Searcharxiv⌕ Search

arXiv subjects

Takayuki Tsuchida

Publications and source records attributed to Takayuki Tsuchida.

At least 19 recordsLinked to original sources
arXiv2025

Integrable semi-discretizations of the sine-Gordon equation in non-characteristic coordinates

Integrable discretizations of the sine-Gordon equation in characteristic (or light-cone) coordinates have been extensively studied after the seminal works of Hirota and Orfanidis in the late 1970s. In contrast, integrable discretizations of the sine-Gordon equation in non-characteristic coordinates have been scarcely studied except the lattice sine-Gordon model proposed by Izergin and Korepin in the early 1980s. In this paper, using the zero-curvature representation, we propose integrable space discretizations of the sine-Gordon equation in three distinct cases of non-characteristic coordinates. For the most interesting case of the sine-Gordon equation in laboratory coordinates, the integrable space discretization is unwieldy; as a remedy, we rewrite the sine-Gordon equation as a two-component evolutionary system and present an aesthetically acceptable space discretization.

nlin.SI↗
arXiv2025

On an integrable discretization of the massive Thirring model in light-cone coordinates and the associated Yang-Baxter map

We propose a fully discrete analog of the massive Thirring model in light-cone coordinates by constructing its Lax-pair representation. This Lax-pair representation can also be used to define a new Yang-Baxter map, so we obtain a Yang-Baxter map that admits a continuous limit. We present most of the results for the general case where the dependent variables are matrix-valued.

nlin.SI↗
arXiv2024

Integrable discretization of the vector/matrix nonlinear Schrödinger equation and the associated Yang-Baxter map

The action of a Bäcklund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary Bäcklund-Darboux (or Zakharov-Shabat dressing) transformation for the nonlinear Schrödinger (NLS) hierarchy as a discrete spectral problem, wherein the two intermediate potentials appearing in the Darboux matrix are considered as a pair of new dependent variables. Then, we associate the discrete spectral problem with a suitable isospectral time-evolution equation, which forms the Lax-pair representation for a space-discrete NLS system. This formulation is valid for the most general case where the two dependent variables take values in (rectangular) matrices. In contrast to the matrix generalization of the Ablowitz-Ladik lattice, our discretization has a rational nonlinearity and admits a Hermitian conjugation reduction between the two dependent variables. Thus, a new proper space-discretization of the vector/matrix NLS equation is obtained; by changing the time part of the Lax pair, we also obtain an integrable space-discretization of the vector/matrix modified KdV (mKdV) equation. Because Bäcklund-Darboux transformations are permutable, we can increase the number of discrete independent variables in a multi-dimensionally consistent way. By solving the consistency condition on the two-dimensional lattice, we obtain a Yang-Baxter map of the NLS type, which can be considered as a fully discrete analog of the principal chiral model for projection matrices.

nlin.SI↗
arXiv2020

Integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system. II

This is a continuation of our previous paper arXiv:1904.07924, which is devoted to the construction of integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system; in this series of papers, we refer to a discretization of one of the two spatial variables as a semi-discretization. In this paper, we construct an integrable semi-discrete Davey-Stewartson system, which is essentially different from the semi-discrete Davey-Stewartson system proposed in the previous paper arXiv:1904.07924. We first obtain integrable semi-discretizations of the two elementary flows that compose the Davey-Stewartson system by constructing their Lax-pair representations and show that these two elementary flows commute as in the continuous case. Then, we consider a linear combination of the two elementary flows to obtain a new integrable semi-discretization of the Davey-Stewartson system. Using a linear transformation of the continuous independent variables, one of the two elementary Davey-Stewartson flows can be identified with an integrable semi-discretization of the $(2+1)$-dimensional Yajima-Oikawa system proposed in https://link.aps.org/doi/10.1103/PhysRevE.91.062902 .

