# Journal of Nonlinear Mathematical Physics

Volume 27, Issue 1, October 2019

**Letter to Editor**

## 1. Euler’s triangle and the decomposition of tensor powers of the adjoint 𝔰𝔩(2)-module

Askold M. Perelomov

Pages: 1 - 6

By considering a relation between Euler’s trinomial problem and the problem of decomposing tensor powers of the adjoint 𝔰𝔩(2)-module I derive some new results for both problems, as announced in arXiv:1902.08065.

**Letter to Editor**

## 2. Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

S.E. Konstein, I.V. Tyutin

Pages: 7 - 11

If G is a finite Coxeter group, then symplectic reflection algebra H := H1,η (G) has Lie algebra 𝔰𝔩2 of inner derivations and can be decomposed under spin: H = H0 ⊕ H1/2 ⊕ H1 ⊕ H3/2 ⊕ ... We show that if the ideals ℐi (i = 1,2) of all the vectors from the kernel of degenerate bilinear forms Bi(x,y)...

**Research Article**

## 3. Remarks on the mass spectrum of two-dimensional Toda lattice of *E*_{8} type

Askold M. Perelomov

Pages: 12 - 16

A simple procedure for obtaining the mass spectrum of 2-dimensional Toda lattice of E8 type is given.

**Research Article**

## 4. Multicomplex solitons

Julia Cen, Andreas Fring

Pages: 17 - 35

We discuss integrable extensions of real nonlinear wave equations with multi-soliton solutions, to their bicomplex, quaternionic, coquaternionic and octonionic versions. In particular, we investigate these variants for the local and nonlocal Korteweg-de Vries equation and elaborate on how multi-soliton...

**Research Article**

## 5. Realizations of the Witt and Virasoro Algebras and Integrable Equations

Qing Huang, Renat Zhdanov

Pages: 36 - 56

In this paper we study realizations of infinite-dimensional Witt and Virasoro algebras. We obtain a complete description of realizations of the Witt algebra by Lie vector fields of first-order differential operators over the space ℝ3. We prove that none of them admits non-trivial central extension, which...

**Research Article**

## 6. Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

Tomas Nilson, Cornelia Schiebold

Pages: 57 - 94

The first main aim of this article is to derive an explicit solution formula for the scalar two-dimensional Toda lattice depending on three independent operator parameters, ameliorating work in [31]. This is achieved by studying a noncommutative version of the 2d-Toda lattice, generalizing its soliton...

**Research Article**

## 7. Interrelations of discrete Painlevé equations through limiting procedures

A. Ramani, B. Grammaticos, T. Tamizhmani

Pages: 95 - 105

We study the discrete Painlevé equations associated to the E7(1) affine Weyl group which can be obtained by the implementation of a special limits of E8(1)-associated equations. This study is motivated by the existence of two E7(1)-associated discrete both having a double ternary dependence in their...

**Research Article**

## 8. The Group-Theoretical Analysis of Nonlinear Optimal Control Problems with Hamiltonian Formalism

Gülden Gün Polat, Teoman Özer

Pages: 106 - 129

In this study, we pay attention to novel explicit closed-form solutions of optimal control problems in economic growth models described by Hamiltonian formalism by utilizing mathematical approaches based on the theory of Lie groups. For this analysis, the Hamiltonian functions, which are used to define...

**Research Article**

## 9. Inverse Scattering Transform and Solitons for Square Matrix Nonlinear Schrödinger Equations with Mixed Sign Reductions and Nonzero Boundary Conditions

Alyssa K. Ortiz, Barbara Prinari

Pages: 130 - 161

The inverse scattering transform (IST) with nonzero boundary conditions at infinity is developed for a class of 2 × 2 matrix nonlinear Schrödinger-type systems whose reductions include two equations that model certain hyperfine spin F = 1 spinor Bose-Einstein condensates, and two novel equations that...

**Research Article**

## 10. SO(4)-symmetry of mechanical systems with 3 degrees of freedom

Sofiane Bouarroudj, Semyon E. Konstein

Pages: 162 - 169

We answered an old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra 𝔬(4) by means of the Poisson brackets? A system which is not centrally symmetric, but has 6 first integrals generating Lie...

**Research Article**

## 11. Fredholm Property of Operators from 2D String Field Theory

Hai-Long Her

Pages: 170 - 184

In a recent study of Landau-Ginzburg model of string field theory by Gaiotto, Moore and Witten, there appears a type of perturbed Cauchy-Riemann equation, i.e. the ζ-instanton equation. Solutions of ζ-instanton equation have degenerate asymptotics. This degeneracy is a severe restriction for obtaining...