Computational and geometric aspects of nonlocal (infinitesimal) symmetries of nonlinear partial differential equations are considered. In particular, the relation of nonlocal symmetries with classical, generalized and internal symmetries is briefly discussed. A nonlocal symmetry for the Kaup–Kupershmidt equation is introduced and studied in some detail. Some explicit particular solutions are found with its help, and a Darboux-like transformation is also obtained.

We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of Mulase's deep algebraic factorization of infinite dimensional groups based on formal pseudo-differential operators, we present two proofs of the well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili (KP) hierarchy in a smooth category. We also generalize these results to a KP hierarchy modelled on formal pseudo-differential operators with coefficients which are series in formal parameters, we describe a rigorous derivation of the Hamiltonian interpretation of the KP hierarchy, and we discuss how solutions depending on formal parameters can lead to sequences of functions converging to a class of solutions of the standard KP-I equation.

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A complete classification of evolution equations u t =F(x, t, u, u x , ..., u x k ) which describe pseudo-spherical surfaces, is given, thus providing a systematic procedure to determine a one-parameter family of linear problems for which the given equation is the integrability condition. It is shown that for every second-order equation which admits a formal symmetry of infinite rank ( formal integrability) such a family exists (kinematic integrability). It is also shown that this result cannot be extended as proven to third-order formally integrable equations. This fact notwithstanding, a special case is proven, and moreover, several equations of interest, including the Harry Dym, cylindrical KdV, and a family of equations solved by inverse scattering by Calogero and Degasperis, are shown to be kinematically integrable. Conservation laws of equations describing pseudo-spherical surfaces are studied, and several examples are given. Academic Press

We study the generalized bosonic string equation, in which f ∈ L 2 (R n ), then there exists a unique real-analytic solution to the Euclidean bosonic string in a Hilbert space H c,∞ (R n ) we define precisely below. Second, we consider the case in which the potential U (x, φ) in the generalized bosonic string equation depends nonlinearly on φ, and we show that this equation admits real-analytic solutions in H c,∞ (R n ) under some symmetry and growth assumptions on U . Finally, we show that the above given equation admits real-analytic solutions in H c,∞ (R n ) if U (x, φ) is suitably near U 0 (x, φ) = φ, even if no symmetry assumptions are imposed.