Journal of Nonlinear Mathematical Physics

Volume 21, Issue 2, March 2014
Research Article

1. Random Lie-point symmetries

Pedro José Catuogno, Luis Roberto Lucinger
Pages: 149 - 165
We introduce the notion of a random symmetry. It consists of taking the action given by a deterministic flow that maintains the solutions of a given differential equation invariant and replacing it with a stochastic flow. This generates a random action, which we call a random symmetry.
Research Article

2. Geometry of rank 2 distributions with nonzero Wilczynski invariants*

Boris Doubrov, Igor Zelenko
Pages: 166 - 187
In the famous 1910 “cinq variables” paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in ℝ5 with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric...
Research Article

3. Non-Abelian Lie algebroids over jet spaces

Arthemy V. Kiselev, Andrey O. Krutov
Pages: 188 - 213
We associate Hamiltonian homological evolutionary vector fields – which are the non-Abelian variational Lie algebroids’ differentials – with Lie algebra-valued zero-curvature representations for partial differential equations.
Research Article

4. Characteristic integrals in 3D and linear degeneracy

E.V. Ferapontov, J. Moss
Pages: 214 - 224
Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and...
Research Article

5. A Perturbative Approach for the Asymptotic Evaluation of the Neumann Value Corresponding to the Dirichlet Datum of a Single Periodic Exponential for the NLS

Guenbo Hwang
Pages: 225 - 247
Boundary value problems for the nonlinear Schrödinger equation formulated on the half-line can be analyzed by the Fokas method. For the Dirichlet problem, the most difficult step of this method is the characterization of the unknown Neumann boundary value. For the case that the Dirichlet datum consists...
Research Article

6. Noether symmetries and the quantization of a Liénard-type nonlinear oscillator

G. Gubbiotti, M.C. Nucci
Pages: 248 - 264
The classical quantization of a Liénard-type nonlinear oscillator is achieved by a quantization scheme (M. C. Nucci. Theor. Math. Phys., 168:994–1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly...
Research Article

7. Time-evolution-proof Scattering Data for the Focusing and Defocusing Zakharov-Shabat Systems

Cornelis van der Mee
Pages: 265 - 277
In this article we give sufficient conditions on the scattering data of a defocusing or focusing Zakharov-Shabat system in order that its potential is square integrable. For a dense subset of integrable as well as square integrable potentials, we show that the scattering data actually satisfy these sufficient...
Research Article

8. On Darboux transformations for the derivative nonlinear Schrödinger equation

Jonathan J.C. Nimmo, Halis Yilmaz
Pages: 278 - 293
We consider Darboux transformations for the derivative nonlinear Schrödinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the...
Research Article

9. Weierstrass integrability for a class of differential systems

Jaume Llibre, Claudia Valls
Pages: 294 - 307
We characterize the differential equations of the form x′=y, y′=an(x)yn+an-1(x)yn-1+⋯+a1(x)y+a0(x), n≥2, an(0)≠0, where aj(x) are meromorphic functions in the variable x for j = 0,…,n that admits either a Weierstrass first integral or a Weierstrass inverse integrating factor.