We show that solution to the Hermite-Padé type I approximation problem leads in a natural way to ... more We show that solution to the Hermite-Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorthms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Padé approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite-Padé problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.
We show that the Wynn recurrence (the missing identity of Frobenius of the Padé approximation the... more We show that the Wynn recurrence (the missing identity of Frobenius of the Padé approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev-Petviashvili equation. This allows, in particular, to present the geometric meaning of the recurrence as a construction of the appropriately constrained quadrangular set of points. The interpretation is valid for a projective line over arbitrary skew field what motivates to consider non-commutative Padé theory. We transfer the corresponding elements, including the Frobenius identities, to the non-commutative level using the quasideterminants. Using an example of the characteristic series of the Fibonacci language we present an application of the theory to the regular languages. We introduce the non-commutative version of the discrete-time Toda lattice equations together with their integrability structure. Finally, we discuss application of the Wynn recurrence in a different context of the geometric theory of discrete analytic functions.
We introduce and solve the non-commutative version of the Hermite-Padé type I approximation probl... more We introduce and solve the non-commutative version of the Hermite-Padé type I approximation problem. Its solution, expressed by quasideterminants, leads in a natural way to a subclass of solutions of the non-commutative Hirota (discrete Kadomtsev-Petviashvili) system and of its linear problem. We also prove integrability of the constrained system, which in the simplest case is the non-commutative discrete-time Toda lattice equation known from the theory of non-commutative Padé approximants and matrix orthogonal polynomials.
We investigate the τ-function of the quadrilateral lattice using the nonlocal ¯ ∂-dressing method... more We investigate the τ-function of the quadrilateral lattice using the nonlocal ¯ ∂-dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach.
With this paper we begin an investigation of difference schemes that possess Darboux transformati... more With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux transformations for the general self adjoint schemes with five and seven neighbouring points. We also introduce a distinguished discretization of the two-dimensional stationary Schrödinger equation, described by a 5-point difference scheme involving two potentials, which admits a Darboux transformation.
We introduce a generalization of the Hopf algebra of quasi-symmetric functions in terms of power ... more We introduce a generalization of the Hopf algebra of quasi-symmetric functions in terms of power series in partially commutative variables. This is the graded dual of the Hopf algebra of coloured non-commutative symmetric functions described as a subalgebra of the Hopf algebra of rooted ordered coloured trees. In the Appendix we discuss the role of partial commutativity in derivation of Weyl commutation relations.
The main objective of this paper is to derive the Enneper-Weierstrass representation of minimal s... more The main objective of this paper is to derive the Enneper-Weierstrass representation of minimal surfaces in $\mathbb{E}^3$ using the soliton surface approach. We exploit the Bryant-type representation of conformally parametrized surfaces in the hyperbolic space $H^3(\lambda)$ of curvature $-\lambda^2$, which can be interpreted as a 2 by 2 linear problem involving the spectral parameter $\lambda$. In the particular case of constant mean curvature-$\lambda$ surfaces a special limiting procedure $(\lambda\rightarrow 0)$, different from that of Umehara and Yamada [33], allows us to recover the Enneper-Weierstrass representation. Applying such a limiting procedure to the previously known cases, we obtain Sym-type formulas. Finally we exploit the relation between the Bryant representation of constant mean curvature-$\lambda$ surfaces and second-order linear ordinary differential equations. We illustrate this approach by the example of the error function equation.
Journal of Physics A: Mathematical and Theoretical, 2021
The discrete non-commutative Darboux system of equations with self-consistent sources is construc... more The discrete non-commutative Darboux system of equations with self-consistent sources is constructed, utilizing both the vectorial fundamental (binary Darboux) transformation and the method of additional independent variables. Then the symmetric reduction of discrete Darboux equations with sources is presented. In order to provide a simpler version of the resulting equations we introduce the τ/σ form of the (commutative) discrete Darboux system. Our equations give, in continuous limit, the version with self-consistent sources of the classical symmetric Darboux system.
Journal of Physics A: Mathematical and Theoretical, 2020
We study double-sided continued fractions whose coefficients are non-commuting symbols. We work w... more We study double-sided continued fractions whose coefficients are non-commuting symbols. We work within the formal approach of the Mal'cev-Neumann series and free division rings. We start with presenting the analogs of the standard results from the theory of continued fractions, including their (right and left) simple fractions decomposition, the Euler-Minding summation formulas, and the relations between nominators and denominators of the simple fraction decompositions. We also transfer to the non-commutative double-sided setting the standard description of the continued fractions in terms of 2 × 2 matrices presenting also a weak version of the Serret theorem. The equivalence transformations between the double continued fractions are described, including also the transformation from generic such fractions to their simplest form. Then we give the description of the double-sided continued fractions within the theory of quasideterminants and we present the corresponding version of the LR and qd-algorithms. We study also (strictly and ultimately) periodic double-sided non-commutative continued fractions and we give the corresponding version of the Euler theorem. Finally we present a weak version of the Galois theorem and we give its relation to the non-commutative KP map, recently studied in the theory of discrete integrable systems.
