Journal of Nonlinear Mathematical Physics

Volume 23, Issue 4, September 2016
Research Article

1. A convenient expression of the time-derivative zn(k)(t) , of arbitrary order k, of the zero zn(t) of a time-dependent polynomial pN(z;t) of arbitrary degree N in z, and solvable dynamical systems

Mario Bruschi, Francesco Calogero
Pages: 474 - 485
Let pN (z; t) be a (monic) time-dependent polynomial of arbitrary degree N in z, and let zn ≡ zn (t) be its N zeros: pN (z;t)=∏n=1N[z-zn(t)] . In this paper we report a convenient expression of the k-th time-derivative zn(k)(t) of the zero zn (t). This formula plays a key role in the...
Research Article

2. New solvable dynamical systems

Francesco Calogero
Pages: 486 - 493
New solvable dynamical systems are identified and the properties of their solutions are tersely discussed.
Research Article

3. On integrability of the Szekeres system. I

Anna Gierzkiewicz, Zdzisław A. Golda
Pages: 494 - 506
The Szekeres system is a four-dimensional system of first-order ordinary differential equations with nonlinear but polynomial (quadratic) right-hand side. It can be derived as a special case of the Einstein equations, related to inhomogeneous and nonsymmetrical evolving spacetime. The paper shows how...
Research Article

4. Algebraic entropy, symmetries and linearization of quad equations consistent on the cube

G. Gubbiotti, C. Scimiterna, D. Levi
Pages: 507 - 543
We discuss the non–autonomous nonlinear partial difference equations belonging to Boll classification of quad graph equations consistent around the cube. We show how starting from the compatible equations on a cell we can construct the lattice equations, its Bäcklund transformations and Lax pairs. By...
Research Article

5. A discrete negative AKNS equation: generalized Cauchy matrix approach

Song-lin Zhao
Pages: 544 - 562
Generalized Cauchy matrix approach is used to investigate a discrete negative Ablowitz–Kaup–Newell–Segur (AKNS) equation. Several kinds of solutions more than multi-soliton solutions to this equation are derived by solving determining equation set. Furthermore, applying an appropriate continuum limit...
Research Article

6. Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons

Xiangke Chang, Jacek Szmigielski
Pages: 563 - 572
In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified...
Research Article

7. Generalized negative flows in hierarchies of integrable evolution equations

Stephen C. Anco, Shahid Mohammad, Thomas Wolf, Chunrong Zhu
Pages: 573 - 606
A one-parameter generalization of the hierarchy of negative flows is introduced for integrable hierarchies of evolution equations, which yields a wider (new) class of non-evolutionary integrable nonlinear wave equations. As main results, several integrability properties of these generalized negative...
Research Article

8. SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

D. Ojeda-Guillén, M. Salazar-Ramírez, R. D. Mota, V. D. Granados
Pages: 607 - 619
We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized...
Research Article

9. Discretization of Liouville type nonautonomous equations preserving integrals

Ismagil Habibullin, Natalya Zheltukhina
Pages: 620 - 642
The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known...