Bilinear Differential-Difference Equations and One-Point Distributions of Some KPZ-Class Models
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909798369329152 |
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| author | Rodriguez, C. Alexander |
| author_facet | Rodriguez, C. Alexander |
| contents | We introduce a collection of nonlinear integrable partial differential-difference equations that are satisfied by the one-point distribution functions of some classical integrable KPZ models. Moreover, these equations can be regarded as reparametrizations or as scaling limits of the Hirota bilinear difference equation (HBDE), a canonical discretization for many important integrable systems such as the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, and the two-dimensional Toda lattice (2DTL). Our contributions are threefold: (i) general Fredholm determinant solutions; (ii) verification that known formulas for classical integrable KPZ models fit within our framework; and (iii) zero-curvature/Lax pair formulations. As an application, we derive formal scaling limits of the equations, including the KP limit under 1:2:3 KPZ scaling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16316 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bilinear Differential-Difference Equations and One-Point Distributions of Some KPZ-Class Models Rodriguez, C. Alexander Probability Mathematical Physics Exactly Solvable and Integrable Systems We introduce a collection of nonlinear integrable partial differential-difference equations that are satisfied by the one-point distribution functions of some classical integrable KPZ models. Moreover, these equations can be regarded as reparametrizations or as scaling limits of the Hirota bilinear difference equation (HBDE), a canonical discretization for many important integrable systems such as the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, and the two-dimensional Toda lattice (2DTL). Our contributions are threefold: (i) general Fredholm determinant solutions; (ii) verification that known formulas for classical integrable KPZ models fit within our framework; and (iii) zero-curvature/Lax pair formulations. As an application, we derive formal scaling limits of the equations, including the KP limit under 1:2:3 KPZ scaling. |
| title | Bilinear Differential-Difference Equations and One-Point Distributions of Some KPZ-Class Models |
| topic | Probability Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2509.16316 |