A twisted Marchenko lemma and applications
Abstract
We develop a twisted Marchenko lemma with applications into non-commutative two-component integrable systems. Starting from a twisted differential equation, we introduce the corresponding two-component Wronski matrix Wn,m(ϕ,ψ). Under the Marchenko lemma conditions, the logarithmic derivative (∂ Wn,m)Wn,m−1 satisfies nonlinear equations, whose two-component projections yield a non-commutative two-component KP equation, while a single-component reduction recovers the non-commutative Burgers equation, KP equation and DS equation. A direct approach based on two-component quasi-Wronskians is developed, from which a higher-order non-commutative two-component KP equation is generated with matrix-valued solutions. Reductions of the seed functions of the non-commutative two-component KP hierarchy yield the one-component counterpart, while a stepwise reduction of those for the two-component KP equation gives the non-commutative DS and NLS equations.
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BibTeX
@misc{zhang2026twisted,
title = {{A twisted Marchenko lemma and applications}},
author = {Jun-Yang Zhang and Shi-Hao Li and Shou-Feng Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28233},
eprint = {2610.28233},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28233},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A twisted Marchenko lemma and applications
AU - Jun-Yang Zhang
AU - Shi-Hao Li
AU - Shou-Feng Shen
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28233
UR - https://arxiv.org/abs/2610.28233
AB - We develop a twisted Marchenko lemma with applications into non-commutative two-component integrable systems. Starting from a twisted differential equation, we introduce the corresponding two-component Wronski matrix W_n,m(ϕ,ψ). Under the Marchenko lemma conditions, the logarithmic derivative (∂ W_n,m)W_n,m^−1 satisfies nonlinear equations, whose two-component projections yield a non-commutative two-component KP equation, while a single-component reduction recovers the non-commutative Burgers equation, KP equation and DS equation. A direct approach based on two-component quasi-Wronskians is developed, from which a higher-order non-commutative two-component KP equation is generated with matrix-valued solutions. Reductions of the seed functions of the non-commutative two-component KP hierarchy yield the one-component counterpart, while a stepwise reduction of those for the two-component KP equation gives the non-commutative DS and NLS equations.
ER -