Representation of zero curvature for the system of nonlinear partial differential equations(x_{alpha ,zar z} = exp (kx)_alpha )and its integrability
✍ Scribed by A. N. Leznov; M. V. Saveliev
- Publisher
- Springer
- Year
- 1979
- Tongue
- English
- Weight
- 246 KB
- Volume
- 3
- Category
- Article
- ISSN
- 0377-9017
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✦ Synopsis
Representation of zero curvature for the system of essentially nonlinear partial differential equations X~,z~ = exp(y,r~=lkc~3x3), 1 <<, ~ <<, r, with an arbitrary numeral matrix k is constructed in an explicit form. On the basis of this representation we give an invariant integration method for the system when k coincides with the Cartan matrix of simple Lie al~bra of rank r. The final solutions depend on 2r arbitrary functions. 1. The system of essentially nonlinear partial differential equations xc~,z~=exp( ~ k~3x~), l <~c~<~r,
(1) 3=1
where k is an arbitrary matrix, is encountered in various branches of theoretical physics and mechanics (field theory, solid-state and plasma physics, the theory of electrolytes, aeromechanics, nonlinear optics, etc., see e.g. [2,3,8] ). As we have shown [1] the cylindrically symmetric selfdual Yang-Mills field configurations in Euclidean space R4 are described by system (1) when k coincides with the Cartan Matrix of simple Lie algebra (5 of rank r of the relevant gauge group G. System (1) is equivalent to Toda's chain with fixed endpoints if k is the Caftan matrix of the series Ar. Note also that a large number of two-dimensional mathematical physics problems considered to date (sine-Gordon, Liouville, Korteweg-de Vries equations, etc., see e.g. [3] ) represent particular issues of system (1) corresponding to a definite choice of matrix k or its continual generalizations for the case of a continuous spectrum of c~ index values.
Recently [4] we have obtained the exact 2r-parametric solutions for system (1) for k being the Cartan matrix of an arbitrary simple Lie algebra (5. The solutions depend on one variable and describe spherically-symmetric instantons and monopoles under the relevant boundary conditions. General solutions for (1) dependent on 2r arbitrary functions are exhibited in [5]. It is worth stressing that the integration method developed m Refs. 4 and 5 is largely an intuitional one and gives no possibility of formulating the results in an mvariant way. In this method, the question concerning the reasons of complete integrability of (1)just for k being the Cartan matrix of a simple Lie algebra remains open. Besides the explicit solutions of system (1) for the algebras E7 and Es are not given in the aforementioned papers.
The purpose of this letter is to construct, on the basis of the representation of zero curvature,