Journal of Nonlinear Mathematical Physics

Volume 25, Issue 1, February 2018
Research Article

1. A Multidimensional Superposition Principle: Classical Solitons IV

Alexander A. Alexeyev
Pages: 1 - 33
This article continues the series of the works of 1998–2007 years devoted to the Multidimensional Superposition Principle, the concept easily explaining both classical soliton and more complex wave interactions in nonlinear PDEs and allowing one, in particular, to construct the general Superposition...
Research Article

2. On Decomposition of the ABS Lattice Equations and Related Bäcklund Transformations

Danda Zhang, Da-jun Zhang
Pages: 34 - 53
The Adler-Bobenko-Suris (ABS) list contains scalar quadrilateral equations which are consistent around the cube, and have D4 symmetry and tetrahedron property. Each equation in the ABS list admits a beautiful decomposition. We revisit these decomposition formulas and by means of them we construct Bäcklund...
Research Article

3. Finite genus solutions for Geng hierarchy

Zhu Li
Pages: 54 - 65
The Geng hierarchy is derived with the aid of Lenard recursion sequences. Based on the Lax matrix, a hyperelliptic curve 𝒦n + 1 of arithmetic genus n+1 is introduced, from which meromorphic function ϕ is defined. The finite genus solutions for Geng hierarchy are achieved according to asymptotic properties...
Research Article

4. The gauge transformation of the modified KP hierarchy

Jipeng Cheng
Pages: 66 - 85
In this paper, we firstly investigate the successive applications of three elementary gauge transformation operators Ti with i = 1,2,3 for the mKP hierarchy in Kupershmidt-Kiso version, and find that the gauge transformation operators Ti can not commute with each other. Then two types of gauge transformation...
Research Article

5. Rosenhain-Thomae formulae for higher genera hyperelliptic curves

Keno Eilers
Pages: 86 - 105
Rosenhain's famous formula expresses the periods of first kind integrals of genus two hyperelliptic curves in terms of θ-constants. In this paper we generalize the Rosenhain formula to higher genera hyperelliptic curves by means of the second Thomae formula for derivative non-singular θ-constants.
Research Article

6. Meromorphic and formal first integrals for the Lorenz system

Kaiyin Huang, Shaoyun Shi, Wenlei Li
Pages: 106 - 121
The Lorenz system x˙=σ(y−x), y˙=rx−y−xz, z˙=−βz+xy, is completely integrable with two functional independent first integrals when σ = 0 and β, r arbitrary. In this paper, we study the integrability of the Lorenz system when σ, β, r take the remaining values. For the case of σβ ≠ 0, we consider the...
Research Article

7. Conservation Laws of The Generalized Riemann Equations

Binfang Gao, Kai Tian, Q. P. Liu, Lujuan Feng
Pages: 122 - 135
Two special classes of conserved densities involving arbitrary smooth functions are explicitly formulated for the generalized Riemann equation at arbitrary N. The particular case when N = 2 covers most of the known conserved densities of the equation, and the result is also extended to the famous Gurevich-Zybin,...
Research Article

8. Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

Qiaozhen Zhu, Jian Xu, Engui Fan
Pages: 136 - 165
In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with 3×3 Lax pairs. The solution can be expressed in terms of the solution of a 3×3 Riemann-Hilbert problem. The relevant jump matrices...
Research Article

9. On the discretization of Laine equations

Kostyantyn Zheltukhin, Natalya Zheltukhina
Pages: 166 - 177
We consider the discretization of Darboux integrable equations. For each of the integrals of a Laine equation we constructed either a semi-discrete equation which has that integral as an n-integral, or we proved that such an equation does not exist. It is also shown that all constructed semi-discrete...