Integrable discretizations of the short pulse equation

In the present paper, we propose integrable semi-discrete and full-discrete analogues of the short pulse (SP) equation. The key of the construction is the bilinear forms and determinant structure of solutions of the SP equation. We also give the determinant formulas of N-soliton solutions of the semi-discrete and full-discrete analogues of the SP equations, from which the multi-loop and multi-breather solutions can be generated. In the continuous limit, the full-discrete SP equation converges to the semi-discrete SP equation, then to the continuous SP equation. Based on the semi-discrete SP equation, an integrable numerical scheme, i.e., a self-adaptive moving mesh scheme, is proposed and used for the numerical computation of the short pulse equation.

💡 Research Summary

The paper presents integrable semi‑discrete and fully‑discrete analogues of the short‑pulse (SP) equation, a nonlinear wave model describing ultra‑short optical pulses. Starting from the known Hirota bilinear form of the continuous SP equation, the authors construct discrete versions that preserve the integrable structure, including Lax pairs, conserved quantities, and determinant (τ‑function) solutions.

Semi‑discrete SP equation – time remains a continuous variable while the spatial coordinate is discretized on a uniform lattice (x_n=n\Delta x). By expressing the field (u_n(t)) as a ratio of τ‑functions and applying Hirota’s D‑operator with a forward/backward difference in space, they obtain the bilinear relation
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