The general form of nonlinear evolution equations integrable by matrix gelfand-dikij spectral problem and their group-theoretical and hamiltonian structures

1984 ◽  
Vol 32 (2) ◽  
pp. 25-60 ◽  
Author(s):  
V. G. Dubrovsky ◽  
B. G. Konopelchenko
Keyword(s):  
Spectral Problem ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Hamiltonian Structures

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.

SOME NEW INTEGRABLE NONLINEAR EVOLUTION EQUATIONS AND THEIR INFINITELY MANY CONSERVATION LAWS

2010 ◽  
Vol 24 (19) ◽  
pp. 2077-2090 ◽  
Author(s):  
XIANGUO GENG ◽  
BO XUE
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Nonlinear Evolution Equations ◽  
Conserved Quantities ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Matrix Spectral Problem

A hierarchy of new nonlinear evolution equations associated with a 3×3 matrix spectral problem with two potentials is derived and its Hamiltonian structures are established with the aid of trace identity. The negative flow of the hierarchy is then discussed. A reduction of this hierarchy and its Hamiltonian structures are constructed. An infinite sequence of conserved quantities of several new soliton equations is obtained.

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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