# Journal of Nonlinear Mathematical Physics

Volume 24, Issue Supplement 1, December 2017

**Research Article**

## 1. Local and Nonlocal Symmetries in Mathematical Physics

Norbert Euler, Enrique G. Reyes

Pages: 1 - 2

**Research Article**

## 2. Solutions of the buoyancy-drag equation with a time-dependent acceleration

Serge E. Bouquet, Robert Conte, Vincent Kelsch, Fabien Louvet

Pages: 3 - 17

We perform the analytic study of the buoyancy-drag equation with a time-dependent acceleration γ(t) by two methods. We first determine its equivalence class under the point transformations of Roger Liouville, and thus for some values of γ(t) define a time-dependent Hamiltonian from which the buoyancy-drag...

**Research Article**

## 3. Symmetry Constraint of the Differential-difference KP Hierarchy and a Second Discretization of the ZS-AKNS System

Kui Chen, Xiao Deng, Da-jun Zhang

Pages: 18 - 35

In this paper we construct a squared-eigenfunction symmetry of the scalar differential-difference KP hierarchy. Through a constraint of the symmetry, Lax triad of the differential-difference KP hierarchy is reduced to a known discrete spectral problem and a semidiscrete AKNS hierarchy. The discrete spectral...

**Research Article**

## 4. 2D reductions of the equation *u*_{yy} = *u*_{tx} + *u*_{y}u_{xx} − *u*_{x}u_{xy} and their nonlocal symmetries

_{yy}

_{tx}

_{y}u

_{xx}

_{x}u

_{xy}

P. Holba, I.S. Krasil'shchik, O.I. Morozov, P. Vojčák

Pages: 36 - 47

We consider the 3D equation uyy = utx + uyuxx − uxuxy and its 2D symmetry reductions: (1) uyy = (uy + y) uxx − uxuxy − 2 (which is equivalent to the Gibbons-Tsarev equation) and (2) uyy = (uy + 2x)uxx + (y − ux)uxy − ux. Using the corresponding reductions of the known Lax pair for the 3D equation, we...

**Research Article**

## 5. Classical Lie symmetries and reductions for a generalized NLS equation in 2+1 dimensions

P. Albares, J. M. Conde, P. G. Estévez

Pages: 48 - 60

A non-isospectral linear problem for an integrable 2+1 generalization of the non linear Schrödinger equation, which includes dispersive terms of third and fourth order, is presented. The classical symmetries of the Lax pair and the related reductions are carefully studied. We obtain several reductions...

**Research Article**

## 6. On Modulated NLS-Ermakov Systems

Colin Rogers, Kwok Chow

Pages: 61 - 74

Spatial modulated coupled nonlinear Schrödinger systems with symmetry reduction to integrable Ermakov and Ermakov-Painlevé subsystems are investigated.

**Research Article**

## 7. Exact solutions and Riccati-type first integrals

J. Mendoza, C. Muriel

Pages: 75 - 89

The λ-symmetry approach is applied to a family of second-order ODEs whose algebra of Lie point symmetries is insufficient to integrate them. The general solution and two functionally independent first integrals of a subclass of the studied equations can be expressed in terms of a fundamental set of solutions...

**Research Article**

## 8. On Lie-point symmetries for Ito stochastic differential equations

G. Gaeta, C. Lunini

Pages: 90 - 102

In the deterministic realm, both differential equations and symmetry generators are geometrical objects, and behave properly under changes of coordinates; actually this property is essential to make symmetry analysis independent of the choice of coordinates and applicable. When trying to extend symmetry...

**Research Article**

## 9. The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra

Anahita Eslami Rad, Jean-Pierre Magnot, Enrique G. Reyes

Pages: 103 - 120

Mulase solved the Cauchy problem of the Kadomtsev-Petviashvili (KP) hierarchy in an algebraic category in “Solvability of the super KP equation and a generalization of the Birkhoff decomposition” (Inventiones Mathematicae, 1988), making use of a delicate factorization of an infinite-dimensional group...

**Research Article**

## 10. Variational symmetries and pluri-Lagrangian systems in classical mechanics

Matteo Petrera, Yuri B. Suris

Pages: 121 - 145

We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, for any Lagrangian system with m commuting variational...

**Research Article**

## 11. The nonlinear pendulum always oscillates

M. C. Nucci

Pages: 146 - 156

It is shown that the nonlinear pendulum equation can be transformed into a linear harmonic oscillator in the phase space thanks to Kerner’s method [12]. Moreover, as a mathematical divertissement, the second-order differential equation determining the phase-space trajectories of the nonlinear pendulum...

**Research Article**

## 12. The heptagon-wheel cocycle in the Kontsevich graph complex

Ricardo Buring, Arthemy V. Kiselev, Nina J. Rutten

Pages: 157 - 173

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on n vertices and 2n − 2 edges, induce – under the orientation mapping – infinitesimal symmetries of classical Poisson...