arXiv subjects

Jarmo Hietarinta

Publications and source records attributed to Jarmo Hietarinta.

At least 19 recordsLinked to original sources

arXiv2025

Almost local integrable models from supersymmetry algebras

Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang-Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in \cite{maity2025non}. In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Examples of the latter include the Iwahori-Hecke algebra and the Birman-Murakami-Wenzl (BMW) algebra. As a result, we can Baxterize the constant Yang-Baxter solutions to yield spectral parameter dependent $R$-matrices. The regularity of these $R$-matrices depends on the representation of SUSY generators. In some cases they are regular in the usual sense and in the remaining they are `almost' regular. In the latter case they are also non-invertible. Nevertheless, we show that they can still help us construct integrable models in all dimensions of the local Hilbert space. These models can be described by Hamiltonian densities that are either local or non-local, depending on the representation chosen for the SUSY generators. We demonstrate this for all constant $4\times 4$ invertible Yang-Baxter solutions. Apart from finding new nearest-neighbor interaction spin $\frac{1}{2}$ systems, we also find their higher spin analogs due to the algebraic [representation independent] approach.

hep-th↗

arXiv2023

Solutions to the constant Yang-Baxter equation: additive charge conservation in three dimensions

We find all solutions to the constant Yang--Baxter equation $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$ in three dimensions, subject to an additive charge-conservation ansatz. This ansatz is a generalisation of (strict) charge-conservation, for which a complete classification in all dimensions was recently obtained. Additive charge-conservation introduces additional sector-coupling parameters -- in 3 dimensions there are $4$ such parameters. In the generic dimension 3 case, in which all of the $4$ parameters are nonzero, we find there is a single 3 parameter family of solutions. We give a complete analysis of this solution, giving the structure of the centraliser (symmetry) algebra in all orders. We also solve the remaining cases with three, two, or one nonzero sector-coupling parameter(s).

math.QA↗

arXiv2023

Degree growth of lattice equations defined on a 3x3 stencil

We study complexity in terms of degree growth of one-component lattice equations defined on a $3\times 3$ stencil. The equations include two in Hirota bilinear form and the Boussinesq equations of regular, modified and Schwarzian type. Initial values are given on a staircase or on a corner configuration and depend linearly or rationally on a special variable, for example $f_{n,m}=\alpha_{n,m}z+\beta_{n,m}$, in which case we count the degree in $z$ of the iterates. Known integrable cases have linear growth if only one initial values contains $z$, and quadratic growth if all initial values contain $z$. Even a small deformation of an integrable equation changes the degree growth from polynomial to exponential, because the deformation will change factorization properties and thereby prevent cancellations.

nlin.SI↗

arXiv2020

Discrete Boussinesq-type equations

We present a comprehensive review of the discrete Boussinesq equations based on their three-component forms on an elementary quadrilateral. These equations were originally found by Nijhoff et al using the direct linearization method and later generalized by Hietarinta using a search method based on multidimensional consistency. We derive from these three-component equations their two- and one-component variants. From the one-component form we derive two different semi-continuous limits as well as their fully continuous limits, which turn out to be PDE's for the regular, modified and Schwarzian Boussinesq equations. Several kinds of Lax pairs are also provided. Finally we give their Hirota bilinear forms and multi-soliton solutions in terms of Casoratians.

nlin.SI↗

arXiv2020

Search for integrable two-component versions of the lattice equations in the ABS-list

We search and classify two-component versions of the quad equations in the ABS list, under certain assumptions. The independent variables will be called $y,z$ and in addition to multilinearity and irreducibility the equation pair is required to have the following specific properties: (1) The two equations forming the pair are related by $y\leftrightarrow z$ exchange. (2) When $z=y$ both equations reduce to one of the equations in the ABS list. (3) Evolution in any corner direction is by a multilinear equation pair. One straightforward way to construct such two-component pairs is by taking some particular equation in the ABS list (in terms of $y$), using replacement $y \leftrightarrow z$ for some particular shifts, after which the other equation of the pair is obtained by property (1). This way we can get 8 pairs for each starting equation. One of our main results is that due to condition (3) this is in fact complete for H1, H3, Q1, Q3. (For H2 we have a further case, Q2, Q4 we did not check.) As for the CAC integrability test, for each choice of the bottom equations we could in principle have $8^2$ possible side-equations. However, we find that only equations constructed with an even number of $y \leftrightarrow z$ replacements are possible, and for each such equation there are two sets of "side" equation pairs that produce (the same) genuine B\"acklund transformation and Lax pair.

