Abstract
In this short note, a particular realization of the vector fields that form a Lie Algebra of symmetries for the Calogero-Bogoyavleskii-Schiff equation is found. The Lie Algebra is examined and the result is a semidirect product of two Lie Groups. The structure of the semidirect product is examined through the table of commutation rules. Two reductions are made with the help of two sets of generators and the final outcome for the solution is related to the elliptic Painlevé P(ξ)-function.
Highlights
The Calogero-Bogoyavleskii-Schiff Nonlinear Partial can be written as [1]: uxt + uxuxy + 1 2 u xx u y 1 4 uxxxy = 0The study of the integrability of this equation has been the subject of a large body of literature concerning specially the solutions obtained by means of a wealth of methods
The structure of the semidirect product is examined through the table of commutation rules
Carnevale [7] and John Weiss himself [8] developed the theory of the Singular Manifold Expansion and its applications
Summary
The study of the integrability of this equation has been the subject of a large body of literature concerning specially the solutions obtained by means of a wealth of methods. For instance the interested reader may wish to check the references [2], [3], [4] and [5]. To achieve this goal the method of the Singular Manifold Expansion has been extensively used.
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