Jarmo Hietarinta

Direct Methods for the Search of the Second Invariant

Abstract We discuss the direct methods that can be used to search for the second invariant of a system defined by the Hamiltonian H = 1 2 (p x 2 ) + p y 2 + A(x, y)p x + B(x, y)p y + V(x, y) . We give an extensive review of those systems that are known to have an invariant that is polynomial in the p s (most of these have A = B = 0). In addition we introduce the field of non-polynomial invariants by giving several new systems that have a rational or transcendental (in the p s) invariant (for these A and B are nonzero). The special case of integrability at a fixed value of the energy is also discussed.

A search for bilinear equations passing Hirota’s three‐soliton condition. II. mKdV‐type bilinear equations

In this paper the results of a search for complex bilinear equations with two‐soliton solutions are presented. The following basic types are discussed: (a) the nonlinear Schrodinger equation B(Dx, ...)G⋅F=0, A(Dx,Dt) F⋅F=GG*, and (b) the Benjamin–Ono equation P(Dx, ...)F⋅F*=0. It is found that the existence of two‐soliton solutions is not automatic, but introduces conditions that are like the usual three‐ and four‐soliton conditions. The search was limited by the degree of A=2, and by degree of P≤4. The main results are the following: (1) (iaD3x+DxDt +iDy+b)G⋅F=0, D2xF⋅F=GG*; (2) (D2x+aD2y +iDt+b)G⋅F =0, DxDy F⋅F=GG*; (3) (iaD3x+D2x +iDt)F⋅F*=0; and (4) (DxDt+i(aDx +bDt))F⋅F*=0.

Multidromion solutions to the Davey-Stewartson equation

Abstract Recently Boiti et al. have shown that the Davey-Stewartson equation can have solutions (called “dromions”) that decay exponentially in all directions. These solutions were later generalized by Fokas and Santini. We construct an N 2 -dromion solution using double-Wronskians in the bilinear formalism. The solution is characterized by a 2 N × 2 N symplectic matrix, and it contains more free parameters than has been obtained so far by other methods.

Faddeev-Hopf knots: dynamics of linked un-knots

Abstract We have studied numerically Faddeev-Hopf knots which are defined as those unit-vector fields in R 3 that have a nontrivial Hopf charge and minimize Faddeevs Lagrangian. A given initial configuration was allowed to relax into a (local) minimum using the first order dissipative dynamics corresponding to the steepest descent method. A linked combination of two un-knots was seen to relax into different minimum energy configurations depending on their charges and their relative handedness and direction. In order to visualize the results we plot certain gauge-invariant iso-surfaces.

Classical versus quantum integrability

Classical integrability and quantum integrability are compared for two degrees of freedom Hamiltonian systems. We use c‐number representatives for quantum operators and the Moyal bracket for the commutator. Three different cases are found: (i) the c‐number representative of the quantum mechanical second invariant is identical to the classical second invariant, (ii) O(ℏ2) corrections are needed in the classical second invariant to obtain the quantum invariant, and (iii) also the potential must be deformed by an O(ℏ2) term. Several examples from the Henon–Heiles and Holt families of integrable potentials are included.

Searching for CAC-maps

Abstract For two-dimensional lattice equations the standard definition of integrability is that it should be possible to extend the map consistently to three dimensions, i.e., that it is “consistent around a cube” (CAC). Recently Adler, Bobenko and Suris conducted a search based on this principle, together with the additional assumptions of symmetry and “the tetrahedron property”. We present here results of a search for CAC lattice equations assuming also the same symmetry properties, but not the tetrahedron property.

A search for integrable two-dimensional hamiltonian systems with polynomial potential

Abstract We report the results of a search for all integrable hamiltonian systems of type H = ( 1 2 )p x 2 + ( 1 2 )p y 2 + V(x,y) , where V is a polynomial in x and y of degree 5 or less and the second invariant is a polynomial in px and py of order 4 or less. Both classical and quantum integrability are discussed.

Solving the two-dimensional constant quantum Yang-Baxter equation

A detailed analysis of the constant quantum Yang–Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk2k3j2j3 Rk1l3j1k3Rl1l2k1k2 in two dimensions is presented, leading to an exhaustive list of its solutions. The set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the Grobner‐basis methods. Each solution was then transformed into a canonical form (based on the various trace matrices of R) for final elimination of duplicates and subcases. If we use homogeneous parametrization the solutions can be combined into 23 distinct cases, modulo the well‐known C, P, and T reflections, and rotations and scalings R=κ(Q⊗Q)R(Q⊗Q)−1.

All solutions to the constant quantum Yang-Baxter equation in two dimensions

Abstract In this Letter we present an exhaustive list of solutions to the constant quantum Yang-Baxter equation R k 1 k 2 j 1 j 2 R l 1 k 3 k 1 j 3 R l 2 l 3 k 2 k 3 = R k 2 k 3 j 2 j 3 R k 1 l 3 j 1 k 3 R l 1 l 2 k 1 k 2 in two dimensions (i.e. all indices ranging over 1, 2 with summation over repeated indices). This set of 64 equations for 16 unknowns was first reduced by hand to manageable subcases which were then solved by computer using Grobner basis methods. For the presentation of the results we propose a canonical form based on the various trace matrices of R .

One-dromion solutions for genetic classes of equations

Abstract We show that dromion solutions (two-dimensional localized bumps that decay exponentially in all directions) exist for generic (2+1)-dimensional bilinear equations of nonlinear Schrodinger and Korteweg-de Vries type.