Dimensional Toda Equation Research Articles

Exact solutions for two dimensional [formula omitted] Toda equation

For the two dimensional Toda equation corresponding to the Kac–Moody algebra Dn(1), the Darboux transformation is constructed. The coefficient matrices of the Lax pair of this equation are of even order. Comparing with the scalars in the Darboux matrices for the two dimensional A2n(2) , Cn(1) and Dn+1(2) Toda equations, the structure of the 2 × 2 blocks in the Darboux matrix for this equation is much more complicated. In the construction of Darboux matrices, it is demanded that the solutions of the Lax pair are in Lag(ℂ2n). With the help of a dense subset of Lag(ℂ2n), the nontrivial blocks of the Darboux matrices are represented by elements of O(n,ℂ). Quite a few algebraic techniques are used to simplify the Darboux matrices and to show that the form of the Darboux matrices only depends on the parity of n.

Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.

Symbolic Computation And New Soliton-like Solutions To The (21)-dimensional Toda Lattice

In this paper, with the aid of symbolic computation, an algebraic algorithm is proposed to construct soliton-like solutions to (2+1)-dimensional differentialdifference equations. The famous (2+1)-dimensional Toda equation is explicitly solved and some new classes of soliton-like solutions are obtained.

Variable-coefficient discrete tanh method and its application to ( [formula omitted])-dimensional Toda equation

In this Letter, a variable-coefficient discrete tanh method is proposed for solving nonlinear differential–difference equations. With the aid of symbolic computation, we choose a ( 2 + 1 )-dimensional Toda equation to illustrate the validity and advantages of the method. As a result, hyperbolic function solutions, trigonometric function solutions and rational solutions with arbitrary functions are obtained. It is shown that the proposed method provides a powerful mathematical tool for nonlinear differential–difference equations in mathematical physics.

Some new solutions derived from the nonlinear (2+1)-dimensional Toda equation–an efficient method of creating solutions

This paper presents a new and efficient approach for constructing exact solutions to nonlinear differential–difference equations (NLDDEs) and lattice equation. By using this method via symbolic computation system MAPLE, we obtained abundant soliton-like and/or period-form solutions to the (2+1)-dimensional Toda equation. It seems that solitary wave solutions are merely special cases in one family. Furthermore, the method can also be applied to other nonlinear differential–difference equations.

On geometric approach to Lie symmetries of differential-difference equations

Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of ( 2 + 1 ) -dimensional Toda equation is investigated by means of our approach.

Explicit Solutions for a (2+1)-Dimensional Toda Lattice with TwoDiscrete Variables

An integrable (2+1)-dimensional Toda lattice with two discretevariables is investigated again, which is produced from a compatiblecondition of the Lax triad. The Darboux transformation for its spectralproblems is found. As an application, explicit solutions of the(2+1)-dimensional Toda equation with two discrete variables are obtained.

An integrable ()-dimensional Toda equation with two discrete variables

Abstract An integrable ( 2 + 1 )-dimensional Toda equation with two discrete variables is presented from the compatible condition of a Lax triad composed of the ZS–AKNS (Zakharov, Shabat; Ablowitz, Kaup, Newell, Segur) eigenvalue problem and two discrete spectral problems. Through the nonlinearization technique, the Lax triad is transformed into a Hamiltonian system and two symplectic maps, respectively, which are integrable in the Liouville sense, sharing the same set of integrals, functionally independent and involutive with each other. In the Jacobi variety of the associated algebraic curve, both the continuous and the discrete flows are straightened out by the Abel–Jacobi coordinates, and are integrated by quadratures. An explicit algebraic–geometric solution in the original variable is obtained by the Riemann–Jacobi inversion.

Soliton-like and periodic form solutions to (2+1)-dimensional Toda equation

In this paper, we present a further extended tanh method for constructing exact solutions to nonlinear difference-differential equation(s) (NDDEs) and Lattice equations. By using this method via symbolic computation system MAPLE , we obtain abundant soliton-like and period-form solutions to the (2 + 1)-dimensional Toda equation. Solitary wave solutions are merely a special case in one family. This method can also be used to other nonlinear difference differential equations.

New rational formal solutions for (1+1)-dimensional Toda equation and another Toda equation

Abstract In this paper, we present a new rational expansion method to uniformly construct a series of exact solutions for nonlinear differential-difference equations. Compared with most existing methods, the proposed method not only recovers some well-known solutions, but also finds some new and general solutions. The efficiency of the method can be demonstrated on (1 + 1)-dimensional Toda equation and another Toda equation.

On the geometric properties of the dispersionless Toda equation

The problem of constructing exact solutions, Noether symmetries, and conservation laws for the (2 + 1)-dimensional dispersionless Toda equation uxy = exp(−uzz) is considered.

Ladder Operator Approach of Special Functions and the N-Soliton Solutions of the 2+1 Dimensional Finite Toda Equation

We consider the generalized ladder operators of the special functions. This scheme can be applied to the problem of obtaining the N -soliton solutions of the 2+1 dimensional finite Toda equation. Analytically the N -soliton solutions can be expressed by the use of Hyper Geometric Functions with two variables.