nlin.SI↗
arXiv2020

Integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system. I

The integrable Davey-Stewartson system is a linear combination of the two elementary flows that commute: $\mathrm{i} q_{t_1} + q_{xx} + 2q\partial_y^{-1}\partial_x (|q|^2) =0$ and $\mathrm{i} q_{t_2} + q_{yy} + 2q\partial_x^{-1}\partial_y (|q|^2) =0$. In the literature, each elementary Davey-Stewartson flow is often called the Fokas system because it was studied by Fokas in the early 1990s. In fact, the integrability of the Davey-Stewartson system dates back to the work of Ablowitz and Haberman in 1975; the elementary Davey-Stewartson flows, as well as another integrable $(2+1)$-dimensional nonlinear Schrödinger equation $\mathrm{i} q_{t} + q_{xy} + 2 q\partial_y^{-1}\partial_x (|q|^2) =0$ proposed by Calogero and Degasperis in 1976, appeared explicitly in Zakharov's article published in 1980. By applying a linear change of the independent variables, an elementary Davey-Stewartson flow can be identified with a $(2+1)$-dimensional generalization of the integrable long wave-short wave interaction model, called the Yajima-Oikawa system: $\mathrm{i} q_{t} + q_{xx} + u q=0$, $u_t + c u_y = 2(|q|^2)_x$. In this paper, we propose a new integrable semi-discretization (discretization of one of the two spatial variables, say $x$) of the Davey-Stewartson system by constructing its Lax-pair representation; the two elementary flows in the semi-discrete case indeed commute. By applying a linear change of the continuous independent variables to an elementary flow, we also obtain an integrable semi-discretization of the $(2+1)$-dimensional Yajima-Oikawa system.

nlin.SI↗
arXiv2018

Integrability of a discrete Yajima-Oikawa system

A space discretization of an integrable long wave-short wave interaction model, called the Yajima-Oikawa system, was proposed in the recent paper arXiv:1509.06996 using the Hirota bilinear method (see also https://link.aps.org/doi/10.1103/PhysRevE.91.062902). In this paper, we propose a Lax-pair representation for the discrete Yajima-Oikawa system as well as its multicomponent generalization also considered in arXiv:1509.06996 and prove that it has an infinite number of conservation laws. We also derive the next higher flow of the discrete Yajima-Oikawa hierarchy, which generalizes a modified version of the Volterra lattice. Relations to two integrable discrete nonlinear Schrödinger hierarchies, the Ablowitz-Ladik hierarchy and the Konopelchenko-Chudnovsky hierarchy, are clarified.

nlin.SI↗
arXiv2018

On a new integrable generalization of the Toda lattice and a discrete Yajima-Oikawa system

We propose a new integrable generalization of the Toda lattice wherein the original Flaschka-Manakov variables are coupled to newly introduced dependent variables; the general case wherein the additional dependent variables are vector-valued is considered. This generalization admits a Lax pair based on an extension of the Jacobi operator, an infinite number of conservation laws and, in a special case, a simple Hamiltonian structure. In fact, the second flow of this generalized Toda hierarchy reduces to the usual Toda lattice when the additional dependent variables vanish; the first flow of the hierarchy reduces to a long wave-short wave interaction model, known as the Yajima-Oikawa system, in a suitable continuous limit. This integrable discretization of the Yajima-Oikawa system is essentially different from the discrete Yajima-Oikawa system proposed in arXiv:1509.06996 (also see https://link.aps.org/doi/10.1103/PhysRevE.91.062902) and studied in arXiv:1804.10224. Two integrable discretizations of the nonlinear Schrödinger hierarchy, the Ablowitz-Ladik hierarchy and the Konopelchenko-Chudnovsky hierarchy, are contained in the generalized Toda hierarchy as special cases.

nlin.SI↗
arXiv2015

Multisoliton solutions of the vector nonlinear Schrödinger equation (Kulish-Sklyanin model) and the vector mKdV equation