nlin.SI↗

arXiv2018

Search for CAC-integrable homogeneous quadratic triplets of quad equations and their classification by BT and Lax

We consider two-dimensional lattice equations defined on an elementary square of the Cartesian lattice and depending on the variables at the corners of the quadrilateral. For such equations the property often associated with integrability is that of "multidimensional consistency" (MDC): it should be possible to extend the equation from two to higher dimensions so that the embedded two-dimensional lattice equations are compatible. Usually compatibility is checked using "Consistency-Around-a-Cube" (CAC). In this context it is often assumed that the equations on the six sides of the cube are the same (up to lattice parameters), but this assumption was relaxed in the classification of Boll \cite{Boll2011}. We present here the results of a search and classification of homogeneous quadratic triplets of multidimensionally consistent lattice equations, allowing different equations on the three orthogonal planes (hence triplets) but using the same equation on parallel planes. No assumptions are made about symmetry or tetrahedron property. The results are then grouped by subset/limit properties, and analyzed by the effectiveness of their B\"acklund transformations, or equivalently, by the quality of their Lax pair (fake or not).

nlin.SI↗

arXiv2018

Elementary introduction to discrete soliton equations

We will give a short introduction to discrete or lattice soliton equations, with the particular example of the Korteweg-de Vries as illustration. We will discuss briefly how B\"acklund transformations lead to equations that can be interpreted as discrete equations on a $\mathbb Z^2$ lattice. Hierarchies of equations and commuting flows are shown to be related to multidimensionality in the lattice context, and multidimensional consistency is one of the necessary conditions for integrability. The multidimensional setting also allows one to construct a Lax pair and a B\"acklund transformation, which in turn leads to a method of constructing soliton solutions. The relationship between continuous and discrete equations is discussed from two directions: taking the continuum limit of a discrete equation and discretizing a continuous equation following the method of Hirota.

nlin.SI↗

arXiv2014

Twisted reductions of integrable lattice equations, and their Lax representations

It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry transformation and apply this idea to autonomous and non-autonomous lattice equations. As results of this approach, we obtain new reductions of the discrete potential Korteweg-de Vries equation, discrete modified Korteweg-de Vries equation and the discrete Schwarzian Korteweg-de Vries equation. We will also describe a direct method for obtaining Lax representations for the reduced equations.

nlin.SI↗

arXiv2012

Hirota's method and the search for integrable partial difference equations. 1. Equations on a 3x3 stencil

Hirota's bilinear method ("direct method") has been very effective in constructing soliton solutions to many integrable equations. The construction of one- and two-soliton solutions is possible even for non-integrable bilinear equations, but the existence of a generic three-soliton solution imposes severe constraints and is in fact equivalent to integrability. This property has been used before in searching for integrable partial differential equations, and in this paper we apply it to two dimensional partial difference equations defined on a 3x3 stencil. We also discuss how the obtained equations are related to projections and limits of the three-dimensional master equations of Hirota and Miwa, and find that sometimes a singular limit is needed.

nlin.SI↗

arXiv2012

Exact Soliton-like Solutions of the Radial Gross-Pitaevskii Equation

We construct exact ring soliton-like solutions of the cylindrically symmetric (i.e., radial) Gross- Pitaevskii equation with a potential, using the similarity transformation method. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings whose time dependence is in the phase dynamics only, or oscillating and bouncing solutions, related to the second Painlevé transcendent. In each case the potential can be chosen to be time-independent.

