Spatially discrete Boussinesq equation: integrability, Darboux transformation, exact solutions and continuum limit

Abstract

The intimate connection between discrete and continuous integrable systems may yield deep insight into some inherent discreteness of physical phenomena. In this paper, a spatially discrete Boussinesq equation is investigated. The integrability of the spatially discrete model is confirmed by showing the existence of Lax pair and infinite number of conservation laws. The Darboux transformation is expressed in terms of Casorati type determinant. Further, by combining the Darboux transformation with different solutions of eigenfunction, we provide a comprehensive approach to construct various types of exact solutions to the spatially discrete Boussinesq equation, such as multi-soliton solutions, periodic solutions, rational solutions and more generally their interaction solutions. Lastly, we prove that the theory of spatially discrete Boussinesq equation including the Lax pair, the conservation laws, the Darboux transformation and the exact solutions converges to the corresponding theory of the Boussinesq equation in the continuum limit.