Motivated by the classical studies on transformations of conjugate nets, we develop the general g... more Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:ZN→RM, N⩽M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Lévy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between “transformations” and “discretizations” is also investigated for quadrilateral lattices. We finally interpret these results within the ∂̄ formalism.
Springer Proceedings in Mathematics & Statistics, 2020
We present geometric interpretation of the discrete Schwarzian Kadomtsev-Petviashvili equation in... more We present geometric interpretation of the discrete Schwarzian Kadomtsev-Petviashvili equation in terms of quadrangular set of points of a projective line. We give also the corresponding interpretation for the projective line considered as a Moebius chain space. In this way we incorporate the conformal geometry interpretation of the equation into the projective geometry approach via Desargues maps.
We give an action of the symmetric group on non-commuting indeterminates in terms of series in th... more We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Mal’cev–Newmann division ring. The action is constructed from the non-Abelian Hirota–Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case, and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.
We investigate the τ-function of the quadrilateral lattice using the nonlocal∂-dressing method, a... more We investigate the τ-function of the quadrilateral lattice using the nonlocal∂-dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach.
We study a potential introduced by Darboux to describe conjugate nets, which within the modern th... more We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a $\tau$-function. We investigate the potential using the non-local $\bar\partial$ dressing method of Manakov and Zakharov, and we show that it can be interpreted as the Fredholm determinant of an integral equation which naturally appears within that approach. Finally, we give some arguments extending that interpretation to multicomponent Kadomtsev-Petviashvili hierarchy.
We review recent results on asymptotic lattices and their integrable reductions. We present the t... more We review recent results on asymptotic lattices and their integrable reductions. We present the theory of general asymptotic lattices in R 3 together with the corresponding theory of their Darboux-type transformations. Then we study the discrete analogues of the Bianchi surfaces and their transformations. Finally, we present the corresponding theory of the discrete analogues of the isothermallyasymptotic (Fubini-Ragazzi) nets.
We show that solution to the Hermite-Padé type I approximation problem leads in a natural way to ... more We show that solution to the Hermite-Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorthms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Padé approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite-Padé problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.
We show that the Wynn recurrence (the missing identity of Frobenius of the Padé approximation the... more We show that the Wynn recurrence (the missing identity of Frobenius of the Padé approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev-Petviashvili equation. This allows, in particular, to present the geometric meaning of the recurrence as a construction of the appropriately constrained quadrangular set of points. The interpretation is valid for a projective line over arbitrary skew field what motivates to consider non-commutative Padé theory. We transfer the corresponding elements, including the Frobenius identities, to the non-commutative level using the quasideterminants. Using an example of the characteristic series of the Fibonacci language we present an application of the theory to the regular languages. We introduce the non-commutative version of the discrete-time Toda lattice equations together with their integrability structure. Finally, we discuss application of the Wynn recurrence in a different context of the geometric theory of discrete analytic functions.
We introduce and solve the non-commutative version of the Hermite-Padé type I approximation probl... more We introduce and solve the non-commutative version of the Hermite-Padé type I approximation problem. Its solution, expressed by quasideterminants, leads in a natural way to a subclass of solutions of the non-commutative Hirota (discrete Kadomtsev-Petviashvili) system and of its linear problem. We also prove integrability of the constrained system, which in the simplest case is the non-commutative discrete-time Toda lattice equation known from the theory of non-commutative Padé approximants and matrix orthogonal polynomials.
We investigate the τ-function of the quadrilateral lattice using the nonlocal ¯ ∂-dressing method... more We investigate the τ-function of the quadrilateral lattice using the nonlocal ¯ ∂-dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach.
With this paper we begin an investigation of difference schemes that possess Darboux transformati... more With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux transformations for the general self adjoint schemes with five and seven neighbouring points. We also introduce a distinguished discretization of the two-dimensional stationary Schrödinger equation, described by a 5-point difference scheme involving two potentials, which admits a Darboux transformation.
We introduce a generalization of the Hopf algebra of quasi-symmetric functions in terms of power ... more We introduce a generalization of the Hopf algebra of quasi-symmetric functions in terms of power series in partially commutative variables. This is the graded dual of the Hopf algebra of coloured non-commutative symmetric functions described as a subalgebra of the Hopf algebra of rooted ordered coloured trees. In the Appendix we discuss the role of partial commutativity in derivation of Weyl commutation relations.
