Special decompositions and linear superpositions of nonlinear systems: BKP and dispersionless BKP equations

TL;DR

This paper investigates special decompositions of the BKP hierarchy, revealing relationships with classical integrable systems and identifying cases where solutions can be linearly superposed, including in the dispersionless BKP hierarchy.

Contribution

It introduces new decomposition solutions of the BKP hierarchy and demonstrates the rare property of linear superposition in certain solutions, especially for the fifth BKP equation.

Findings

Decomposition solutions relate BKP hierarchy to classical integrable systems

Identification of linear superposition solutions in the fifth BKP equation

Extension of superposition solutions to the dispersionless BKP hierarchy

Abstract

The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is explored. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.