There exist two natural vector generalizations of the completely integrable nonlinear Schrödinger (NLS) equation in $1+1$ dimensions: the well-known Manakov model and the lesser-known Kulish-Sklyanin model. In this paper, we propose a binary Darboux (or Zakharov-Shabat dressing) transformation that can be directly applied to the Kulish-Sklyanin model. By deriving a simple closed expression for iterations of the binary Darboux transformation, we obtain an explicit formula for the $N$-soliton solution of the Kulish-Sklyanin model under vanishing boundary conditions. Because the third-order symmetry of the vector NLS equation can be reduced to a vector generalization of the modified KdV (mKdV) equation, we can also obtain multisoliton (or multi-breather) solutions of the vector mKdV equation in closed form.

nlin.SI↗
arXiv2015

On a new integrable discretization of the derivative nonlinear Schrödinger (Chen-Lee-Liu) equation

We propose a general integrable lattice system involving some free parameters, which contains known integrable lattice systems such as the Ablowitz-Ladik discretization of the nonlinear Schrödinger (NLS) equation as special cases. With a suitable choice of the parameters, it provides a new integrable space-discretization of the derivative NLS equation known as the Chen-Lee-Liu equation. Analogously to the continuous case, the space-discrete Chen-Lee-Liu system possesses a Lax pair and admits a complex conjugation reduction between the two dependent variables. Thus, we obtain a proper space-discretization of the Chen-Lee-Liu equation defined on the three lattice sites $n-1$, $n$, $n+1$ for the first time. Considering a negative flow of the discrete Chen-Lee-Liu hierarchy, we obtain a proper discretization of the massive Thirring model in light-cone coordinates. Multicomponent generalizations of the obtained discrete equations are straightforward because the performed computations are valid for the general case where the dependent variables are vector- or matrix-valued.

nlin.SI↗
arXiv2014

Comment on "Discretisations of constrained KP hierarchies"

In the recent paper (R. Willox and M. Hattori, arXiv:1406.5828), an integrable discretization of the nonlinear Schrödinger (NLS) equation is studied, which, they think, was discovered by Date, Jimbo and Miwa in 1983 and has been completely forgotten over the years. In fact, this discrete NLS hierarchy can be directly obtained from an elementary auto-Bäcklund transformation for the continuous NLS hierarchy and has been known since 1982. Nevertheless, it has been rediscovered again and again in the literature without attribution, so we consider it meaningful to mention overlooked original references on this discrete NLS hierarchy.

nlin.SI↗
arXiv2014

Comment on "A counterpart of the WKI soliton hierarchy associated with so(3,R)"

In the recent paper (Wen-Xiu Ma, Solomon Manukure and Hong-Chan Zheng, arXiv:1405.1089), the authors proposed an integrable hierarchy different from the well-known Wadati-Konno-Ichikawa (WKI) hierarchy. However, using a simple linear change of dependent variables, one can check that their hierarchy is equivalent to the WKI hierarchy. For the same reason, some "new integrable" hierarchies proposed by Wen-Xiu Ma and coworkers in recent e-prints are equivalent to the already known ones.

nlin.SI↗
arXiv2013

Exact solutions of multicomponent nonlinear Schrödinger equations under general plane-wave boundary conditions

We construct exact soliton solutions of integrable multicomponent nonlinear Schrödinger (NLS) equations under general nonvanishing boundary conditions. Different components of the vector (or matrix) dependent variable can approach plane waves with different wavenumbers and frequencies at spatial infinity. We apply Bäcklund-Darboux transformations to the cubic NLS equations with a self-focusing nonlinearity, a self-defocusing nonlinearity or a mixed focusing-defocusing nonlinearity. Both bright-soliton solutions and dark-soliton solutions are obtained, depending on the signs of the nonlinear terms and the type of Bäcklund-Darboux transformation. The multicomponent solitons generally possess internal degrees of freedom and provide highly nontrivial generalizations of the scalar NLS solitons. The main step in the construction of the multicomponent solitons is to compute the matrix exponential of a constant non-diagonal matrix arising from the Lax pair. With a suitable re-parametrization of the non-diagonal matrix, the matrix exponential can be computed explicitly in closed form for the most interesting cases such as the two-component vector NLS equation. In particular, we do not resort to Cardano's formula in diagonalizing a $3 \times 3$ matrix, so our expressions for the multicomponent solitons are in some sense more explicit and useful than those obtained in [Q-H. Park and H. J. Shin, Phys. Rev. E 61 (2000) 3093].

