# Journal of Nonlinear Mathematical Physics

Volume 27, Issue 4, September 2020

**Letter to Editor**

## 1. On the hierarchies of the fully nonlinear Möbius-invariant and symmetry-integrable evolution equations of order three

Marianna Euler, Norbert Euler

Pages: 521 - 528

This is a follow-up paper to the results published in Studies in Applied Mathematics 143, 139–156 (2019), where we reported a classification of 3rd- and 5th-order semi-linear symmetry-integrable evolution equations that are invariant under the Möbius transformation, which includes a list of fully nonlinear...

**Research Article**

## 2. Minimal surfaces associated with orthogonal polynomials

Vincent Chalifour, Alfred Michel Grundland

Pages: 529 - 549

This paper is devoted to a study of the connection between the immersion functions of two-dimensional surfaces in Euclidean or hyperbolic spaces and classical orthogonal polynomials. After a brief description of the soliton surfaces approach defined by the Enneper-Weierstrass formula for immersion and...

**Research Article**

## 3. Study on geometric structures on Lie algebroids with optimal control applications

Esmaeil Peyghan, Liviu Popescu

Pages: 550 - 580

We construct ρ£-covariant derivatives in π*π as the generalization of covariant derivative in π*π to £πE. Moreover, we introduce Berwald and Yano derivatives as two important classes of ρ£-covariant derivatives in π*π and we study properties of them. Finally, we solve an optimal control problem using...

**Research Article**

## 4. Nonlocal symmetries and group invariant solutions for the coupled variable-coefficient Newell-Whitehead system

Yarong Xia, Ruoxia Yao, Xiangpeng Xin

Pages: 581 - 591

Starting from the Lax pairs, the nonlocal symmetries of the coupled variable-coefficient Newell-Whitehead system are obtained. By introducing an appropriate auxiliary dependent variable, the nonlocal symmetries are localized to Lie point symmetries and the coupled variable-coefficient Newell-Whitehead...

**Research Article**

## 5. Asymptotics behavior for the integrable nonlinear Schrödinger equation with quartic terms: Cauchy problem

Lin Huang

Pages: 592 - 615

We consider the Cauchy problem of integrable nonlinear Schrödinger equation with quartic terms on the line. The first part of the paper considers the Riemann-Hilbert formula via the unified method(also known as the Fokas method). The second part of the paper establishes asymptotic formulas for the solution...

**Research Article**

## 6. On the discretization of Darboux Integrable Systems

Kostyantyn Zheltukhin, Natalya Zheltukhina

Pages: 616 - 632

We study the discretization of Darboux integrable systems. The discretization is done using x-, y-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.

**Research Article**

## 7. Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation

Xiaoxue Xu, Cewen Cao, Guangyao Zhang

Pages: 633 - 646

Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be used to derive an integrable symplectic correspondence. Resorting...

**Research Article**

## 8. Decomposition of 2-Soliton Solutions for the Good Boussinesq Equations

Vesselin Vatchev

Pages: 647 - 663

We consider decompositions of two-soliton solutions for the good Boussinesq equation obtained by the Hirota method and the Wronskian technique. The explicit forms of the components are used to study the dynamics of 2-soliton solutions. An interpretation in the context of eigenvalue problems arising from...

**Research Article**

## 9. Integrability conditions of a weak saddle in generalized Liénard-like complex polynomial differential systems

Jaume Giné, Claudia Valls

Pages: 664 - 678

We consider the complex differential system
x˙=x+yf(x), y˙=−y+xf(y),
where f is the analytic function f(z)=∑j=1∞ajzj with aj ∈ ℂ. This system has a weak saddle at the origin and is a generalization of complex Liénard systems. In this work we study its local analytic integrability.

**Research Article**

## 10. Symmetry classification of scalar Ito equations with multiplicative noise

Giuseppe Gaeta, Francesco Spadaro

Pages: 679 - 687

We provide a symmetry classification of scalar stochastic equations with multiplicative noise. These equations can be integrated by means of the Kozlov procedure, by passing to symmetry adapted variables.

**Research Article**

## 11. Gambier lattices and other linearisable systems

Basil Grammaticos, Alfred Ramani

Pages: 688 - 696

We propose two different appraoches to extending the Gambier mapping to a two-dimensional lattice equation. A first approach relies on a hypothesis of separate evolutions in each of the two directions. We show that known equations like the Startsev-Garifullin-Yamilov equation, the Hietarinta equation,...

**Research Article**

## 12. Trigonal Toda Lattice Equation

Shigeki Matsutani

Pages: 697 - 704

In this article, we give the trigonal Toda lattice equation,
−12d3dt3qℓ(t)=eqℓ+1(t)+eqℓ+ζ3(t)++eqℓ+ζ32(t)−3eqℓ(t),
for a lattice point ℓ ∈ [ζ3] as a directed 6-regular graph where ζ3=e2π−1/3, and its elliptic solution for the curve y(y − s) = x
3, (s ≠ 0).