This paper establishes the existence of global solutions of the nonlinear equation of Kadomtsev-Petviashvili ut+uu,+u,,x=D-luyy, (1) which are periodic in x and y and D-' denotes the primitive in x. Zakharov [l], Chen [2], and Fokas [3] have established the existence of an infinite sequence of integrals of motion F,(U), n > 1, and shown that the Poisson bracket of F,Ju) with F,(U) vanishes for all n and n' and smooth periodic U. Except for the special solutions of Novikov and Krichever [4], it was not known if there were general global solutions of (1). We show that solutions of (1) exist for all time, are uniquely determined by their initial values, and that these values can be prescribed almost arbitrarily.
To formulate this result, we introduce the following notation. The norm in the space L, is denoted by IrJp, in L, by IjvII =(vIz, and max Iuj = max., 144 y)l. Let D-lo = f; u and V, signify the set of periodic functions with IloIIym= f 5 IID~-~(D,1D,)'41-=~ for m = 0, 1, 2 ,.... n=O I=0 with fh u(x, y) dx = 0.
A global solution of ( 1) is a functon u(x, y, t) such that 1) u(t)11 ",, m 2 3 is bounded for all t and u satisfies (1).
THEOREM.
Zf the initial data g is an element of V, with small II g I(, then there exists a unique global solution u(x, y, t) to (1) from V, with I( u(t)11 v3 < c where c depends only on II g II "). The solution u(t) in Vz is Lipchitz continuous in gE V3 locally uniformly in t.
This result is established in the next section. We give the apriori bound 217