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Comment on: "St\"{a}ckel and Eisenhart lifts, Haantjes geometry and Gravitation"
A. V. Tsiganov
https://arxiv.org/abs/2511.05765 https://arxiv.org/pdf/2511.05765 https://arxiv.org/html/2511.05765
arXiv:2511.05765v1 Announce Type: new
Abstract: One of the oldest methods for constructing integrable Hamiltonian systems, proposed by Jacobi, recently is being presented as a novel St\"{a}ckel lift construction related with Haantjes geometry. It may cause some confusion.
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Travelling wave solutions of equations in the Burgers Hierarchy
Amitava Choudhuri, Modhan Mohan Panja, Supriya Chatterjee, Benoy Talukdar
https://arxiv.org/abs/2511.06333 https://arxiv.org/pdf/2511.06333 https://arxiv.org/html/2511.06333
arXiv:2511.06333v1 Announce Type: new
Abstract: We emphasize that construction of travelling wave solutions for partial differential equations is a problem of considerable interest and thus introduce a simple algebraic method to generate such solutions for equations in the Burgers hierarchy. Our method based on a judicious use of the well known Cole-Hopf transformation is found to work satisfactorily for higher Burgers equations for which the direct method of integration is inapplicable. For Burgers equation we clearly demonstrate how does the diffusion term in the equation counteract the nonlinearity to result in a smooth wave. We envisage a similar study for higher equations in the Buggers hierarchy and establish that (i) as opposed to the solution of the Burgers equation, the purely nonlinear terms of these equations support smooth solutions and more interestingly (ii) the complete solutions of all higher-order equations are identical.
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Residual Symmetry Reductions and Painlev\'e Solitons
Yan Li, Ya-Rong Xia, Ruo-Xia Yao, S. Y. Lou
https://arxiv.org/abs/2511.09077 https://arxiv.org/pdf/2511.09077 https://arxiv.org/html/2511.09077
arXiv:2511.09077v1 Announce Type: new
Abstract: This letter introduces the novel concept of Painlev\'e solitons -- waves arising from the interaction between Painlev\'e waves and solitons in integrable systems. Painlev\'e solitons may also be viewed as solitons propagating against a Painlev\'e wave background, in analogy with the established notion of elliptic solitons, which refer to solitons on an elliptic wave background. By employing a novel symmetry decomposition method aided by nonlocal residual symmetries, we explicitly construct (extended) Painlev\'e II solitons for the Korteweg-de Vries (KdV) equation and (extended) Painlev\'e IV solitons for the Boussinesq equation.
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Generalized discrete integrable operator and integrable hierarchy
Huan Liu
https://arxiv.org/abs/2511.05046 https://arxiv.org/pdf/2511.05046 https://arxiv.org/html/2511.05046
arXiv:2511.05046v1 Announce Type: new
Abstract: We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A central finding is their inherent connection to higher-order pole solutions of integrable hierarchies, contrasting sharply with standard operators linked to simple poles. This work not only provides explicit resolvent formulas for matrix kernels and differential operator analogues but also offers discrete integrable structures that encode higher-order behaviour.
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