Papers by Jarmo Hietarinta
Journal of Difference Equations and Applications, 2013
Hirota's bilinear method ("direct method") has been very effective in constructing soliton soluti... more 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 (PDE), and in this paper we apply it to two dimensional partial difference equations (P∆E) defined on a 3 × 3 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.
Physics Letters A, 1985
We present the complete singularity analysis for an N-dimensional system of quartic oscillators. ... more We present the complete singularity analysis for an N-dimensional system of quartic oscillators. By extending our results to two-dimensional systems, we show that there are N-dimensional quartic potentials which possess the Painlevk property. These are also found to be integrable, in accordance with the Ablowitz-Ramani-Segur conjecture.
Lecture Notes in Physics
ABSTRACT We give an introduction to Hirota’s bilinear method, which is particularly efficient for... more ABSTRACT We give an introduction to Hirota’s bilinear method, which is particularly efficient for constructing multisoliton solutions to integrable nonlinear evolution equations. We discuss in detail how the method works for equations in the Korteweg–de Vries class and then go through some other classes of equations. Finally we discuss how the existence of multisoliton solutions can be used as an integrability condition and therefore as a method of searching for possible new integrable equations.
Proceedings of the 1991 international symposium on Symbolic and algebraic computation, 1991
The search for integrable PDE's has been an active research subject with computer algebra as a ne... more The search for integrable PDE's has been an active research subject with computer algebra as a necessary tool. In this paper we describe a search method based on the requirement that standard type three-and four-soliton solution exist in the bilinear formalism of Hirota. The existence on N-soliton solutions can be formulated as a requirement that a certain high degree polynomial in N x A4 variables vanishes on an affine manifold defined by N polynomials of M vi~riables each. An exhaustive search has been carried out for certain classes of typical equations and several new equations have been found. Permission to copy without fee all or part of this material is granted provided that the copias are not made or distributed for diract commercial advantage, tha ACM copyright notice and the titla of the publication and its date appear, and notice is given that copying ia by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requiresa foe and/or specific permission.
Physical Review D, 1980
We give a second-quantized model in which, using link operators, quarks and antiquarks are perman... more We give a second-quantized model in which, using link operators, quarks and antiquarks are permanently confined without color van der Waals forces between hadrons. We also sketch how to include short-range interactions between hadrons, quark-antiquark creation and ...
Lecture Notes in Physics
Integrable dynamical systems have been studied quite actively in the recents years. Most of the t... more Integrable dynamical systems have been studied quite actively in the recents years. Most of the time the system has been a classical one. Here we will discussthedifferences ofclassical and quantum integrability from an algebraic viewpoint. (For more details see [1].)
Chaos Soliton Fractal, 2000
Symmetries and Integrability of Difference Equations
Preliminaries Singularity confinement and algebraic entropy Integrability in 2D Preliminaries Sin... more Preliminaries Singularity confinement and algebraic entropy Integrability in 2D Preliminaries Singularity confinement and algebraic entropy Integrability in 2D Generalities Singularity confinement in projective space Singularity confinement vs. growth of complexity Jarmo Hietarinta Definitions of Integrability Preliminaries Singularity confinement and algebraic entropy Integrability in 2D Generalities Singularity confinement in projective space Singularity confinement vs. growth of complexity Singularity analysis for difference equations Grammaticos, Ramani, and Papageorgiou, [Phys. Rev. Lett. 67 (1991) 1825] proposed The Singularity Confinement Criterion as an analogue of the Painleve test. Idea: If the dynamics leads to a singularity then after a few steps one should be able to get out of it (confinement), and this should take place without loss of information.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2003
A normal human heart rate shows complex fluctuations in time, which is natural, because the heart... more A normal human heart rate shows complex fluctuations in time, which is natural, because the heart rate is controlled by a large number of different feedback control loops. These unpredictable fluctuations have been shown to display fractal dynamics, long-term correlations, and 1/f noise. These characterizations are statistical and they have been widely studied and used, but much less is known about the detailed time evolution (dynamics) of the heart-rate control mechanism. Here we show that a simple one-dimensional Langevin-type stochastic difference equation can accurately model the heart-rate fluctuations in a time scale from minutes to hours. The model consists of a deterministic nonlinear part and a stochastic part typical to Gaussian noise, and both parts can be directly determined from the measured heart-rate data. Studies of 27 healthy subjects reveal that in most cases, the deterministic part has a form typically seen in bistable systems: there are two stable fixed points an...
