Two-dimensional Toda Lattice Equation Research Articles

A class of solutions of the two-dimensional Toda lattice equation

A method is proposed to systematically generate solutions of the two-dimensional Toda lattice equation in terms of previously known solutions ϕ(x,y) of the two-dimensional Laplace's equation. The solitonic solution of Nakamura's [J. Phys. Soc. Jpn. 52 (1983) 380] is shown to correspond to one particular choice of ϕ(x,y).

Tropical limit of matrix solitons and entwining Yang\u2013Baxter maps

We consider a matrix refactorization problem, i.e., a “Lax representation,” for the Yang–Baxter map that originated as the map of polarizations from the “pure” 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang–Baxter equation, but satisfies a mixed version of the Yang–Baxter equation together with the Yang–Baxter map. Such maps have been called “entwining Yang–Baxter maps” in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is not in general a Yang–Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang–Baxter equation holds, by exploring the pure 3-soliton solution in the “tropical limit,” where the 3-soliton interaction decomposes into 2-soliton interactions. Here, this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang–Baxter maps as in the KP case, indicating a kind of universality.

Matrix Integral Solutions to the q-Difference Two-Dimensional Toda Lattice Equation and Its Pfaffianized System

Matrix integrals used in random matrix theory have been revealed to provide τ-functions for several hierarchies of soliton equations. In this article, we extend this relation by showing that such i...

Mixed lump–soliton solutions to the two-dimensional Toda lattice equation via symbolic computation

Based on symbolic computation, we derive explicit rational and rational–exponential solutions to two-dimensional Toda lattice equation via the direct bilinear method. For two types of solutions, the auxiliary functions are taken as the ansatzs with the quadratic function and the combination of quadratic and exponential functions, respectively. It is shown that the purely rational solution depicts a lump with the locality in all directions in the space. The mixed rational–exponential solutions are classified to two kinds of interaction solution: (i) The first one describes the interaction of one lump with one line soliton, in which fission and fusion phenomena occurs between both local waves. (ii) The second one corresponds to the interaction of one lump with two resonant line solitons, which indicates a rogue wave arising from the resonant soliton pair.

Integrable discretization and numerical simulations of the generalized coupled integrable dispersionless equations

ABSTRACTIn this paper, we study the generalized coupled integrable dispersionless (GCID) equations and construct two integrable discrete analogues including a semi-discrete system and a full-discrete one. The results are based on the relations among the GCID equations, the sine-Gordon equation and the two-dimensional Toda lattice equation. We also present the N-soliton solutions to the semi-discrete and fully discrete systems in the form of Casorati determinant. In the continuous limit, we show that the fully discrete GCID equations converge to the semi-discrete GCID equations, then further to the continuous GCID equations. By using the integrable semi-discrete system, we design two numerical schemes to the GCID equations and carry out several numerical experiments with solitons and breather solutions.

The modified semi-discrete two-dimensional Toda lattice with self-consistent sources

In this paper, we derive the Grammian determinant solutions to the modified semi-discrete two-dimensional Toda lattice equation, and then construct the semi-discrete two-dimensional Toda lattice equation with self-consistent sources via source generation procedure. The algebraic structure of the resulting coupled modified differential–difference equation is clarified by presenting its Grammian determinant solutions and Casorati determinant solutions. As an application of the Grammian determinant and Casorati determinant solution, the explicit one-soliton and two-soliton solution of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources are given. We also construct another form of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources which is the Backlund transformation for the semi-discrete two-dimensional Toda lattice equation with self-consistent sources.

Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation

We introduce a so-called coprimeness-preserving non-integrable extension to the two-dimensional Toda lattice equation. We believe that this equation is the first example of such discrete equations defined over a three-dimensional lattice. We prove that all the iterates of the equation are irreducible Laurent polynomials of the initial data and that every pair of two iterates is co-prime, which indicate confined singularities of the equation. By reducing the equation to two- or one-dimensional lattices, we obtain coprimeness-preserving non-integrable extensions to the one-dimensional Toda lattice equation and the Somos-4 recurrence.

Periodic-wave solutions of the two-dimensional Toda lattice equation by a direct method

Hirota bilinear method is proposed to directly construct periodic wave solutions in terms of Riemann theta functions for $(2+1)$ -dimensional Toda lattice equations. The asymptotic properties of the periodic waves are analyzed in detail, including one-periodic and two-periodic solutions. Furthermore, the curves of the solutions are plotted to analyze the solutions. It is shown that well-known soliton solutions can be reduced from the periodic wave solutions.

On integrability of a noncommutative q-difference two-dimensional Toda lattice equation

In our previous work (C.X. Li and J.J.C. Nimmo, 2009 [18]), we presented a generalized type of Darboux transformations in terms of a twisted derivation in a unified form. The twisted derivation includes ordinary derivatives, forward difference operators, super derivatives and q-difference operators as its special cases. This result not only enables one to recover the known Darboux transformations and quasideterminant solutions to the noncommutative KP equation, the non-Abelian two-dimensional Toda lattice equation, the non-Abelian Hirota–Miwa equation and the super KdV equation, but also inspires us to investigate quasideterminant solutions to q-difference soliton equations. In this paper, we first construct the bilinear Bäcklund transformations for the known bilinear q-difference two-dimensional Toda lattice equation (q-2DTL) and then derive a Lax pair whose compatibility gives a formally different nonlinear q-2DTL equation and finally obtain its quasideterminant solutions by iterating its Darboux transformations.

