# Journal of Nonlinear Mathematical Physics

Volume 26, Issue 2, March 2019

**Research Article**

## 1. Liouvillian integrability of a general Rayleigh-Duffing oscillator

Jaume Giné, Claudia Valls

Pages: 169 - 187

We give a complete description of the Darboux and Liouville integrability of a general Rayleigh-Duffing oscillator through the characterization of its polynomial first integrals, Darboux polynomials and exponential factors.

**Research Article**

## 2. Generalized Solvable Structures and First Integrals for ODEs Admitting an 𝔰𝔩(2, ℝ) Symmetry Algebra

Paola Morando, Concepción Muriel, Adrián Ruiz

Pages: 188 - 201

The notion of solvable structure is generalized in order to exploit the presence of an 𝔰𝔩(2, ℝ) algebra of symmetries for a kth-order ordinary differential equation ℰ with k > 3. In this setting, the knowledge of a generalized solvable structure for ℰ allows us to reduce ℰ to a family of second-order...

**Research Article**

## 3. A construction of Multidimensional Dubrovin-Novikov Brackets

Ian A. B. Strachan

Pages: 202 - 213

A method for the construction of classes of examples of multi-dimensional, multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This is based on an extension of the original construction of Gelfand and Dorfman which gave examples of Novikov algebras in terms of structures defined...

**Research Article**

## 4. Delta shock waves in conservation laws with impulsive moving source: some results obtained by multiplying distributions

C.O.R. Sarrico

Pages: 214 - 227

The present paper concerns the study of a Riemann problem for the conservation law ut + [ϕ(u)]x = kδ(x − vt) where x, t, k, v and u = u(x, t) are real numbers. We consider ϕ an entire function taking real values on the real axis and δ stands for the Dirac measure. Within a convenient space of distributions...

**Research Article**

## 5. A Hamiltonian yielding damped motion in an homogeneous magnetic field: quantum treatment

François Leyvraz, Francesco Calogero

Pages: 228 - 239

In earlier work, a Hamiltonian describing the classical motion of a particle moving in two dimensions under the combined influence of a perpendicular magnetic field and of a damping force proportional to the particle velocity, was indicated. Here we derive the quantum propagator for the Hamiltonian in...

**Research Article**

## 6. Bilinear identities for the constrained modified KP hierarchy

Huizhan Chen, Lumin Geng, Na Li, Jipeng Cheng

Pages: 240 - 254

In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ, the corresponding identities are written into the Hirota forms. Also, we give the explicit solution forms of ρ and σ.

**Research Article**

## 7. Analytical Cartesian solutions of the multi-component Camassa-Holm equations

Hongli An, Liying Hou, Manwai Yuen

Pages: 255 - 272

Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b(t) + A(t)x and no extra constraint on the dimension N is required. The advantage of our method...

**Research Article**

## 8. Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations with Polynomial Right-Hand Sides

Francesco Calogero, Farrin Payandeh

Pages: 273 - 280

The evolution equations mentioned in the title of this paper read as follows:
x˜n=P(n)(x1,x2), n=1,2,
where ℓ is the “discrete-time” independent variable taking integer values (ℓ = 0, 1, 2, ...), xn ≡ xn(ℓ) are the 2 dependent variables, x˜n≡xn(ℓ+1), and the 2 functions P(n)(x1, x2), n = 1, 2, are...

**Research Article**

## 9. Differential Equations Invariant Under Conditional Symmetries

Decio Levi, Miguel A. Rodríguez, Zora Thomova

Pages: 281 - 293

Nonlinear PDE’s having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples starting from the Boussinesq and including non-autonomous...

**Research Article**

## 10. *N* = 2 Supercomplexification of the Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt Equations

Ziemowit Popowicz

Pages: 294 - 312

The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated....

**Research Article**

## 11. Moving Boundary Problems for Heterogeneous Media. Integrability via Conjugation of Reciprocal and Integral Transformations

Colin Rogers

Pages: 313 - 325

The combined action of reciprocal and integral-type transformations is here used to sequentially reduce to analytically tractable form a class of nonlinear moving boundary problems involving heterogeneity. Particular such Stefan problems arise in the description of the percolation of liquids through...