On tronquée solutions of the first Painlevé hierarchy

It is well known that, due to Boutroux, the first Painlevé equation admits solutions characterized by divergent asymptotic expansions near infinity in specified sectors of the complex plane. In this paper, we show that such solutions exist for higher order analogues of the first Painlevé equation (the first Painlevé hierarchy) as well.     

Value distribution of meromorphic solutions of certain difference Painlevé equations

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painlevé I and II equations are estimated. And the forms of rational solutions of the difference Painlevé II equation and the autonomous difference Painlevé I equation are also given. It is also proved that the non-autonomous difference Painlevé I equation has no rational solution.     

Exact solutions of equations for the Burgers hierarchy

Some classes of the rational, periodic and solitary wave solutions for the Burgers hierarchy are presented. The solutions for this hierarchy are obtained by using the generalized Cole–Hopf transformation.     

On the Gevrey order of formal solutions of nonlinear difference equations

We prove a Maillet type theorem for formal solutions of nonlinear difference systems, relating the Gevrey order of the formal solutions to the lowest level of an associated, linear difference operator.     

Rational formal solutions of differential-difference equations

In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansätz. With the help of symbolic computation Maple, we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.     

Order one equations with the Painlevé property

Differential equations with the Painleve property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order one equations appeared before, the connection with the classical theory did not. Using techniques from algebraic geometry we provide the link between the classical and the modern treatment, and with the help of differential Galois theory a new classification is derived, both for characteristic 0 and p.     

Lie point symmetries, partial Noether operators and first integrals of the Painlevé-Gambier equations

We utilize the Lie-Tressé linearization method to obtain linearizing point transformations of certain autonomous nonlinear second-order ordinary differential equations contained in the Painlevé-Gambier classification. These point transformations are constructed using the Lie point symmetry generators admitted by the underlying Painlevé-Gambier equations. It is also shown that those Painlevé-Gambier equations which have a few Lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature. Moreover, by making use of the partial Lagrangian approach we obtain time dependent and time independent first integrals for these Painlevé-Gambier equations which have not been reported in the earlier literature. A comparison of the results obtained in this paper is made with the ones obtained using the generalized Sundman linearization method.     

On linear equations with general polynomial solutions

We provide necessary and sufficient conditions for which an nth-order linear differential equation has a general polynomial solution. We also give necessary conditions that can directly be ascertained from the coefficient functions of the equation.     

ON GENERATING EQUATIONS FOR THE KAUP-NEWELL HIERARCHY

It is shown that the Kanp-Newell hierarchy can be derived from the so-called gen- erating equations which are Lax integrable.Positive and negative flows in the hierarchy are derived simultaneously.The generating equations and mutual commutativity of these flows en- able us to construct new Lax integrable equations.     

On L_w^2 -quasi-derivatives for solutions of perturbed general quasi-differential equations

This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equation of nth order with complex coefficients M[y] – wy = wf(t, y ^[0],... , y ^[n–1]), t [a, b) provided that all rth quasi-derivatives of solutions of M[y] – wy = 0 and all solutions of its normal adjoint are in and under suitable conditions on the function f.     

Universal characters, integrable chains and the Painlevé equations

The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see (Adv. Math. 74 (1989) 57). We prove that the universal character solves the Darboux chain. The N-periodic closing of the chain is equivalent to the Painlevé equation of type . Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters.     

Autonomous self-similar ordinary differential equations and the Painlevé connection

We demonstrate an intimate connection between nonlinear higher-order ordinary differential equations possessing the two symmetries of autonomy and self-similarity and the leading-order behaviour and resonances determined in the application of the Painlevé Test. Similar behaviour is seen for systems of first-order differential equations. Several examples illustrate the theory. In an integrable case of the ABC system the singularity analysis reveals a positive and a negative resonance and the method of leading-order behaviour leads naturally to a Laurent expansion containing both.     

On the comparison principle for unbounded solutions of elliptic equations with first order terms

We prove a comparison principle for unbounded weak sub/super solutions of the equation

$\lambda u?\text{div}(A(x)Du)=H(x,Du)\text{ in }\mathrm{\Omega }$

where $A(x)$ is a bounded coercive matrix with measurable ingredients, $\lambda \ge 0$ and $\xi ?H(x,\xi )$ has a super linear growth and is convex at infinity. We improve earlier results where the convexity of $H(x,?)$ was required to hold globally.