Journal of Nonlinear Mathematical Physics

Volume 26, Issue 4, July 2019
Research Article

1. Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

Francesco Calogero, Farrin Payandeh
Pages: 509 - 519
In this paper we identify certain peculiar systems of 2 discrete-time evolution equations, x˜n=F(n)(x1,x2),   n=1,2, which are algebraically solvable. Here ℓ is the “discrete-time” independent variable taking integer values (ℓ = 0, 1, 2,...), xn ≡ xn(ℓ) are 2 dependent variables, and x˜n≡xn(ℓ+1) are...
Research Article

2. Constructing discrete Painlevé equations: from E8(1) to A1(1) and back

Shulin Lyu, James Griffin, Yang Chen
Pages: 520 - 535
The ‘restoration method’ is a novel method we recently introduced for systematically deriving discrete Painlevé equations. In this method we start from a given Painlevé equation, typically with E8(1) symmetry, obtain its autonomous limit and construct all possible QRT-canonical forms of mappings that...
Research Article

3. On Slant Magnetic Curves in S-manifolds

Güvenç Şaban, Cihan Özgür
Pages: 536 - 554
We consider slant normal magnetic curves in (2n + 1)-dimensional S-manifolds. We prove that γ is a slant normal magnetic curve in an S-manifold (M2m+s, φ, ξα, ηα, g) if and only if it belongs to a list of slant φ-curves satisfying some special curvature equations. This list consists of some specific...
Research Article

4. A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations

Alexander G. Rasin, Jeremy Schiff
Pages: 555 - 568
We study the simple-looking scalar integrable equation fxxt − 3(fx ft − 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities),...
Research Article

5. On the global dynamics of the Newell–Whitehead system

Claudia Valls
Pages: 569 - 578
In this paper by using the Poincaré compactification in ℝ3 we make a global analysis of the model x′ = z, y′ = b(x−dy), z′ = x(x2 −1)+y+cz with b ∈ ℝ and c, d ∈ ℝ+, here known as the three-dimensional Newell–Whitehead system. We give the complete description of its dynamics on the sphere at infinity....
Research Article

6. An exact solution for geophysical internal waves with underlying current in modified equatorial β-plane approximation*

Dong Su, Hongjun Gao
Pages: 579 - 603
In this paper, a modification of the standard geophysical equatorial β-plane model equations, incorporating a gravitational-correction term in the tangent plane approximation, is derived. We present an exact solution to meet the modified governing equations, whose form is explicit in the Lagrangian framework...
Research Article

7. Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations

M.E. Fels, E. Yaşar
Pages: 604 - 649
For a scalar evolution equation ut = K(t, x, u, ux, ..., u2m+1) with m ≥ 1, the cohomology space H1,2(ℛ∞) is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for ut = K for which the equation is Hamiltonian is also shown...
Research Article

8. Systems of Hamilton-Jacobi equations

Julio Cambronero, Javier Pérez Álvarez
Pages: 650 - 658
In this article we develop a generalization of the Hamilton-Jacobi theory, by considering in the cotangent bundle an involutive system of dynamical equations.