We investigate non-degenerate Lagrangians of the form $$ \int f(u_x, u_y, u_t) dx dy dt $$ such that the corresponding Euler-Lagrange equations $ (f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0 $ are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f, are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the `master-Lagrangian' corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball. We demonstrate how the knowledge of the symmetry group allows one to linearise the integrability conditions.
Deep Dive into Integrable Lagrangians and modular forms.
such that the corresponding Euler-Lagrange equations $ (f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0 $ are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f, are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the `master-Lagrangian’ corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball. We demonstrate how the knowledge of the symmetry group allows one to linearise the integrability conditions.
In this paper we investigate integrable three-dimensional Euler-Lagrange equations, (f ux ) x + (f uy ) y + (f ut ) t = 0, (1.1) corresponding to Lagrangian densities of the form f (u x , u y , u t ). Familiar examples include the dispersionless KP equation u xt -u x u xx = u yy with the Lagrangian density f = 1 3 u 3 x + u 2 y -u x u t ; this equation, also known as the Khokhlov-Zabolotskaya equation, arises in non-linear acoustics [19]. Another example, u xx + u yy = e ut u tt , is known as the Boyer-Finley equation [3]: it appears as a symmetry reduction of the self-duality equations, and corresponds to the Lagrangian density f = u 2
x + u 2 y -2e ut . The paper [8] provides a system of partial differential equations for the Lagrangian density f (a, b, c) (we set a = u x , b = u y , c = u t ) which are necessary and sufficient for the integrability of the equation (1.1) by the method of hydrodynamic reductions as proposed in [7]. These conditions can be represented in a remarkable compact form:
Theorem 1 [8]. For a non-degenerate Lagrangian, the Euler-Lagrange equation (1.1) is integrable by the method of hydrodynamic reductions if and only if the density f satisfies the relation
here d 3 f and d 4 f are the symmetric differentials of f . The Hessian H and the 4 × 4 matrix M are defined as follows:
(1.
The differential dM = M a da + M b db + M c dc is a matrix-valued form
A Lagrangian is said to be non-degenerate iff H = 0 (we point out that the equations H = 0 and detM = 0 have been discussed in the literature, see [6] and references therein).
Both sides of the relation (1.2) are homogeneous symmetric quartics in da, db, dc. Equating similar terms we obtain expressions for all fourth order partial derivatives of the density f in terms of its second and third order derivatives (15 equations altogether). The resulting overdetermined system for f is in involution, and its solution space is 20-dimensional: indeed, the values of partial derivatives of f up to order 3 at a point (a 0 , b 0 , c 0 ) amount to 20 arbitrary constants. Thus, we are dealing with a 20-dimensional moduli space of integrable Lagrangians.
In Sect. 2 we prove that the integrability conditions (1.2) are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians possesses an open orbit.
Explicit formulae for integrable Lagrangians in terms of modular forms are constructed in Sect. 3. We first consider Lagrangian densities of the form f = u x u y g(u t ), which can be viewed as a deformation of the integrable density f = u x u y u t found in [8]. By virtue of the integrability conditions (1.2), the function g has to satisfy the fourth order ODE
which inherits a remarkable Gl(2, R)-invariance. We prove that he ‘generic’ solution of this ODE is given by the series
notice that under the substitution u t = 2πiz the right hand side of this formula becomes a special modular form of weight one and level three, known as the Eisenstein series E 1,3 (z). We point out that modular forms and non-linear ODEs related to them appear in a variety of problems in mathematical physics, see e.g. [1,2,4,5,9,10,14,15,18] and references therein.
Lagrangian densities of the form g(u x , u y )u t and the general case f (u x , u y , u t ) are discussed in Sect. 3.2 and 3.3, respectively. Here the ‘generic’ solution is an automorphic form of two (three) variables.
The first main observation, overlooked in [8], is the invariance of the integrability conditions (1.2) under projective transformations of the form
here l, l 1 , l 2 , l 3 are arbitrary (inhomogeneous) linear forms in a, b, c. Introducing the quartic form
one can verify that F = l 4 F , which establishes the SL(4, R)-invariance of the integrability conditions (1.2). Combined with obvious symmetries of the form
this provides a 20-dimensional symmetry group of the problem.
Remark. The class of Euler-Lagrange equations (1.1) is form-invariant under a point group generated by arbitrary linear transformations of the variables x, y, t and u. Obviously, point transformations preserve the integrability. Since the prolongation of these transformations to the variables a, b, c and f is given by (2.4), this explains the SL(4, R)-invariance of the integrability conditions (1.2).
The main result of this section is the following Theorem 2. The action of the symmetry group on the 20-dimensional moduli space of integrable Lagrangians possesses an open orbit.
The infinitesimal generators of the symmetry group (2.4), (2.5) 3 projective transformations of a, b, c, f :
moreover, we have 5 extra generators corresponding to the transformations (2.5):
The main idea of the proof is to prolong these infinitesimal generators to the 20-dimensional moduli space of solutions of the involutive system (1.2). We point out that, since all fourth order derivatives of f are explicitly known, this moduli space can be identified with the values of f and its
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