The main objective of this paper is to derive the Enneper-Weierstrass representation of minimal s... more The main objective of this paper is to derive the Enneper-Weierstrass representation of minimal surfaces in $\mathbb{E}^3$ using the soliton surface approach. We exploit the Bryant-type representation of conformally parametrized surfaces in the hyperbolic space $H^3(\lambda)$ of curvature $-\lambda^2$, which can be interpreted as a 2 by 2 linear problem involving the spectral parameter $\lambda$. In the particular case of constant mean curvature-$\lambda$ surfaces a special limiting procedure $(\lambda\rightarrow 0)$, different from that of Umehara and Yamada [33], allows us to recover the Enneper-Weierstrass representation. Applying such a limiting procedure to the previously known cases, we obtain Sym-type formulas. Finally we exploit the relation between the Bryant representation of constant mean curvature-$\lambda$ surfaces and second-order linear ordinary differential equations. We illustrate this approach by the example of the error function equation.
Journal of Physics A: Mathematical and Theoretical, 2021
The discrete non-commutative Darboux system of equations with self-consistent sources is construc... more The discrete non-commutative Darboux system of equations with self-consistent sources is constructed, utilizing both the vectorial fundamental (binary Darboux) transformation and the method of additional independent variables. Then the symmetric reduction of discrete Darboux equations with sources is presented. In order to provide a simpler version of the resulting equations we introduce the τ/σ form of the (commutative) discrete Darboux system. Our equations give, in continuous limit, the version with self-consistent sources of the classical symmetric Darboux system.
Journal of Physics A: Mathematical and Theoretical, 2020
We study double-sided continued fractions whose coefficients are non-commuting symbols. We work w... more We study double-sided continued fractions whose coefficients are non-commuting symbols. We work within the formal approach of the Mal'cev-Neumann series and free division rings. We start with presenting the analogs of the standard results from the theory of continued fractions, including their (right and left) simple fractions decomposition, the Euler-Minding summation formulas, and the relations between nominators and denominators of the simple fraction decompositions. We also transfer to the non-commutative double-sided setting the standard description of the continued fractions in terms of 2 × 2 matrices presenting also a weak version of the Serret theorem. The equivalence transformations between the double continued fractions are described, including also the transformation from generic such fractions to their simplest form. Then we give the description of the double-sided continued fractions within the theory of quasideterminants and we present the corresponding version of the LR and qd-algorithms. We study also (strictly and ultimately) periodic double-sided non-commutative continued fractions and we give the corresponding version of the Euler theorem. Finally we present a weak version of the Galois theorem and we give its relation to the non-commutative KP map, recently studied in the theory of discrete integrable systems.
Motivated by the classical studies on transformations of conjugate nets, we develop the general g... more Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:ZN→RM, N⩽M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Lévy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between “transformations” and “discretizations” is also investigated for quadrilateral lattices. We finally interpret these results within the ∂̄ formalism.
Springer Proceedings in Mathematics & Statistics, 2020
We present geometric interpretation of the discrete Schwarzian Kadomtsev-Petviashvili equation in... more We present geometric interpretation of the discrete Schwarzian Kadomtsev-Petviashvili equation in terms of quadrangular set of points of a projective line. We give also the corresponding interpretation for the projective line considered as a Moebius chain space. In this way we incorporate the conformal geometry interpretation of the equation into the projective geometry approach via Desargues maps.
We give an action of the symmetric group on non-commuting indeterminates in terms of series in th... more We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Mal’cev–Newmann division ring. The action is constructed from the non-Abelian Hirota–Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case, and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.
We investigate the τ-function of the quadrilateral lattice using the nonlocal∂-dressing method, a... more We investigate the τ-function of the quadrilateral lattice using the nonlocal∂-dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach.
We study a potential introduced by Darboux to describe conjugate nets, which within the modern th... more We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a $\tau$-function. We investigate the potential using the non-local $\bar\partial$ dressing method of Manakov and Zakharov, and we show that it can be interpreted as the Fredholm determinant of an integral equation which naturally appears within that approach. Finally, we give some arguments extending that interpretation to multicomponent Kadomtsev-Petviashvili hierarchy.
We review recent results on asymptotic lattices and their integrable reductions. We present the t... more We review recent results on asymptotic lattices and their integrable reductions. We present the theory of general asymptotic lattices in R 3 together with the corresponding theory of their Darboux-type transformations. Then we study the discrete analogues of the Bianchi surfaces and their transformations. Finally, we present the corresponding theory of the discrete analogues of the isothermallyasymptotic (Fubini-Ragazzi) nets.
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Papers by Adam Doliwa