Abstract
The Helmholtz solution of the inverse problem for the variational calculus is used to study the analytic or Lagrangian structure of a number of nonlinear evolution equations. The quasilinear equations in the KdV hierarchy constitute a Lagrangian system. On the other hand, evolution equations with nonlinear dispersive terms (FNE) are non-Lagrangian. However, the method of Helmholtz can be judiciously exploited to construct Lagrangian system of such equations. In all cases the derived Lagrangians are gauge equivalent to those obtained earlier by the use of Hamilton’s variational principle supplemented by the methodology of integer-programming problem. The free Hamiltonian densities associated with the so-called gauge equivalent Lagrangians yield the equation of motion via a new canonical equation similar to that of Zakharov, Faddeev and Gardner. It is demonstrated that the Lagrangian system of FNE equations supports compacton solutions.
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PACS: 47.20.Ky; 42.81.Dp
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Ghosh, S., Das, U. & Talukdar, B. Inverse Variational Problem for Nonlinear Evolution Equations. Int J Theor Phys 44, 363–373 (2005). https://doi.org/10.1007/s10773-005-3645-x
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DOI: https://doi.org/10.1007/s10773-005-3645-x