On symplectic and isospectral integration of the stationary Landau-Lifshitz (Neumann oscillator) equation

In this paper we discuss numerical integration of the stationary Landau–Lifshitz (LL) equation. Using a Lax pair representation of the LL equation, we propose an isospectral algorithm that preserves the geometric structure of the system. The algorithm computes a discrete flow of a pair of matrices satisfying Lax–type equations and projects the flow on the phase space of the system. Since the stationary LL equation is equivalent to an integrable Hamiltonian system on the cotangent bundle of the unit sphere, we show that it can be also integrated by a symplectic method for constrained Hamiltonian systems. Comparison of the two methods demonstrates that they are similar in terms of accuracy and stability over long–time integration, but the isospectral method is much faster since it avoids solving a system of nonlinear equations required at each iteration of the symplectic algorithm

On symplectic and isospectral integration of the stationary Landau-Lifshitz (Neumann oscillator) equation

In this paper we discuss numerical integration of the stationary Landau–Lifshitz (LL) equation. Using a Lax pair representation of the LL equation, we propose an isospectral algorithm that preserves the geometric structure of the system. The algorithm computes a discrete flow of a pair of matrices satisfying Lax–type equations and projects the flow on the phase space of the system. Since the stationary LL equation is equivalent to an integrable Hamiltonian system on the cotangent bundle of the unit sphere, we show that it can be also integrated by a symplectic method for constrained Hamiltonian systems. Comparison of the two methods demonstrates that they are similar in terms of accuracy and stability over long–time integration, but the isospectral method is much faster since it avoids solving a system of nonlinear equations required at each iteration of the symplectic algorithm

Sturm-Liouvilleov problem

Klasična Sturm-Liouvilleova jednadžba, nazvana po Jacquesu Sturmu i Josephu Liouvilleu, običcna je diferencijalna jednadžba drugog reda posebnoga oblika. Vrijednost lambda nije odredena i pronalaženje te vrijednosti za koju postoje netrivijalna rješenja jednadžbe i koja zadovoljavaju rubne uvjete dio je problema koji nazivamo Sturm-Liouvilleov problem. Pokazat ćemo da se proizvoljni linearni operator drugog reda može transformirati u Sturm- Liouvilleov operator tj. da je Sturm-Liouvilleov operator kanonski oblik diferencijalnog operatora drugog reda. Vlastite vrijednosti regularnog Sturm-Liouvilleovog problema su realne, prebrojive i tvore strogo rastući niz lamda1 < lambda2 < : : : limes kojega je beskonačno. Također, za svaku vlastitu vrijednost lambda n postoji odgovarajuća vlastita funkcija un(x) jedinstveno određena do na multiplikativnu konstantu koja ima točno n nultočaka u intervalu [a; b]. Ovo je jedan od fundamentalnih rezultata za Sturm-Liouvilleove operatore

On the Birkhoff factorization problem for the Heisenberg magnet and nonlinear Schroedinger equations

A geometrical description of the Heisenberg magnet (HM) equation with classical spins is given in terms of flows on the quotient space $G/H_+$ where $G$ is an infinite dimensional Lie group and $H_+$ is a subgroup of $G$. It is shown that the HM flows are induced by an action of $\mathbb{R}^2$ on $G/H_+$, and that the HM equation can be integrated by solving a Birkhoff factorization problem for $G$. For the HM flows which are Laurent polynomials in the spectral variable we derive an algebraic transformation between solutions of the nonlinear Schroedinger (NLS) and Heisenberg magnet equations. The Birkhoff factorization for $G$ is treated in terms of the geometry of the Segal-Wilson Grassmannian $Gr(H)$. The solution of the problem is given in terms of a pair of Baker functions for special subspaces of $Gr(H)$. The Baker functions are constructed explicitly for subspaces which yield multisoliton solutions of NLS and HM equations.Comment: To appear in Journal of Mathematical Physic

Differential algebras on kappa-Minkowski space and action of the Lorentz algebra

We propose two families of differential algebras of classical dimension on kappa-Minkowski space. The algebras are constructed using realizations of the generators as formal power series in a Weyl super-algebra. We also propose a novel realization of the Lorentz algebra so(1,n-1) in terms of Grassmann-type variables. Using this realization we construct an action of so(1,n-1) on the two families of algebras. Restriction of the action to kappa-Minkowski space is covariant. In contrast to the standard approach the action is not Lorentz covariant except on constant one-forms, but it does not require an extra cotangent direction.Comment: 16 page

Differential structure on kappa-Minkowski space, and kappa-Poincare algebra

We construct realizations of the generators of the $\kappa$-Minkowski space and $\kappa$-Poincar\'{e} algebra as formal power series in the $h$-adic extension of the Weyl algebra. The Hopf algebra structure of the $\kappa$-Poincar\'{e} algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on $\kappa$-Minkowski space which is compatible with the action of the Lorentz algebra. In contrast to the conventional bicovariant calculus, the space of one-forms has the same dimension as the $\kappa$-Minkowski space.Comment: 20 pages. Accepted for publication in International Journal of Modern Physics

Gauge transformations and symmetries of integrable systems

We analyze several integrable systems in zero-curvature form within the framework of $SL(2,\R)$ invariant gauge theory. In the Drienfeld-Sokolov gauge we derive a two-parameter family of nonlinear evolution equations which as special cases include the Kortweg-de Vries (KdV) and Harry Dym equations. We find residual gauge transformations which lead to infinintesimal symmetries of this family of equations. For KdV and Harry Dym equations we find an infinite hierarchy of such symmetry transformations, and we investigate their relation with local conservation laws, constants of the motion and the bi-Hamiltonian structure of the equations. Applying successive gauge transformatinos of Miura type we obtain a sequence of gauge equivalent integrable systems, among them the modified KdV and Calogero KdV equations.Comment: 18pages, no figure Journal versio

Noncommutative Differential Forms on the kappa-deformed Space

We construct a differential algebra of forms on the kappa-deformed space. For a given realization of the noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of one-forms and nilpotent exterior derivatives. We derive explicit expressions for the exterior derivative and one-forms in covariant and noncovariant realizations. We also introduce higher-order forms and show that the exterior derivative satisfies the graded Leibniz rule. The differential forms are generally not graded-commutative, but they satisfy the graded Jacobi identity. We also consider the star-product of classical differential forms. The star-product is well-defined if the commutator between the noncommutative coordinates and one-forms is closed in the space of one-forms alone. In addition, we show that in certain realizations the exterior derivative acting on the star-product satisfies the undeformed Leibniz rule.Comment: to appear in J. Phys. A: Math. Theo

Generalized kappa-deformed spaces, star-products, and their realizations

In this work we investigate generalized kappa-deformed spaces. We develop a systematic method for constructing realizations of noncommutative (NC) coordinates as formal power series in the Weyl algebra. All realizations are related by a group of similarity transformations, and to each realization we associate a unique ordering prescription. Generalized derivatives, the Leibniz rule and coproduct, as well as the star-product are found in all realizations. The star-product and Drinfel'd twist operator are given in terms of the coproduct, and the twist operator is derived explicitly in special realizations. The theory is applied to a Nappi-Witten type of NC space