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Register a new userSecond-order integrable Lagrangians and WDVV equations
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We investigate integrability of Euler-Lagrange equations associated with 2D second-order Lagrangians of the form begin{equation*} int f(u_{xx},u_{xy},u_{yy}) dxdy. end{equation*} 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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Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalised) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.
We show that reductions of KP hierarchies related to the loop algebra of $SL_n$ with homogeneous gradation give solutions of the Darboux-Egoroff system of PDEs. Using explicit dressing matrices of the Riemann-Hilbert problem generalized to include a set of commuting additional symmetries, we construct solutions of the Witten--Dijkgraaf--E. Verlinde--H. Verlinde equations.
We classify integrable Hamiltonian equations in 3D with the Hamiltonian operator d/dx, where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality w such that w_x=u_y. Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density h). We show that the generic integrable density is expressed in terms of the Weierstrass elliptic functions. Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.
N=4 superconformal multi-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial differential equations linear in U and generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F. Putting U=0 yields a class of models (with zero central charge) which are encoded by the finite Coxeter root systems. We extend these WDVV solutions F in two ways: the A_n system is deformed n-parametrically to the edge set of a general orthocentric n-simplex, and the BCF-type systems form one-parameter families. A classification strategy is proposed. A nonzero central charge requires turning on U in a given F background, which we show is outside of reach of the standard root-system ansatz for indecomposable systems of more than three particles. In the three-body case, however, this ansatz can be generalized to establish a series of nontrivial models based on the dihedral groups I_2(p), which are permutation symmetric if 3 divides p. We explicitly present their full prepotentials.
We study higher order KdV equations from the GL(2,$mathbb{R}$) $cong$ SO(2,1) Lie group point of view. We find elliptic solutions of higher order KdV equations up to the ninth order. We argue that the main structure of the trigonometric/hyperbolic/elliptic $N$-soliton solutions for higher order KdV equations is the same as that of the original KdV equation. Pointing out that the difference is only the time dependence, we find $N$-soliton solutions of higher order KdV equations can be constructed from those of the original KdV equation by properly replacing the time-dependence. We discuss that there always exist elliptic solutions for all higher order KdV equations.
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