Multidimensional Soliton Integrodifferential Systems

Summary

The article presents preliminary results on applications of the theory of noncommutative KdV equation u _t =∂³ u−3(u∂u+ (∂u) u) (recently developed in [Traves, 2007]) to algebras of matrices, first of finite rank and then of infinite rank. The resulting differential equations in these algebras can only make sense in a noncommutative setup, as the basic “space derivation” is commutation with another (fixed) matrix. The infinite rank situation is reinterpreted, via Hermite expansion, in the algebra of bounded linear operators on Schwartz space S(ℝⁿ) Special choices of the “space derivation” as commutation with partial differential operators can be identified to evolution equations whose linear part is partial differential (in ℝ²ⁿ⁺¹) and the nonlinear part is integrodifferential: a partial differential operator (in ℝ²ⁿ) acting on the square of the unknown u in the sense of Volterra composition. The choice of the harmonic oscillator D _x ²+x ² (when n=1) is particularly amenable to Hermite expansion approach. Existence and uniqueness of global solutions in the Cauchy problem can be proved for special initial data (in ℝ²⁺¹)

2000 AMS Subject Classification: Primary: 37J35, Secondary: 35Q53

*This paper is in final form and no version of it will be submitted for publication elsewhere.