Ultradiscrete Plücker relation specialized for soliton solutions

We propose an ultradiscrete analog of the Plücker relation specialized for soliton solutions. It is expressed by an ultradiscrete permanent which is obtained by ultradiscretizing the permanent, that is, the signature-free determinant. Using this relation, we also show soliton solutions to the ultradiscrete Kadomtsev–Petviashvili equation and the ultradiscrete two-dimensional Toda lattice equation, respectively.

Integrable discretizations for the short-wave model of the Camassa–Holm equation

The link between the short-wave model of the Camassa–Holm equation (SCHE) and bilinear equations of the two-dimensional Toda lattice equation is clarified. The parametric form of the N-cuspon solution of the SCHE in Casorati determinant is then given. Based on the above finding, integrable semi-discrete and full-discrete analogues of the SCHE are constructed. The determinant solutions of both semi-discrete and fully discrete analogues of the SCHE are also presented.

A kind of explicit Riemann theta functions periodic waves solutions for discrete soliton equations

In this paper, Riemann theta functions are used to construct one-theta function and two-theta functions solutions to a class of Hirota bilinear equations, such as extended version of the discrete mKdV equation and deautonomization of the two-dimensional Toda lattice equation. To get the Riemann theta function periodic waves solutions (the quasi-periodic solutions), this method is direct and simple which use only the identities of the theta functions.

Commutativity of Pfaffianization and Bäcklund Transformations: Two Differential-Difference Systems

We generalize the idea of the Commutativity of pfaffianization and Backlund transformations to two differential-difference systems: the two-dimensional Toda lattice equation and the differential-difference KP equations. By means of which, Backlund transformation formulae for their coupled systems are presented respectively.

An extended two-dimensional Toda lattice hierarchy and two-dimensional Toda lattice with self-consistent sources

We extend the two-dimensional Toda lattice hierarchy (2DTLH) by its squared eigenfunction symmetries. This extended 2DTLH (ex2DTLH) includes the two-dimensional Toda lattice equation with self-consistent sources (2DTLSCS) as its first nontrivial equation. Lax representation of ex2DTLH is also presented. With the help of the Lax representation, we construct a nonauto-Bäcklund Darboux transformation (DT) for 2DTLSCS by applying the variation of constants to 2DTLSCS auto-Bäcklund DT. This DT enables us to find many solutions to 2DTLSCS, including solitons, rational solutions, positons, negatons, and complexitons.

Construction of q-discrete two-dimensional Toda lattice equation with self-consistent sources

The q-discrete two-dimensional Toda lattice equation with self-consistent sources is presented through the source generalization procedure. In addition, the Grammtype determinant solutions of the system are obtained. Besides, a bilinear Bäcklund transformation (BT) for the system is given.

Commutativity of Pfaffianization and Bäcklund transformations for a special two-dimensional lattice

In this Letter, we first present the Wronskian solutions to the Backlund transformation of a special two-dimensional lattice by Blaszak and Szum [M. Blaszak, A. Szum, J. Math. Phys. 42 (2001) 225] and then derive the coupled system for the Backlund transformation through Pfaffianization. It is shown the coupled system so obtained is nothing but the Backlund transformation for the coupled system of the special lattice [G.F. Yu, C.X. Li, J.X. Zhao, X.B. Hu, J. Nonlinear Math. Phys. 12 (2005) 316]. This implies that Pfaffianization and Backlund transformations are commutative for the special lattice. Moreover, since the two-dimensional Toda lattice equation forms one part of the special lattice, it is obvious that the commutativity is also applicable to it.

2D Toda lattice equation with self-consistent sources: Casoratian type solutions, bilinear Bäcklund transformation and Lax pair

The two-dimensional Toda lattice equation with self-consistent sources is proposed based on its bilinear forms. Casoratian-type solutions and Bäcklund transformation (BT) for the bilinear forms are presented. Starting from the BT, a Lax pair is derived for the 2D Toda lattice with self-consistent sources.

Pfaffianization of the q-difference version of the two-dimensional Toda lattice equation

We apply the pfaffianization procedure to the q-difference version of the two-dimensional Toda lattice equation to produce the pfaffian analogue of the q-difference two-dimensional Toda lattice equation. A solution to the pfaffianized system is expressed in terms of the q-exponential functions.

Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete analogues

We present a class of solutions of the two-dimensional Toda lattice equation, its fully discrete analogue and its ultra-discrete limit. These solutions demonstrate the existence of soliton resonance and web-like structure in discrete integrable systems such as differential-difference equations, difference equations and cellular automata (ultra-discrete equations).

Gramm-type Pfaffian solutions to three differential-difference coupled systems

In this paper, we first derive the Grammian determinant solutions to the differential-difference KP equation and the semi-discrete Toda equation. Then we present the Gramm-type Pfaffian solutions to the Pfaffianized systems corresponding to the two-dimensional Toda lattice equation, the differential-difference KP equation and the semi-discrete Toda equation.