An integrable semi-discretization of the two-component Hunter-Saxton equation
Abstract
In this paper, we propose an integrable semi-discretization of the two-component Hunter--Saxton (2-HS) equation, the short-wave limit of the two-component Camassa--Holm (2-CH) equation. At the continuous level, we show that the 2-HS equation can be derived from a new bilinear formulation, distinct from the conventional one in the literature, via a pseudo 2-reduction and a hodograph transformation. For the semi-discrete construction, we first discretize the underlying bilinear equations in the spatial direction. We then impose the pseudo 2-reduction and apply a discrete hodograph transformation to obtain the semi-discrete system in the physical variables. To the best of our knowledge, the resulting system is the first integrable semi-discretization that preserves the two-component structure of the 2-HS equation. We construct the N-soliton solutions of the continuous and semi-discrete systems in Wronskian and Casoratian forms, respectively. The integrability of the semi-discrete system is inherited from the underlying integrable hierarchy and is further verified by a Lax pair.
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BibTeX
@misc{hori2026integrable,
title = {{An integrable semi-discretization of the two-component Hunter-Saxton equation}},
author = {Ayako Hori and Yuta Tanaka and Ken-ichi Maruno and Yasuhiro Ohta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.18701},
eprint = {2606.18701},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.18701},
}
RIS
TY - UNPB
TI - An integrable semi-discretization of the two-component Hunter-Saxton equation
AU - Ayako Hori
AU - Yuta Tanaka
AU - Ken-ichi Maruno
AU - Yasuhiro Ohta
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2606.18701
UR - https://arxiv.org/abs/2606.18701
AB - In this paper, we propose an integrable semi-discretization of the two-component Hunter--Saxton (2-HS) equation, the short-wave limit of the two-component Camassa--Holm (2-CH) equation. At the continuous level, we show that the 2-HS equation can be derived from a new bilinear formulation, distinct from the conventional one in the literature, via a pseudo 2-reduction and a hodograph transformation. For the semi-discrete construction, we first discretize the underlying bilinear equations in the spatial direction. We then impose the pseudo 2-reduction and apply a discrete hodograph transformation to obtain the semi-discrete system in the physical variables. To the best of our knowledge, the resulting system is the first integrable semi-discretization that preserves the two-component structure of the 2-HS equation. We construct the N-soliton solutions of the continuous and semi-discrete systems in Wronskian and Casoratian forms, respectively. The integrability of the semi-discrete system is inherited from the underlying integrable hierarchy and is further verified by a Lax pair.
ER -