Lattice Boussinesq Equation Research Articles

On the Lagrangian multiform structure of the extended lattice Boussinesq system

The lattice Boussinesq (lBSQ) equation is a member of the lattice Gel'fand-Dikii (lGD) hierarchy, introduced in \cite{NijPapCapQui1992}, which is an infinite family of integrable systems of partial difference equations labelled by an integer $N$, where $N=2$ represents the lattice Korteweg-de Vries (KdV) system, and $N=3$ the Boussinesq system. In \cite{Hiet2011} it was shown that, written as three-component system, the lBSQ system allows for extra parameters which essentially amounts to building the lattice KdV inside the lBSQ. In this paper we show that, on the level of the Lagrangian structure, this boils down to a linear combination of Lagrangians from the members of the lGD hierarchy as was established in \cite{LobbNijGD2010}. The corresponding Lagrangian multiform structure is shown to exhibit a `double zero' structure.

A new method to generate non-autonomous discrete integrable systems via convergence acceleration algorithms

In this paper, we propose a new algebraic method to construct non-autonomous discrete integrable systems. The method starts from constructing generalizations of convergence acceleration algorithms related to discrete integrable systems. Then the non-autonomous version of the corresponding integrable systems are derived. The molecule solutions of the systems are also obtained. As an example of the application of the method, we propose a generalization of the multistep ϵ-algorithm, and then derive a non-autonomous discrete extended Lotka–Volterra equation. Since the convergence acceleration algorithm from the lattice Boussinesq equation is just a particular case of the multistep ϵ-algorithm, we have therefore arrived at a generalization of this algorithm. Finally, numerical experiments on the new algorithm are presented.

Conservation laws of some lattice equations

We derive infinitely many conservation laws for some multi-dimensionally consistent lattice equations from their Lax pairs. These lattice equations are the Nijhoff-Quispel-Capel equation, lattice Boussinesq equation, lattice nonlinear Schr\"{o}dinger equation, modified lattice Boussinesq equation, Hietarinta's Boussinesq-type equations, Schwarzian lattice Boussinesq equation and Toda-modified lattice Boussinesq equation.

Parameter Extension and the Quasi-Rational Solution of a Lattice Boussinesq Equation

For a lattice Boussinesq equation, we introduce a simple-parameter invertible transformation by which the equation is transformed into an extended version. This new equation admits solitons and nonzero quasi-rational solutions, both in Casoratian form. These solutions can be reverted to those of the lattice Boussinesq equation.

Soliton Taxonomy for a Modification of the Lattice Boussinesq Equation

Integrable multi-component lattice equations of the Boussinesq family have been known for some time. Recently some new equations of this type were found using the Consistency-Around-the-Cube approach. Here we investigate one of these models, B-2, and in particular the consequences of a nonzero deformation parameter $b_0>0$, which allows special kinds of solitons in the parameter range $-b_0/3<k<b_0$.

Q-difference and confluent forms of the lattice Boussinesq equation and the relevant convergence acceleration algorithms

A q-difference lattice Boussinesq equation and its confluent form are proposed. With the help of determinant identity, their molecule solutions are constructed. It is shown that the q-difference lattice Boussinesq equation can be used as a numerical convergence acceleration algorithm to compute the approximation limt → ∞f(t). Numerical examples with the application of this algorithm are presented.

Convergence Acceleration Algorithm via an Equation Related to the Lattice Boussinesq Equation

The molecule solution of an equation related to the lattice Boussinesq equation is derived with the help of determinantal identities. It is shown that this equation can for certain sequences be used as a numerical convergence acceleration algorithm. Numerical examples with applications of this algorithm are presented.

Multisoliton solutions to the lattice Boussinesq equation

The lattice Boussinesq equation (BSQ) is a three-component difference-difference equation defined on an elementary square of the two-dimensional lattice, having three-dimensional consistency. We write the equations in the Hirota bilinear form and construct their multisoliton solutions in terms of Casoratians, following the methodology in our previous papers. In the construction it turns out that instead of the usual discretization of the exponential as [(a+k)/(a−k)]n, we need two different terms [(a−ωk)/(a−k)]n and [(a−ω2k)/(a−k)]n, where ω is a cubic root of unity ≠1.

The lattice Gel'fand-Dikii hierarchy

Using the direct linearization method the authors construct lattice versions of the hierarchy of Gel'fand-Dikii equations, the first members being the lattice KdV equation and the lattice Boussinesq equation. The (local) initial value problem on the lattice of this family of equations is formulated, giving rise to integrable multi-dimensional mappings. The involutivity of the invariants of these mappings is established on the basis of a novel classical tau -matrix structure.