Singular Solutions to the Quantum Yang–Baxter Equations

Let X and A be weak Hopf algebras in the sense of Li (1998 Li , F. ( 1998 ). Weak Hopf algebras and some new solutions of the quantum Yang–Baxter equation . J. Algebra 208 ( 1 ): 72 – 100 .[Crossref], [Web of Science ®] , [Google Scholar]). As in the case of Hopf algebras (Majid, 1990 Majid , S. ( 1990 ). Quasitriangular Hopf algebras and Yang–Baxter equations . Internat. J. Modern Phys. A 5 : 1 – 91 . [Google Scholar]), a weak bicrossed coproduct X∞ _R A is constructed by means of good regular R-matrices of the weak Hopf algebras X and A. Using this, we provide a new framework of obtaining singular solutions of the quantum Yang–Baxter equation by constructing weak quasitriangular structures over X∞ _R A when both X and A admit a weak quasitriangular structure. Finally, two explicit examples are given.     

Solutions of the Hom-Yang–Baxter Equation from Monoidal Hom-(Co)Algebra Structures

A method for constructing solutions of the Hom-Yang–Baxter equations is presented. Thus, methods yields a so-called α-involutory solution of the Hom-Yang–Baxter equation for every monoidal Hom-(co)algebra structure on a space. Characterizations for solutions of Hom-Yang–Baxter equations arising from monoidal Hom-(co)algebra structures are given, and a monoidal Hom-(co)algebra structure which produces such a solution is constructed.     

Indecomposable involutive set-theoretic solutions of the Yang–Baxter equation

We describe the indecomposable involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation as dynamical extensions of non-degenerate left cycle sets. Moreover we characterize the indecomposable dynamical extensions and we produce several examples. As an application we construct a family of finite indecomposable solutions whose structure groups have not the unique product property.     

Coxeter-like groups for set-theoretic solutions of the Yang–Baxter equation

We attach with every finite, involutive, nondegenerate set-theoretic solution of the Yang–Baxter equation a finite group that plays for the associated structure group the role that a finite Coxeter group plays for the associated Artin–Tits group.     

A new family of set-theoretic solutions of the Yang–Baxter equation

A new family of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation is constructed. Two subfamilies, consisting of irretractable square-free solutions, are new counterexamples to Gateva-Ivanova’s Strong Conjecture [7 Gateva-Ivanova, T. (2004). A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation. J. Math. Phys. 45(10):3828–3858.[Crossref], [Web of Science ®] , [Google Scholar]]. They are in addition to those obtained by Vendramin [15 Vendramin, L. (2016). Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova. J. Pure Appl. Algebra 220:2064–2076.[Crossref], [Web of Science ®] , [Google Scholar]] and [1 Bachiller, D., Cedó, F., Jespers, E., Okniński, J. (2017). A family of irretractable square-free solutions of the Yang-Baxter equation. Forum Math. (to appear). [Google Scholar]].     

Spectral-Parameter Dependent Yang–Baxter Operators and Yang–Baxter Systems From Algebra Structures

For any algebra, two families of colored Yang–Baxter operators are constructed, thus producing solutions to the two-parameter quantum Yang–Baxter equation. An open problem about a system of functional equations is stated. The matrix forms of these operators for two and three dimensional algebras are computed. A FRT bialgebra for one of these families is presented. Solutions for the one-parameter quantum Yang–Baxter equation are derived and a Yang–Baxter system constructed.     

Traveling-Wave Solutions of the Kolmogorov–Petrovskii–Piskunov Equation

We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov–Petrovskii–Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction–diffusion processes in the theory of combustion, mathematical biology, and other areas of natural sciences. A new efficiently numerically implementable analytical representation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples.     

𝒪-Operators of Loday Algebras and Analogues of the Classical Yang–Baxter Equation

We introduce notions of 𝒪-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota–Baxter operators. The invertible 𝒪-operators give a sufficient and necessary condition on the existence of the 2 ⁿ⁺¹ operations on an algebra with the 2 ⁿ operations in an associative cluster. The analogues of the classical Yang–Baxter equation in these algebras can be understood as the 𝒪-operators associated to certain dual bimodules. As a byproduct, the constraint conditions (invariances) of nondegenerate bilinear forms on these algebras are given.     

Stabilization of Solutions of the Nonlinear Fokker–Planck Equation

One considers the equation $$ \mathrm{div}\left( {{u^{\sigma }}Du} \right)+b(x)Du-{u_t}=f(x)g(u),\quad x\in {{\mathbb{R}}^n},\quad t\in \left( {0,\infty } \right), $$ where $ b:{{\mathbb{R}}^n}\to {{\mathbb{R}}^n} $ and $ f:{{\mathbb{R}}^n}\to [0,\infty ) $ are locally bounded measurable functions, g: (0,∞)?→?(0,∞) is continuous and nondecreasing, One obtains the conditions ensuring that its positive solutions stabilize to zero as t?→?∞.     

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which in the case of the Benjamin–Ono equation, are the mean, a spatial phase, a temporal phase, and the real part of one of the Fourier modes at t=0.     

Low-Regularity Solutions of the Periodic Camassa–Holm Equation

We prove local existence and uniqueness of weak solutions of the Camassa–Holm equation with periodic boundary conditions in various spaces of low-regularity which include the periodic peakons. The proof uses the connection of the Camassa–Holm equation with the geodesic flow on the diffeomorphism group of the circle with respect to the L ² metric.     

Bifurcation Features of Periodic Solutions of the Mackey–Glass Equation

Computational Mathematics and Mathematical Physics - Bifurcations of periodic solutions of the well-known Mackey–Glass equation from its unique equilibrium state under varying equation...     

Asymptotic Analysis of Multilump Solutions of the Kadomtsev–Petviashvili-I Equation

We construct lump solutions of the Kadomtsev–Petviashvili-I equation using Grammian determinants in the spirit of the works by Ohta and Yang. We show that the peak locations depend on the real roots of the Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. We also prove that if the time goes to ?∞, then all the peak locations are on a vertical line, while if the time goes to ∞, then they are all on a horizontal line, i.e., a π/2 rotation is observed after interaction.     

Global Dissipative Multipeakon Solutions of the Camassa–Holm Equation

We show how to construct globally defined dissipative multipeakon solutions of the Camassa–Holm equation. The construction includes in particular the case with peakon-antipeakon collisions. The solutions are dissipative in the sense that the associated energy is decreasing in time.