nlin.SI↗
arXiv2012

A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices

We refine and develop the inverse scattering theory on a lattice in such a way that the Ablowitz-Ladik lattice and derivative NLS lattices as well as their matrix analogs can be solved in a unified way. The inverse scattering method for the (matrix analog of the) Ablowitz-Ladik lattice is simplified to the same level as that for the continuous NLS system. Using the linear eigenfunctions of the Lax pair for the Ablowitz-Ladik lattice, we can construct solutions of the derivative NLS lattices such as the discrete Gerdjikov-Ivanov (also known as Ablowitz-Ramani-Segur) system and the discrete Kaup-Newell system. Thus, explicit solutions such as the multisoliton solutions for these systems can be obtained by solving linear summation equations of the Gel'fand-Levitan-Marchenko type. The derivation of the discrete Kaup-Newell system from the Ablowitz-Ladik lattice is based on a new method that allows us to generate new integrable systems from known systems in a systematic manner. In an appendix, we describe the reduction of the matrix Ablowitz-Ladik lattice to a vector analog of the modified Volterra lattice from the point of view of the inverse scattering method.

nlin.SI↗
arXiv2011

New reductions of integrable matrix PDEs: $Sp(m)$-invariant systems

We propose a new type of reduction for integrable systems of coupled matrix PDEs; this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix. Via this reduction, we obtain a new integrable system of coupled derivative mKdV equations and a new integrable variant of the massive Thirring model, in addition to the already known systems. We also discuss integrable semi-discretizations of the obtained systems and present new soliton solutions to both continuous and semi-discrete systems. As a by-product, a new integrable semi-discretization of the Manakov model (self-focusing vector NLS equation) is obtained.

nlin.SI↗
arXiv2011

On a (2+1)-dimensional generalization of the Ablowitz-Ladik lattice and a discrete Davey-Stewartson system

We propose a natural (2+1)-dimensional generalization of the Ablowitz-Ladik lattice that is an integrable space discretization of the cubic nonlinear Schroedinger (NLS) system in 1+1 dimensions. By further requiring rotational symmetry of order 2 in the two-dimensional lattice, we identify an appropriate change of dependent variables, which translates the (2+1)-dimensional Ablowitz-Ladik lattice into a suitable space discretization of the Davey-Stewartson system. The space-discrete Davey-Stewartson system has a Lax pair and allows the complex conjugation reduction between two dependent variables as in the continuous case. Moreover, it is ideally symmetric with respect to space reflections. Using the Hirota bilinear method, we construct some exact solutions such as multidromion solutions.

nlin.SI↗
arXiv2011

Systematic method of generating new integrable systems via inverse Miura maps

We provide a new natural interpretation of the Lax representation for an integrable system; that is, the spectral problem is the linearized form of a Miura transformation between the original system and a modified version of it. On the basis of this interpretation, we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by Miura transformations. Thus, this method can be used to generate new integrable systems from known systems through inverse Miura maps; it can be applied to both continuous and discrete systems in 1+1 dimensions as well as in 2+1 dimensions. The effectiveness of the method is illustrated using examples such as the nonlinear Schroedinger (NLS) system, the Zakharov-Ito system (two-component KdV), the three-wave interaction system, the Yajima-Oikawa system, the Ablowitz-Ladik lattice (integrable space-discrete NLS), and two (2+1)-dimensional NLS systems.

nlin.SI↗