Search

Hierarchies of evolution equations associated with certain symmetric spaces are presented. Main examples of such symmetric spaces are Hermitian symmetric spaces and the obtained evolution equations include generalized nonlinear Schrodinger equations. Those results are obtained by extending the Lie algebra in the Ablowitz–Kaup–Newell–Segur scheme from sl(2,C) to more general one.

Nonlocal symmetries of the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy are introduced and it is shown that the symmetry algebra of the AKNS hierarchy is isomorphic to the loop algebra sl(2,C)⊗C[λ, λ−1]. As a special case, the symmetry algebra of the nonlinear Schrodinger equation is determined and is shown to be isomorphic to the loop algebra su(2)⊗R[λ, λ−1] or g⊗R[λ, λ−1] corresponding to the sign of the nonlinear term, where g is a noncompact real form of sl(2,C).

The Lax forms of many soliton· like equations such as the Burgers equation, the Dym equation, etc. are given. Relaxing the conditions imposed on the formal pseudo·differential operator in Sato theory of the KP-hierarchy, we obtain hierarchies of soliton-like equations. The Burgers hierarchy or the Dym hierarchy is obtained by a reduction of the original hierarchy of differential equations. We can also obtain many other hierarchies of soliton-like equations.

This paper treats geodesic triangles on two-dimensional orientable Riemannian manifolds $M$. Fixing two vertices $A$ and $B$, we can consider the area and the interior angles of the geodesic triangle $\Delta PAB$ as smooth functions of $P$. Applying the Laplace operator to these functions, we obtain formulas for the area and interior angles of $\Delta APAB$. It is shown that if $M$ is of constant curvature, the area and interior angles of geodesic triangles are harmonic.

A relation between certain evolution equations defined by Lax equations is shown. This gives a transformation from the solution space of the Dym hierarchy to the solution space of the modified KdV hierarchy and also gives transformations between the solution spaces of related equations.

A new approach using the Green function instead of the method of Fourier expansion is presented to determine the position-charge relationship for radiation detectors employing a charge division on the resistive anode bordered by an additional resistive line. The resistive anode with a polygonal shape is considered as a general case. Theoretical consideration shows that the position of radiation incidence in the position-sensitive detectors is represented in the integral form containing the Green function if the total charge generated in the detector is assumed to be collected over an infinite time. Anodes with the geometries of a triangle, square, circle or spherical triangle are described as practical shapes. It is shown that, when the shape of the anode is (spherical) triangular or square, the relationship between the position and charge is determined by a harmonic function obtained from a simple geometric consideration.

Let E, F be real analytic vector bundles over a real analytic compact manifold M and D: F(E)-+r(F) a real analytic polynomial differential operator satisfying D(0) = 0. Let s(i) be a parametrized family of cross sections of E, where t moves in some neighborhood of 0 in a euclidean space. We say that s(i) is a deformation of the solution 0 if s(0) = 0 and D(s(0) = 0. In the present paper, we will show the existence of deformations of the solution 0 under some conditions. Namely, let L be the linearized differential operator of D at 0 and assume: (1) the equation D(s) = 0 is involutive, (2) L is elliptic, (3) H(M9 0) = 0, where 0 is the solution sheaf of the equation L(s) = 0. Then we can prove that there is a deformation 5(0 which is complete at t = 0 in an appropriate sense. (Theorem 1, 2) We would like to point out the analogy between the above result and a theorem in [3] on the existence of deformations of complex structures. In fact, the arguments proceed along almost the same line as in [3]. In § 1, we prove some propositions which are needed in the later sections. In §2, we construct the deformation s(0 and prove its completeness in §3. Finally the author wishes to express his hearty gratitude to Professor N. Tanaka for his constant encouragement and valuable suggestions during the preparation of this paper.