Journal of Nonlinear Mathematical Physics

We construct the recursion operators for the non-commutative Burgers equations using their Lax operators. We investigate the existence of any integrable mixed version of left- and right-handed Burgers equations on higher symmetry grounds.

We show that the geodesic flow on the infinite-dimensional group of diffeomorphisms of the circle, endowed with a fractional Sobolev metric at the identity, is described by the generalized Constantin–Lax–Majda equation with parameter a=−12.

The objective of this paper is twofold: (a) to find a natural example of a perturbed Lagrangian that has different partial Noether operators with symmetries different from those of the underlying Lagrangian. First we regard the Schwarzschild spacetime as a perturbation of the Minkowski spacetime and...

It is known that the complex spin group Spin(n, ℂ) is the universal covering group of complex orthogonal group SO(n, ℂ). In this work we construct a new kind of spinors on some classes of Kahler–Norden manifolds. The structure group of such a Kahler–Norden manifold is SO(n, ℂ) and has a lifting to Spin(n,...

By using the self-dual Yang–Mills (SDYM) equation as an example, we study a method for relating symmetries and recursion operators of two partial differential equations connected to each other by a non-auto-Bäcklund transformation. We prove the Lie-algebra isomorphism between the symmetries of the SDYM...

The paper discusses fractional integrals and derivatives appearing in the so-called (q, h)-calculus which is reduced for h = 0 to quantum calculus and for q = h = 1 to difference calculus. We introduce delta as well as nabla version of these notions and present their basic properties. Furthermore, we...

We use an algorithm based on the notion of transvectants from classical invariant theory in determining the form of Stanley decomposition of the ring of invariants for the coupled Takens–Bogdanov systems when the Stanley decompositions of the Jordan blocks of the linear part are known at each stage....

We show that an ultradiscrete analogue of the third Painlevé equation admits discrete Riccati type solutions. We derive these solutions by considering a framework in which the ultradiscretization process arises as a restriction of a non-archimedean valuation over a field. Using this framework we may...

We exhibit the solution of the initial-value problem for the system of 2N + 2 oscillators characterized by the Hamiltonian H(p^0,pˇ0,p^_,pˇ_,q^0,qˇ0,q^_,qˇ_)=12[p^02−pˇ02+Ω2(q^02−qˇ02)]        +q^0−Ωqˇ02b∑n=1N[p^n2−pˇn2+ωn2(q^n2−qˇn2)]+p^0−Ωq^0b∑n=1N[−p^npˇn+ωn2q^nqˇn] where N is an arbitrary positive...

A simple technique is identified to manufacture solvable nonlinear dynamical systems, and in particular three classes whose generic solutions are, respectively, isochronous, multi-periodic, or asymptotically isochronous.

In the present paper we propose a new proof of the Grosset–Veselov formula connecting one-soliton solution of the Korteweg–de Vries equation to the Bernoulli numbers. The approach involves Eulerian numbers and Riccati's differential equation.