Nonlinear Sciences

We first construct a (2+1) -dimensional negative AKNS hierarchy and then we give all possible local and (discrete) nonlocal reductions of these equations. We find Hirota bilinear forms of the negative AKNS hierarchy and give one- and two-soliton solutions. By using the soliton solutions of the negative AKNS hierarchy we find one-soliton solutions of the local and nonlocal reduced equations.

In this work we continue to study negative AKNS( N ) that is AKNS( −N ) system for N=3,4 . We obtain all possible local and nonlocal reductions of these equations. We construct the Hirota bilinear forms of these equations and find one-soliton solutions. From the reduction formulas we obtain also one-soliton solutions of all reduced equations.

We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in the AKNS hierarchy of integrable equations. Our analysis is based on the $\dbar$-steepest descent method. We consider RHPs arising from the inverse scattering transform of the AKNS hierarchy with $H^{1,1}(\R)$ initial data. The analysis will be divided into three regions: fast decay region, oscillating region and self-similarity region (the Painlevé region). The resulting formulas can be directly applied to study the long-time asymptotic of the solutions of integrable equations such as NLS, mKdV and their higher-order generalizations.

A new approach to the Euler-Bernoulli beam based on an inhomogeneous matrix string problem is presented. Three ramifications of the approach are developed: (1) motivated by an analogy with the Camassa-Holm equation a class of isospectral deformations of the beam problem is formulated; (2) a reformulation of the matrix string problem in terms of a certain compact operator is used to obtain basic spectral properties of the inhomogeneous matrix string problem with Dirichlet boundary conditions; (3) the inverse problem is solved for the special case of a discrete Euler-Bernoulli beam. The solution involves a non-commutative generalization of Stieltjes' continued fractions, leading to the inverse formulas expressed in terms of ratios of Hankel-like determinants.

We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalised sextic Freud weight ω(x;t,λ)=|x | 2λ+1 exp(− x 6 +t x 2 ),x∈R, with parameters λ>−1 and t∈R . We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalised hypergeometric functions 1 F 2 ( a 1 ; b 1 , b 2 ;z) . We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalised quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalised hypergeometric functions.

We consider the generalized Painlevé--Ince equation, x ¨ +αx x ˙ +β x 3 =0 and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as β= α 2 /9 the given differential equation is maximally symmetric and well-known that it pass the Painlevé test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlevé--Ince equation fails at the Painlevé test, except if we apply the singularity analysis for the new second-order differential equation which follows from the change of variable x=1/y. We conclude that the Painlevé--Ince equation is integrable is terms of Lie symmetries and of the Painlevé test.

We define a nonlinear q -difference system mathcal P N,( M − , M + ) as monodromy preserving deformations of a certain linear equation. We study its relation to a series mathcal F N,M defined as a certain generalization of q -hypergeometric functions.

A method is proposed to systematically generate solutions of the two-dimensional Toda lattice equation in terms of previously known solutions ϕ(x,y) of the two-dimensional Laplace's equation. The two-dimensional solution of Nakamura's [J. Phys. Soc. Jpn. \textbf{52}, 380 (1983)] is shown to correspond to one particular choice of ϕ(x,y) .

We demonstrate that a Poisson structure can always be associated to a general nonautonomous 3D vector field of ODEs by means of a diffeomorphism that preserves both the orientation and the volume of phase-space. The only prerequisite is the existence of one constant of motion.

A method for the construction of classes of examples of multi-dimensional, multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This is based on an extension of the original construction of Gelfand and Dorfman which gave examples of Novikov algebras in terms of structures defined from commutative, associative algebras. Given such an algebra, the construction involves only linear algebra.