cond-mat.quant-gas↗

arXiv2011

Integrable lattice equations with vertex and bond variables

We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as non-autonomous "Yang-Baxter maps". We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with algebraic entropy as well as multidimensional consistency

nlin.SI↗

arXiv2011

Soliton Taxonomy for a Modification of the Lattice Boussinesq Equation

Integrable multi-component lattice equations of the Boussinesq family have been known for some time. Recently some new equations of this type were found using the Consistency-Around-the-Cube approach. Here we investigate one of these models, B-2, and in particular the consequences of a nonzero deformation parameter $b_0>0$, which allows special kinds of solitons in the parameter range $-b_0/3<k<b_0$.

nlin.SI↗

arXiv2011

Weak Lax pairs for lattice equations

We consider various 2D lattice equations and their integrability, from the point of view of 3D consistency, Lax pairs and Bäcklund transformations. We show that these concepts, which are associated with integrability, are not strictly equivalent. In the course of our analysis, we introduce a number of black and white lattice models, as well as variants of the functional Yang-Baxter equation.

nlin.SI↗

arXiv2010

Boussinesq-like multi-component lattice equations and multi-dimensional consistency

We consider quasilinear, multi-variable, constant coefficient, lattice equations defined on the edges of the elementary square of the lattice, modeled after the lattice modified Boussinesq (lmBSQ) equation, e.g., $\tilde y z=\tilde x-x$. These equations are classified into three canonical forms and the consequences of their multidimensional consistency (Consistency-Around-the-Cube, CAC) are derived. One of the consequences is a restriction on form of the equation for the $z$ variable, which in turn implies further consistency conditions, that are solved. As result we obtain a number of integrable multi-component lattice equations, some generalizing lmBSQ.

nlin.SI↗

arXiv2009

Multisoliton solutions to the lattice Boussinesq equation

The lattice Boussinesq equation (BSQ) is a three-component difference-difference equation defined on an elementary square of the 2D lattice, having 3D consistency. We write the equations in the Hirota bilinear form and construct their multisoliton solutions in terms of Casoratians, following the methodology in our previous papers. In the construction it turns out that instead of the usual discretization of the exponential as $[(a+k)/(a-k)]^n$ we need two different terms $[(a-ωk)/(a-k)]^n$ and $[(a-ω^2 k)/(a-k)]^n$, where $ω$ is a cubic root of unity $\neq 1$.

nlin.SI↗

arXiv2009

Unwinding in Hopfion vortex bunches

We investigate the behaviour of parallel Faddeev-Hopf vortices under energy minimization in a system with physically relevant, but unusual boundary conditions. The homotopy classification is no longer provided by the Hopf invariant, but rather by the set of integer homotopy invariants proposed by Pontrjagin. The nature of these invariants depends on the boundary conditions. A set of tightly wound parallel vortices of the usual Hopfion structure is observed to form a bunch of intertwined vortices or unwind completely, depending on the boundary conditions.

hep-th↗

arXiv2009

Soliton Solutions for ABS Lattice Equations II: Casoratians and Bilinearization

In Part I [arXiv:0902.4873 [nlin.SI]] soliton solutions to the ABS list of multi-dimensionally consistent difference equations (except Q4) were derived using connection between the Q3 equation and the NQC equations, and then by reductions. In that work central role was played by a Cauchy matrix. In this work we use a different approach, we derive the $N$-soliton solutions following Hirota's direct and constructive method. This leads to Casoratians and bilinear difference equations. We give here details for the H-series of equations and for Q1; the results for Q3 have been given earlier.

nlin.SI↗

arXiv2009

Soliton Solutions for ABS Lattice Equations: I Cauchy Matrix Approach

In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1+1 dimensions. In the scalar (i.e. single-field) case there now exist classification results by Adler, Bobenko and Suris (ABS) leading to some new examples in addition to the lattice equations "of KdV type" that were known since the late 1970s and early 1980s. In this paper we review the construction of soliton solutions for the KdV type lattice equations and use those results to construct N-soliton solutions for all lattice equations in the ABS list except for the elliptic case of Q4, which is left to a separate treatment.

nlin.SI↗