From the special 2 + 1 Toda lattice to the Kadomtsev-Petviashvili equation

The nonlinearization of the eigenvalue problems associated with the Toda hierarchy and the coupled Korteweg-de Vries (KdV) hierarchy leads to an integrable symplectic map S and an integrable Hamiltonian system (H0), respectively. It is proved that S and (H0) have the same integrals {Hk}. The quasi-periodic solution of the (2 + 1)-dimensional Kadomtsev-Petviashvili equation is split into three Hamiltonian systems (H0),(H1),(H2), while that of the special (2 + 1)-dimensional Toda equation is separated into (H0),(H1) plus the discrete flow generated by the symplectic map S. A clear evolution picture of various flows is given through the `window' of Abel-Jacobi coordinates. The explicit theta-function solutions are obtained by resorting to this separation technique.

SYMMETRIES OF THE(2 +1) DIMENSIONAL DISCRETE Toda EQUATION

We extend the formal series symmetry theory proposed in Refs. [1-4] to the (2 + 1) dimensional discrete Toda equation. Two sets of infinitely many symmetries of this equation are found. Every one set of symmetries constitute a generalized W∞ algebra which contains the usual W∞ algebra as its subalgebra.

On the inverse scattering transform of the 2 + 1 Toda equation

The method of solution to the 2 + 1 dimensional Toda equation is described in some detail. This equation reduces directly to the well know Toda lattice in 1 + 1 dimension and, by an appropriate asymptotic reduction, to the Kadomtsev-Petviashvili equation in a continuous limit. The solution exhibits a number of interesting aspects depending on certain choices of signs, of which there are four, in the equation. For two choices of sign the equation is well posed and linearly stable/unstable. For the other choices of sign the equation is linearly illposed. In these cases we can relate the solution to a boundary value problem and give a formal construction of the solution. For one choice of signs in the illposed case an analogue of the Sommerfeld radiation condition is developed in order to identify a unique solution. In general the method of solution of the “Toda molecule” equation requires an implementation of the dbar technique to cases where the associated eigenfunctions possess both smooth regions of nonholomorphicity and a discontinuity across a curve, which in this problem is the unit circle. Special lump type solutions and solutions depending on suitable arbitrary functions are presented.

On the method of solution to the 2+1 Toda equation

The method of solution to the (2+1)-dimensional Toda equation is discussed. This equation reduces to the well-known Toda lattice in 1+1 dimensions and via an appropriate asymptotic reduction to the 2+1 Kadomtsev-Petviashvili equation in the continuous limit. The solution exhibits a number of interesting aspects depending on the choice of parameters such as linear instability with a bounded growth rate and ill-posedness. In the ill-posed case we can relate the solution to a boundary value problem and give a formal construction of the solution. For one choice of signs, an analogue of the Sommerfeld radiation condition is used to identify a unique solution. In general, the method of solution of the Toda “molecule” equation requires an implementation of the dbar technique to cases where the associated eigenfunctions exhibit both smooth regions of nonholomorphicity and a discontinuity across a curve, which in this problem is the unit circle.

The 3+1 Dimensional Toda Molecule Equation and Its Multiple Soliton Solutions

We consider the 3+1 dimensional Toda equation Δ log V n - V n +1 +2 V n - V n -1 =0 where Δ ≡(∂/∂ x ) 2 +(∂/∂ y ) 2 +(∂/∂ z ) 2 . The boundary condition is molecule type which is V n =0 at finite n corresponding to both ends of the finite chain. We take the form of the solution to be V n ( x , y , z )= V n ( r , θ, φ)= r -2 V n (θ,φ) where r , θ and φ are usual spherical coordinates. We have found explicit multiple soliton solutions for V n (θ, φ), which correspond to the superpositions of arbitrary number of axially symmetric solutions with each symmetry axis directing to arbitrary different directions.

Exact Solutions of the 2+1 Dimensional Toda Equation under Periodic Boundary Condition

We consider the 2+1 dimensional Toda lattice equation, Δ log (1+ V n )- V n +1 +2 V n - V n -1 =0, ( Δ ≡∂ 2 /∂ x 2 +∂ 2 /∂ y 2 =∂ 2 /∂ r 2 + r -1 ∂/∂ r + r -2 ∂ 2 /∂θ 2 ), under the periodic boundary condition V n = V n + N with N being arbitrary integer. We have obtained exact solutions with finite period N . When the period N is infinity ( N =∞), the present solution coincides with the previously known cylindrical soliton of Toda equation. When the period N is finite, the present solution represents the cylindrical-soliton-like wave but deformed from pure cylindrical symmetry.

The 3+1 Dimensional Toda Equation and its Exact Solutions

Exact solutions of the 3+1 dimensional Toda equation, Δlog V n - V n +1 +2 V n - V n -1 =0, where Δ≡∂ 2 /∂ x 2 +∂ 2 /∂ y 2 +∂ 2 /∂ z 2 have been obtained. The boundary condition is the “molecule” condition which is vanishing V n at both ends of the finite chain, V N = V - N =0 or V N = V - N -1 =0. In the large molecule and small angle limit, the present solution reduces to the 2+1 dimensional cylindrical Toda soliton.