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On a matrix constrained CKP hierarchy
Song Li, Kelei Tian, Zhiwei Wu
https://arxiv.org/abs/2510.09054 https://arxiv.org/pdf/2510.09054

On a matrix constrained CKP hierarchy
The algebraic structures of integrable hierarchies play an important role in the study of soliton equations. In this paper, we use splitting theory to give a matrix representation of a constrained CKP hierarchy, which can be considered as a generalization of the $\hat{A}_{2n}^{(2)}$-KdV hierarchy and the constrained KP hierarchy. An equivalent construction in terms of the pseudo-differential operator is discussed. Darboux transformations, scaling transformation and tau functions $\ln τ_f$ for …

Self-replication and Computational Universality
Jordan Cotler, Cl\'ement Hongler, Barbora Hudcov\'a
https://arxiv.org/abs/2510.08342 https://arxiv.…

Chaotic variability in a model of coupled ice streams
Kolja Kypke, Peter Ashwin, Peter Ditlevsen
https://arxiv.org/abs/2510.12525 https://arxiv.org/pdf/251…

Chaotic variability in a model of coupled ice streams
Regions of fast-flowing ice in ice sheets, known as ice streams, have been theorized to be able to exhibit build-up/surge oscillatory variability due to thermomechanical coupling at the base of the ice. A simple model of three coupled ice streams is constructed to replicate the spatial configuration of a single ice stream being bisected into two termini. The model is constructed to mimic existing branching ice streams in northern Greenland. This model is shown to exhibit both steady-flow and bu…

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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Asymmetric rational reductions of 2D-Toda hierarchy and a generalized Frobenius manifold
Haonan Qu, Qiulan Zhao
https://arxiv.org/abs/2510.04151 https://ar…

Asymmetric rational reductions of 2D-Toda hierarchy and a generalized Frobenius manifold
We study the local bihamiltonian structures of the asymmetric rational reductions of the 2D-Toda hierarchy (RR2T) of types $(2,1)$ and $(1,2)$ at the full-dispersive level, and construct a three-dimensional generalized Frobenius manifold with non-flat unity associated with the $(2,1)$-type. Furthermore, we explicitly relate the $(2,1)$-type RR2T to the bi-graded Toda and constrained KP hierarchies via linear reciprocal and Miura-type transformations.

Focusing mKdV equation: Two-phase solutions and their stability analysis
Liming Ling, Xuan Sun
https://arxiv.org/abs/2510.12073 https://arxiv.org/pdf/2510.…

Focusing mKdV equation: Two-phase solutions and their stability analysis
In this work, we primarily focus on the two-phase solutions and their stability to the focusing mKdV equation. By employing the algebro-geometric approach in combination with an effective integration method, we construct explicit two-phase solutions and their corresponding wave-functions expressed in terms of the Riemann theta function. The spectral stability of two-phase solutions is examined via a modified squared-eigenfunction approach, and their stability with respect to subharmonic perturb…

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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Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations
Liming Ling, Huajie Su
https://arxiv.org/abs/2510.12129 https://arxi…

Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations
We prove that the $N$-solitons, including breathers and multi-hump solitons, of the coupled nonlinear Schrödinger (CNLS) equations are nonlinearly stable in the Sobolev space $H^{N}$. Moreover, $(N_{1},N_{2})$-solitons of the coupled modified Korteweg--de Vries (CmKdV) equations are shown to be nonlinearly stable in the Sobolev space $H^{2N_{1}+N_{2}}$. The number of negative eigenvalues of the second variation of the Lyapunov functional is $N$ for $N$-solitons of the CNLS equations, and $N_{1…

Invariant volume form for 3D QRT maps
Jaume Alonso, Yuri B. Suris
https://arxiv.org/abs/2510.11468 https://arxiv.org/pdf/2510.11468

Invariant volume form for 3D QRT maps
Recently, we proposed a three-dimensional generalization of QRT maps. These novel maps can be associated with pairs of pencils of quadrics in $\mathbb P^3$. By construction, these maps have two rational integrals (parameters of both pencils). In the present paper, we find an invariant volume form for these maps, thus finally establishing their integrability.