scholarly journals Equations with Peakon Solutions in the Negative Order Camassa-Holm Hierarchy

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Fengfeng Dong ◽  
Lingjun Zhou
Keyword(s):  
Spectral Problem ◽  
Inverse Scattering ◽  
Continued Fractions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Lax Pair ◽  
Nonlinear Evolution Equations ◽  
Negative Order

The negative order Camassa-Holm (CH) hierarchy consists of nonlinear evolution equations associated with the CH spectral problem. In this paper, we show that all the negative order CH equations admit peakon solutions; the Lax pair of the N-order CH equation given by the hierarchy is compatible with its peakon solutions. Special peakon-antipeakon solutions for equations of orders -3 and -4 are obtained. Indeed, for N≤-2, the peakons of N-order CH equation can be constructed explicitly by the inverse scattering approach using Stieltjes continued fractions. The properties of peakons for N-order CH equation when N is odd are much different from the CH peakons; we present the case N=-3 as an example.

Darboux transformation and exact solutions for a (2 + 1)-dimensional integrable system of nonlinear evolution equations

2020 ◽  
Vol 34 (34) ◽  
pp. 2050392
Author(s):  
Zhen Chuan Zhou ◽  
Xiao Ming Zhu
Keyword(s):  
Exact Solutions ◽  
Spectral Problem ◽  
Integrable System ◽  
Darboux Transformation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Lax Pair ◽  
Nonlinear Evolution Equations ◽  
Recursion Operator

In this paper, starting from a spectral problem, we construct a [Formula: see text]-dimensional integrable system of nonlinear evolution equations. Based on the Lax pair, the recursion operator and Darboux transformation for the whole hierarchy were constructed. As an application, some exact solutions for the hierarchy are obtained by using the Darboux transformation.

An extension of the coupled derivative nonlinear Schrödinger hierarchy

2018 ◽  
Vol 32 (02) ◽  
pp. 1850016
Author(s):  
Siqi Xu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Compatibility Condition ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Nonlinear Schrödinger ◽  
Matrix Spectral Problem

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

scholarly journals Matrix spectral problem with multiple-order jumps and poles

1996 ◽  
Vol 37 (3) ◽  
pp. 320-333 ◽  
Author(s):  
Zhuhan Jiang
Keyword(s):  
Spectral Problem ◽  
Spectral Data ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Temporal Dimension ◽  
Explicit Representations ◽  
Inverse Spectral Method ◽  
Matrix Spectral Problem

AbstractThe inverse spectral method for a general N × N spectral problem for solving nonlinear evolution equations in one spacial and one temporal dimension is extended to include multi-boundary jumps and high-order poles and their explicit representations. It therefore provides a formalism to generate soliton solutions that correspond to higher-order poles of the spectral data.

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