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]].     

Matched pairs approach to set theoretic solutions of the Yang–Baxter equation

We study set-theoretic solutions $(X,r)$ of the Yang–Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterisation of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid $S(X,r)$ and we show that r extends as a solution $r_S$ on $S(X,r)$ as a set. Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids S of the required type, including iterated products $S?S?S$ equivalent to $r_S$ a solution, and extensions $(S?T,r_{S?T})$. Examples include a general ‘double’ construction $(S?S,r_{S?S})$ and some concrete extensions, their actions and graphs based on small sets.     

Combinatorial Yang–Baxter maps arising from the tetrahedron equation

We survey the matrix product solutions of the Yang–Baxter equation recently obtained from the tetrahedron equation. They form a family of quantum R-matrices of generalized quantum groups interpolating the symmetric tensor representations of U_q(A_n?1⁽¹⁾) and the antisymmetric tensor representations of \({U_{ - {q^{ - 1}}}}\left( {A_{n - 1}^{\left( 1 \right)}} \right)\). We show that at q = 0, they all reduce to the Yang–Baxter maps called combinatorial R-matrices and describe the latter by an explicit algorithm.     

Folding of Set-Theoretical Solutions of the Yang–Baxter Equation

We establish a correspondence between the invariant subsets of a non-degenerate symmetric set-theoretical solution of the quantum Yang–Baxter equation and the parabolic subgroups of its structure group, equipped with its canonical Garside structure. Moreover, we introduce the notion of a foldable solution, which extends the one of a decomposable solution.     

Soliton interactions and Yang–Baxter maps for the complex coupled short-pulse equation

The complex coupled short-pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so-called self-symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long-time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well-known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.     

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.     

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.     

Existence of weak solutions of the g-Kelvin–Voight equation

The 2D $g$-Navier–Stokes equations have the form $\frac{?u}{?t}?\nu \Delta u+u.?u+?p=f\text{in }\Omega$ with the continuity equation $?.\left(gu\right)=0\text{in }\Omega$ in a bounded domain $\Omega ?{\mathbb{R}}^2$ where $g=g\left(x_{1,}{x}_2\right)$ is a smooth real valued function defined on $\Omega$. We use the method described by Roh [J. Roh, $g$-Navier Stokes equations, Ph.D. Thesis, University of Minnesota, 2001] for the derivation of $g$-Kelvin–Voight equations represented by $\frac{?u}{?t}?\nu {\Delta }_{g}u+\frac{\nu }{g}(?g??)u?\alpha {\Delta }_g{u}_t+\frac{\alpha }{g}(?g??)u_t+u??u+?p=f\left(x\right)\text{ in }\Omega$$?.\left(gu\right)=0\text{in }\Omega$ We discuss the existence and uniqueness of weak solutions of $g$-Kelvin–Voight equations by the use of the well known Feado–Galerkin method.     

Primitive solutions of the Korteweg–de Vries equation

Theoretical and Mathematical Physics - We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions. These...     

Exact solutions of the generalized Sinh–Gordon equation

In this paper, we successfully derive a new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.     

On analytical solutions of the Black–Scholes equation

This work presents a theoretical analysis for the Black–Scholes equation. Given a terminal condition, the analytical solution of the Black–Scholes equation is obtained by using the Adomian approximate decomposition technique. The mathematical technique employed in this work also has significance in studying some other problems in finance theory.     

Numerical conservative solutions of the Hunter–Saxton equation

BIT Numerical Mathematics - In the article a convergent numerical method for conservative solutions of the Hunter–Saxton equation is derived. The method is based on piecewise linear...