Second-order integrable Lagrangians and WDVV equations

Letters in Mathematical Physics - Tập 111 - Trang 1-33 - 2021
E. V. Ferapontov1,2, M. V. Pavlov3, Lingling Xue3
1Department of Mathematical Sciences, Loughborough University, Leicestershire, UK
2Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa, Russian Federation
3Department of Applied Mathematics, Ningbo University, Ningbo, People’s Republic of China

Tóm tắt

We investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

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