Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line, with the initial function ϕ in the Sobolev space of order s>3/2 and the solution u(t) staying in the same space, s=∞ being included For the proper KdV equation, existence of global solutions follows if s≥2. The proof is based on the theory of abstract quasilinear evolution equations developed elsewhere.
KeywordsInitial Data Evolution Equation Sobolev Space Number Theory Algebraic Geometry
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