TY - JOUR

T1 - On generating integrable dynamical systems in 1+1 and 2+1 dimensions by using semisimple lie algebras

AU - Zhang, Yufeng

AU - Tam, Honwah

AU - Wu, Lixin

N1 - Funding Information:
This work was supported by the Innovation Team of Jiangsu Province hosted by the Chinese University of Mining and Technology (2014), the National Natural Science Foundation of China (Grant No. 11371361), the Natural Science Foundation of Shandong Province (Grant No. ZR2013AL016), and Hong Kong Research Grant Council (Grant No. HKBU202512).

PY - 2015

Y1 - 2015

N2 - We deduce a set of integrable equations under the framework of zero curvature equations and obtain two sets of integrable soliton equations, which can be reduced to some new integrable equations including the generalised nonlinear Schrödinger (NLS) equation. Under the case where the isospectral functions are oneorder polynomials in the parameter λ, we generate a set of rational integrable equations, which are reduced to the loop soliton equation. Under the case where the derivative λt of the spectral parameter λ is a quadratic algebraic curve in λ, we derive a set of variable-coefficient integrable equations. In addition, we discretise a pair of isospectral problems introduced through the Lie algebra given by us for which a set of new semi-discrete nonlinear equations are available; furthermore, the semi-discrete MKdV equation and the Hirota lattice equation are followed to produce, respectively. Finally, we apply the Lie algebra to introduce a set of operator Lax pairs with an operator, and then through the Tu scheme and the binomial-residue representation method proposed by us, we generate a 2+1-dimensional integrable hierarchy of evolution equations, which reduces to a generalised 2+1-dimensional Davey-Stewartson (DS) equation.

AB - We deduce a set of integrable equations under the framework of zero curvature equations and obtain two sets of integrable soliton equations, which can be reduced to some new integrable equations including the generalised nonlinear Schrödinger (NLS) equation. Under the case where the isospectral functions are oneorder polynomials in the parameter λ, we generate a set of rational integrable equations, which are reduced to the loop soliton equation. Under the case where the derivative λt of the spectral parameter λ is a quadratic algebraic curve in λ, we derive a set of variable-coefficient integrable equations. In addition, we discretise a pair of isospectral problems introduced through the Lie algebra given by us for which a set of new semi-discrete nonlinear equations are available; furthermore, the semi-discrete MKdV equation and the Hirota lattice equation are followed to produce, respectively. Finally, we apply the Lie algebra to introduce a set of operator Lax pairs with an operator, and then through the Tu scheme and the binomial-residue representation method proposed by us, we generate a 2+1-dimensional integrable hierarchy of evolution equations, which reduces to a generalised 2+1-dimensional Davey-Stewartson (DS) equation.

KW - Hirota lattice equation

KW - Lie algebra

KW - Semidiscrete equation

UR - http://www.scopus.com/inward/record.url?scp=84958523275&partnerID=8YFLogxK

U2 - 10.1515/zna-2015-0321

DO - 10.1515/zna-2015-0321

M3 - Journal article

AN - SCOPUS:84958523275

SN - 0932-0784

VL - 70

SP - 975

EP - 977

JO - Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences

JF - Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences

IS - 11

ER -