# Journal of Nonlinear Mathematical Physics

Volume 26, Issue 1, December 2018

**Research Article**

## 1. A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Xianguo Geng, Yunyun Zhai, Bo Xue, Jiao Wei

Pages: 1 - 23

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which...

**Research Article**

## 2. The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble

Shulin Lyu, James Griffin, Yang Chen

Pages: 24 - 53

We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function w(x,t)=xαe−x−tx, x ∈ [0, ∞), α > −1, t ≥ 0, are greater than s. This probability is expressed as the quotient of Dn(s, t) and its value at s = 0, where Dn(s, t) denotes the determinant of...

**Research Article**

## 3. Solving the constrained modified KP hierarchy by gauge transformations

Huizhan Chen, Lumin Geng, Na Li, Jipeng Cheng

Pages: 54 - 68

In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator. For this, by selecting the special generating...

**Research Article**

## 4. On *μ*-symmetries, *μ*-reductions, and *μ*-conservation laws of Gardner equation

Özlem Orhan, Özer Teoman

Pages: 69 - 90

In this study, we represent an application of the geometrical characterization of μ-prolongations of vector fields to the nonlinear partial differential Gardner equation with variable coefficients. First, μ-symmetries and the corresponding μ-symmetry classification are investigated and then μ-reduction...

**Research Article**

## 5. A 3-Lie algebra and the dKP Hierarchy

Min-Ru Chen, Ying Chen, Zhao-Wen Yan, Jian-Qin Mei, Xiao-Li Wang

Pages: 91 - 97

In terms of a 3-Lie algebra and the classical Poisson bracket {Bn,L} of the dKP hierarchy, a special 3-bracket {Bm,Bn,L} is proposed. When m = 0 or m = 1, the 3-lax equation ∂L∂t={Bm,Bn,L} is the dKP hierarchy and the corresponding proof is given. Meanwhile, for the generalized case (m,n), the generalized...

**Research Article**

## 6. On a canonical nine-body problem. Integrable Ermakov decomposition via reciprocal transformations

Colin Rogers

Pages: 98 - 106

Here, a recently introduced nine-body problem is shown to be decomposable via a novel class of reciprocal transformations into a set of integrable Ermakov systems. This Ermakov decomposition is exploited to construct more general integrable nine-body systems in which the canonical nine-body system is...

**Research Article**

## 7. Rational solutions to Q3_{δ} in the Adler-Bobenko-Suris list and degenerations

_{δ}

Song-lin Zhao, Da-jun Zhang

Pages: 107 - 132

We derive rational solutions in Casoratian form for the Nijhoff-Quispel-Capel (NQC) equation by using the lattice potential Korteweg-de Vries (lpKdV) equation and two Miura transformations between the lpKdV and the lattice potential modified KdV (lpmKdV) and the NQC equation. This allows us to present...

**Research Article**

## 8. Exact Pollard-like internal water waves

Mateusz Kluczek

Pages: 133 - 146

In this paper we construct a new solution which represents Pollard-like three-dimensional nonlinear geophysical internal water waves. The Pollard-like solution includes the effects of the rotation of Earth and describes the internal water wave which exists at all latitudes across Earth and propagates...

**Research Article**

## 9. Time-independent Hamiltonians describing systems with friction: the “cyclotron with friction”

Francesco Calogero, François Leyvraz

Pages: 147 - 154

As is well-known, any ordinary differential equation in one dimension can be cast as the Euler–Lagrange equation of an appropriate Lagrangian. Additionally, if the initial equation is autonomous, the Lagrangian can always be chosen to be time-independent. In two dimensions, however, the situation is...

**Research Article**

## 10. Inverse scattering on the half-line for generalized ZS-AKNS system with general boundary conditions

Mansur I. Ismailov, Bulent Yilmaz

Pages: 155 - 167

The inverse scattering problem for a first order system of three equations on the half-line with nonsingular diagonal matrix multiplying the derivative and general boundary conditions is considered. It is focused the case of two repeated diagonal elements of diagonal matrix. The scattering matrix on...