Physical review. E, Statistical, nonlinear, and soft matter physics, 2003
Higher-order and multicomponent generalizations of the nonlinear Schrödinger equation are importa... more Higher-order and multicomponent generalizations of the nonlinear Schrödinger equation are important in various applications, e.g., in optics. One of these equations, the integrable Sasa-Satsuma equation, has particularly interesting soliton solutions. Unfortunately, the construction of multisoliton solutions to this equation presents difficulties due to its complicated bilinearization. We discuss briefly some previous attempts and then give the correct bilinearization based on the interpretation of the Sasa-Satsuma equation as a reduction of the three-component Kadomtsev-Petviashvili hierarchy. In the process, we also get bilinearizations and multisoliton formulas for a two-component generalization of the Sasa-Satsuma equation (the Yajima-Oikawa-Tasgal-Potasek model), and for a (2+1)-dimensional generalization.
AIP Conference Proceedings, 2010
ABSTRACT In this paper we discuss solutions in Casoratian form for H1, which is the simplest memb... more ABSTRACT In this paper we discuss solutions in Casoratian form for H1, which is the simplest member in ABS list of lattice equations. By investigating the condition satisfied by the Casoratian basic column we propose a generalization, which yields solutions which are different form solitons. These solutions can be considered as limit solutions of solitons. Similar generalizations can apply to other lattice equations in ABS list, such as H2, H3 and Q1. Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Keywords (in text query field) Abstract Text Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
The Painlevé Property, 1999
In these lectures we discuss how the Painlevé equations can be written in terms of entire functio... more In these lectures we discuss how the Painlevé equations can be written in terms of entire functions, and then in the Hirota bilinear (or multilinear) form. Hirota's method, which has been so useful in soliton theory, is reviewed and connections from soliton equations to Painlevé equations through similarity reductions are discussed from this point of view. In the main part we discuss how singularity structure of the solutions and formal integration of the Painlevé equations can be used to find a representation in terms of entire functions. Sometimes the final result is a pair of Hirota bilinear equations, but for P V I we need also a quadrilinear expression. The use of discrete versions of Painlevé equations is also discussed briefly. It turns out that with discrete equations one gets better information on the singularities, which can then be represented in terms of functions with a simple zero.
Scattering, 2002
Publisher Summary Certain partial differential equations have localized stable traveling-wave sol... more Publisher Summary Certain partial differential equations have localized stable traveling-wave solutions. If the underlying system is integrable, there are infinitely many conserved quantities and this in turn implies that these traveling waves—solitons—in fact scatter elastically. The chapter focuses on the scattering properties of solitons. Integrability is a very strong property and has many important consequences, including the possibility of solving the initial value problem using the inverse scattering transform method. It is, however, possible to study soliton scattering even if the system is not completely integrable. There are many other soliton equations with two-soliton solution (2SS) of different structure, but still have very much the same behavior under scattering. Some coupled soliton equations have two different kinds of soliton solutions. One of such equations—Hirota–Satsuma equation—is presented.
A unit-vector-field defined everywhere in the 3D-space (and pointing to the same direction far aw... more A unit-vector-field defined everywhere in the 3D-space (and pointing to the same direction far away) can be partially characterized by the topological Hopf charge. Further geometric information is given by the curve at which the vector field points to a direction opposite to the direction at infinity. This curve can form knots. Given an initial topologically nontrivial vector-field we follow
Physics Letters A, 1998
The correspondence between the integrability of classical mechanical systems and their quantum co... more The correspondence between the integrability of classical mechanical systems and their quantum counterparts is not a 1-1, although some close correspondencies exist. If a classical mechanical system is integrable with invariants that are polynomial in momenta one can construct a corresponding commuting set of differential operators. Here we discuss some 2-or 3-dimensional purely quantum integrable systems (the 1-dimensional counterpart is the Lame equation). That is, we have an integrable potential whose amplitude is not free but rather proportional toh 2 , and in the classical limit the potential vanishes. Furthermore it turns out that some of these systems actually have N + 1 commuting differential operators, connected by a nontrivial algebraic relation. Some of them have been discussed recently by A.P. Veselov et. al. from the point of view of Baker-Akheizer functions.
Physical Review Letters, 1978
We show that pseudoparticle solutions with integer winding number do not generate mass terms or b... more We show that pseudoparticle solutions with integer winding number do not generate mass terms or breaking of chiral SU(N) in theories with a non-trivial flavor group.
Physical Review D, 1982
We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator M(p,... more We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator M(p, q) one can associate (in a given ordering rule) a c-number function A (p, q). It is this function that appears in the action of the phase-space path integral. A quantum canonical transformation A-+A ' can now be expressed as an integral transformation A (p, q) = dp dqW(p, q;p, q)Pi (p, q). The kernel W is constructed explicitly for point transformations and for the p =q, q =p reflection by studying changes of variables in the path integral. The ordering dependence of W is displayed. The invariance of commutation rules is also discussed.
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Papers by Jarmo Hietarinta