SECOND-ORDER QUASILINEAR PDEs AND CONFORMAL STRUCTURES IN PROJECTIVE SPACE

Тип публикации: статья из журнала

Год издания: 2010

Идентификатор DOI: 10.1142/S0129167X10006215

Ключевые слова: Multidimensional dispersionless integrable systems, hydrodynamic reductions, integrability, conformal structures, dispersionless Lax pairs, conservation laws, conformal structures, conservation laws, dispersionless Lax pairs, hydrodynamic reductions, integrability, Multidimensional dispersionless integrable systems

Аннотация: We investigate second-order quasilinear equations of the form f(ij)u(xi)x(j) = 0, where u is a function of n independent variables x(1),..., x(n), and the coefficients f(ij) depend on the first-order derivatives p(1) = u(x1),..., p(n) = u(xn) only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n vertical bar 1, R), which acts by projective transformations on the space P(n) with coordinates p(1),..., p(n). The coefficient matrix f(ij) defines on P(n) a conformal structure f(ij) (p) dp(i)dp(j). The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived, implying that the moduli space of integrable equations is 20-dimensional. Any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. The integrability conditions imply that the conformal structure f(ij) (p) dp(i)dp(j) is conformally flat, and possesses infinitely many three-conjugate null coordinate systems parametrized by three arbitrary functions of one variable. Integrable equations provide examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.

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Издание

Журнал: INTERNATIONAL JOURNAL OF MATHEMATICS

Выпуск журнала: Vol. 21, Is. 6

Номера страниц: 799-841

ISSN журнала: 0129167X

Место издания: SINGAPORE

Издатель: WORLD SCIENTIFIC PUBL CO PTE LTD

Авторы

  • Burovskiy P.A. (Siberian Federal University)
  • Ferapontov E.V. (Department of Mathematical Sciences,Loughborough University)
  • Tsarev S.P. (Siberian Federal University)

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