$\mathrm{PGL}(3)$-invariant integrable systems from factorisation of linear differential and difference operators
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| Format: | Preprint |
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2026
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| _version_ | 1866915874158411776 |
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| author | Nijhoff, Frank Peng, Linyu Zhang, Cheng Zhang, Da-jun |
| author_facet | Nijhoff, Frank Peng, Linyu Zhang, Cheng Zhang, Da-jun |
| contents | In this paper, we present a unified approach to constructing continuous and discrete $\mathrm{PGL}(3)$-invariant integrable systems, formulated in terms of the common dependent variables $z_1,z_2$, from linear spectral problems and their factorisation. Starting from third-order spectral problems, we first provide explicit forms of the differential and difference invariants, generalising the Schwarzian derivative and cross-ratio to the rank-$3$ setting. The factorisation induces dualities among linear spectral problems, underlying the exact discretisation and multi-dimensional consistency of the associated Boussinesq systems. Then, we derive both continuous and discrete $\mathrm{PGL}(3)$-invariant Boussinesq systems, representing natural rank-$3$ generalisations of the Schwarzian KdV and cross-ratio equations. A geometric lifting-decoupling mechanism is developed to explain the reduction of these systems to the $\mathrm{PGL}(2)$-invariant Boussinesq equations. Finally, we derive a ${\mathrm{PGL}}(3)$-invariant system of generating PDEs together with its Lagrangian structure, in which the lattice parameters serve as independent variables, providing the generating PDE system for the Boussinesq hierarchy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_18386 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $\mathrm{PGL}(3)$-invariant integrable systems from factorisation of linear differential and difference operators Nijhoff, Frank Peng, Linyu Zhang, Cheng Zhang, Da-jun Exactly Solvable and Integrable Systems Mathematical Physics In this paper, we present a unified approach to constructing continuous and discrete $\mathrm{PGL}(3)$-invariant integrable systems, formulated in terms of the common dependent variables $z_1,z_2$, from linear spectral problems and their factorisation. Starting from third-order spectral problems, we first provide explicit forms of the differential and difference invariants, generalising the Schwarzian derivative and cross-ratio to the rank-$3$ setting. The factorisation induces dualities among linear spectral problems, underlying the exact discretisation and multi-dimensional consistency of the associated Boussinesq systems. Then, we derive both continuous and discrete $\mathrm{PGL}(3)$-invariant Boussinesq systems, representing natural rank-$3$ generalisations of the Schwarzian KdV and cross-ratio equations. A geometric lifting-decoupling mechanism is developed to explain the reduction of these systems to the $\mathrm{PGL}(2)$-invariant Boussinesq equations. Finally, we derive a ${\mathrm{PGL}}(3)$-invariant system of generating PDEs together with its Lagrangian structure, in which the lattice parameters serve as independent variables, providing the generating PDE system for the Boussinesq hierarchy. |
| title | $\mathrm{PGL}(3)$-invariant integrable systems from factorisation of linear differential and difference operators |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2603.18386 |