Discrete Boussinesq-type equations

Abstract

We present a comprehensive review of the discrete Boussinesq equations based on their three-component forms on an elementary quadrilateral. These equations were originally found by Nijhoff et al using the direct linearization method and later generalized by Hietarinta using a search method based on multidimensional consistency. We derive from these three-component equations their two- and one-component variants. From the one-component form we derive two different semi-continuous limits as well as their fully continuous limits, which turn out to be PDE's for the regular, modified and Schwarzian Boussinesq equations. Several kinds of Lax pairs are also provided. Finally we give their Hirota bilinear forms and multi-soliton solutions in terms of Casoratians.

arXiv comment45 pages, to appear in the CRC press book "Nonlinear Systems and Their Remarkable Mathematical Structures, Vol. 3", Norbert Euler and Da-jun Zhang (eds.) Added Equation (3.9) and some references

References in corpus (6)

• Soliton Solutions for ABS Lattice Equations II: Casoratians and Bilinearization
• Notes on solutions in Wronskian form to soliton equations: KdV-type
• Symbolic Computation of Lax Pairs of Partial Difference Equations Using Consistency Around the Cube
• Multisoliton solutions to the lattice Boussinesq equation
• Soliton Taxonomy for a Modification of the Lattice Boussinesq Equation
• Search for CAC-integrable homogeneous quadratic triplets of quad equations and their classification by BT and Lax

Cited by in corpus (3)

• Algebro-geometric solutions to the lattice potential modified Kadomtsev--Petviashvili equation
• Discrete Lax pairs and hierarchies of integrable difference systems
• Search for integrable two-component versions of the lattice equations in the ABS-list