Journal of Nonlinear Mathematical Physics

Volume 22, Issue 1, December 2014
Research Article

1. New solutions with peakon creation in the Camassa–Holm and Novikov equations

Marcus Kardell
Pages: 1 - 16
In this article we study a new kind of unbounded solutions to the Novikov equation, found via a Lie symmetry analysis. These solutions exhibit peakon creation, i.e., these solutions are smooth up until a certain finite time, at which a peak is created. We show that the functions are still weak solutions...
Research Article

2. The compatibility of additional symmetry and gauge transformations for the constrained discrete Kadomtsev-Petviashvili hierarchy

Maohua Li, Jipeng Cheng, Jingsong He
Pages: 17 - 31
In this paper, the compatibility between the gauge transformations and the additional symmetry of the constrained discrete Kadomtsev-Petviashvili hierarchy is given, which preserves the form of the additional symmetry of the cdKP hierarchy, up to shifting of the corresponding additional flows by ordinary...
Research Article

3. Exact solutions of the Gerdjikov-Ivanov equation using Darboux transformations

Halis Yilmaz
Pages: 32 - 46
We study the Gerdjikov-Ivanov (GI) equation and present a standard Darboux transformation for it. The solution is given in terms of quasideterminants. Further, the parabolic, soliton and breather solutions of the GI equation are given as explicit examples.
Research Article

4. Reduction by λ –symmetries and σ –symmetries: a Frobenius approach

Paola Morando
Pages: 47 - 59
Different kinds of reduction for ordinary differential equations, such as λ –symmetry and σ –symmetry reductions, are recovered as particular cases of Frobenius reduction theorem for distribution of vector fields. This general approach provides some hints to tackle the reconstruction problem and to solve...
Research Article

5. On the Darboux integrability of the Painlevé II equations

Jaume Llibre, Claudia Valls
Pages: 60 - 75
In this paper we prove the non–existence of Darboux first integrals for the Painlevé II equations x.=y−z2−x2,y.=α+12+2xy,ż=1 for all values of α ∊ ℂ \ {αn: n = 2,4,…}. These αn are real and larger than −1/2.
Research Article

6. The Asymptotic Behaviors of Solutions to the Perturbed Riemann Problem near the Singular Curve for the Chromatography System*

Chun Shen
Pages: 76 - 101
The Riemann problem for a simplified chromatography system is considered and the global Riemann solutions are constructed in all kinds of situations. In particular, the zero rarefaction wave, the zero shock wave and the zero delta shock wave are discovered in the Riemann solutions in some limit situations,...
Research Article

7. On a spectral analysis of scattering data for the Camassa-Holm equation

Chueh-Hsin Chang, Tony Wen-Hann Sheu
Pages: 102 - 116
Physical details of the Camassa–Holm (CH) equation that are difficult to obtain in space-time simulation can be explored by solving the Lax pair equations within the direct and inverse scattering analysis context. In this spectral analysis of the completely integrable CH equation we focus solely on the...
Research Article

8. Mean-Field Limit of a Microscopic Individual-Based Model Describing Collective Motions

Carlo Bianca, Christian Dogbe
Pages: 117 - 143
This paper is mainly concerned with a mean-field limit and long time behavior of stochastic microscopic interacting particles systems. Specifically we prove that a class of ODE modeling collective interactions in animals or pedestrians converges in the mean-field limit to the solution of a non-local...
Research Article

9. A Vector Fokas-Lenells System from the Coupled Nonlinear Schrödinger Equations

MengXia Zhang, ShaoLing He, ShuQiang Lv
Pages: 144 - 154
With the aid of the spectral gradient method of Fuchssteiner, the compatible pair of Hamiltonian operators for the coupled NLS hierarchy is rediscovered. This result enables us to construct a hierarchy, which contains a vector generalization of Fokas-Lenells system. The vector Fokas-Lenells system is...
Research Article

10. A three-component Camassa-Holm system with cubic nonlinearity and peakons

Baoqiang Xia, Ruguang Zhou, Zhijun Qiao
Pages: 155 - 169
In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peaked solitons (peakons). The 3CH model is proven to be integrable in the sense of Lax pair, Hamiltonian structure, and conservation laws. We show that this system admits peakons and multi-peakon solutions....