strange nonchaotic dynamics and chaotic behaviour are investigated numerically. The creation of strange nonchaotic attractors as well as the onset of chaos are studied through an analysis of Poincaré surfaces, a construction of the bifurcation diagram and a new method for computing Lyapunov exponents that exploits the underlying symplectic structure of Hamiltonian dynamics [Phys. Rev. Lett. 74 (1995) 70].     

Implicit linear time‐dependent differential‐difference equations and applications

The paper deals with the differential‐difference equation in a Banach space. The operator coefficient of the delay‐free derivative is allowed to be degenerate. Existence and uniqueness theorems are proved under the main assumption that for every the point is a polar singularity of the resolvent . The results are applied to evolution problems of microwave circuits. Copyright © 2000 John Wiley & Sons, Ltd.