TY - JOUR
T1 - Discrete Crum’s Theorems and Lattice KdV-Type Equations
AU - Zhang, Cheng
AU - Peng, Linyu
AU - Zhang, Da jun
N1 - Funding Information:
This research is supported by the National Natural Science Foundation of China (Grant Nos. 11601312, 11631007, and 11875040), the Shanghai Young Eastern Scholar program (2016–2019), the JSPS (Grant-in-Aid for Scientific Research No. 16KT0024), Waseda University (Special Research Project Nos. 2017K-170, 2019C-179, 2019E-036, and 2019R-081), and the MEXT Top Global University Project.
Publisher Copyright:
© 2020, Pleiades Publishing, Ltd.
PY - 2020/2/1
Y1 - 2020/2/1
N2 - We develop Darboux transformations (DTs) and their associated Crum’s formulas for two Schrödinger-type difference equations that are themselves discretized versions of the spectral problems of the KdV and modified KdV equations. With DTs viewed as a discretization process, classes of semidiscrete and fully discrete KdV-type systems, including the lattice versions of the potential KdV, potential modified KdV, and Schwarzian KdV equations, arise as the consistency condition for the differential/difference spectral problems and their DTs. The integrability of the underlying lattice models, such as Lax pairs, multidimensional consistency, τ-functions, and soliton solutions, can be easily obtained by directly applying the discrete Crum’s formulas.
AB - We develop Darboux transformations (DTs) and their associated Crum’s formulas for two Schrödinger-type difference equations that are themselves discretized versions of the spectral problems of the KdV and modified KdV equations. With DTs viewed as a discretization process, classes of semidiscrete and fully discrete KdV-type systems, including the lattice versions of the potential KdV, potential modified KdV, and Schwarzian KdV equations, arise as the consistency condition for the differential/difference spectral problems and their DTs. The integrability of the underlying lattice models, such as Lax pairs, multidimensional consistency, τ-functions, and soliton solutions, can be easily obtained by directly applying the discrete Crum’s formulas.
KW - Darboux transformation
KW - discrete Crum’s theorem
KW - discrete Schrödinger equation
KW - exact discretization
KW - lattice KdV equations
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U2 - 10.1134/S0040577920020038
DO - 10.1134/S0040577920020038
M3 - Article
AN - SCOPUS:85083275266
SN - 0040-5779
VL - 202
SP - 165
EP - 182
JO - Theoretical and Mathematical Physics(Russian Federation)
JF - Theoretical and Mathematical Physics(Russian Federation)
IS - 2
ER -