SOLUTIONS OF INITIAL VALUE PROBLEMS FOR A PAIR OF LINEAR FIRST ORDER ORDINARY DIFFERENTIAL SYSTEMS WITH INTERFACE-SPATIAL CONDITIONS

Solutions of initial value problems associated with a pair of ordinary differential systems (L1,L2) defined on two adjacent intervals 11 and 12 and satisfying certain interface-spatial conditions at the common end (interface) point are studied.


Introduction
In the studies of acoustic waveguides in ocean [1], optical fiber transmission [4], soliton theory [3], etc., we encounter a new class of problems of the type dfkl Llf1 --.aPk--Of 1 defined on an interval I 1 and m k X-"--, df2_ L2f2 v.,_aI-2f2 defined on an adjacent interval I2, where O1, 0 2 are constants, intervals I and 12 have common end (interface) point t c, and the functions fl, f2 are required to satisfy certain interface conditions at t c. In most of the cases, the complete set of physical conditions on the system gives rise to self adjoint eigenvalue problems associated with the pair (L1,L2). In some cases, however, the physical conditions at the interface may be inadequate to describe the problem in a mathematically sound manner. In such a situation, when the problem is formulated mathematically, it becomes ill-posed, and therefore cannot be solved effectively (uniquely) using existing methods. With the introduction of interface-spatial conditions (entirely a new concept), we shall be able to convert these ill-posed problems into wellposed problems and this justifies their mathematical study.
In a series of papers, we wish to develop a unified approach to these interface-spatial problems for both the regular and the singular cases. In the present paper, for the first time, we shall study the initial value problems (IVPs) for a pair of linear first order ordinary differential systems satisfying certain interface-spatial conditions. Before proving the main theorems, we introduce a few notations and make some assumptions.
For any compact interval J of N and for any non-negative integer k, let Ca(J) denote the space of /a-times continuously differentiable complex-valued functions defined on J. If I is a non-compact interval of R, CI(I) denotes the collection of all complex-valued functions f defined on I whose restriction f lj to any compact subinterval J of I belongs to Ck(J). Let ACk(I) denote the space of all complex-valued functions I which have (/-1) derivatives on I, and, the (k-t) th derivative is absolutely continuous over each compact subinterval of I. Let 11 (a, el, 12 -[c, b), -oo_<a<c<b_< +oo, and let f(J) denote the jth derivative off. For a matrix A, let R(A) and p(A) denote the range and rank of A. Let C n denote the complex n-dimensional space.
Let Al(t) (A2(t)) be matrix valued functions of order n x n (m x m), whose entries belong to C(I1) (Cd(/2)). Let bl(t (b2(t)) be a vector-valued function of order n x 1 (mx 1), whose entries are integrable over every compact subinterval of I 1 (I2).
Without loss of generality, we assume n>_m. Let A and B be mxn and mxm matrices with complex entries respectively, and R(A)-R(B). Consequently, p(A)p(B)-'d( <_ m).

Main Theorems
Theorem 1: (a) If either bl(t 5 0, b2(t 5 O, or C is a nonzero vector, then the IFSVP(I) has an IFCo fundamental system consisting of "md + d'+ 1" linearly independent IFS solutions of IFSIVP(I). If bl(t --0, b2(t _= 0, and C is a zero vector, then the IFSIVP(I) has an fundamental system consisting of "md + d'" linearly independent IFS solutions of IFSIVP(I).
(b) If either bl(t 5 O, b2(t 5 0, or D is a nonzero vector, then the IFSIVP(II) has a fundamental system consisting of "n d + d' + 1" linearly independent IFS solutions of IFSIVP(II). If bl(t _= 0, b2(t 0, and D is a zero vector, then the IFSIVP(II) has an IFS fundamental system consisting of "nd + d'" linearly independent IFS solutions of IFSIVP(II).
Therefore, there exist constants a (i-1,..., m-d) such that Thus, by the uniqueness of the solution of IVPs for a system of ordinary differential equations, we Then (K1,K2) is an IFS solution of IFSIVP(I). Consequently, we get B(r2(c K2(c)) 0.
This proves the linear independency of (Yil, Yi2)s.

Physical Examples
Example 1 Acoustic waveguides in ocean [1]: The following problem is encountered in the study of acoustic waves in the ocean consisting of two layers: an outer layer of finite depth and an inner layer of infinite depth: d2fl + kf1 AI 1 0 < t < d 1 Ll f l dt 2 d2 f 2 + kf2 A f2, dl < < t <_ + c, Here ill, f12 are constant densities of the two layers, kl,k 2 are the constants which depend upon the frequency constant and the constant sound velocities Cl, c 2 of the two layers, respectively, is an unknown constant, d I denotes the depth of the outer layer, and fl,f2 stand for the depth eigenfunctions.
In this example, the interface conditions at t-d I of the two layers can be written in the Hence, by Theorem 3 and Remark 2, there exist a unique IFS solution for any IFSIVP associated with (22)-(23) and (25). Also, By Theorem 4 and Remark 2, there exist exactly two linearly independent IFS solutions of problems (22)-(23) Here r]l and q2 are the refractive indices of the core and cladding, respectively, is the wave propagation constant, u is an integer k 0 w/c, c is the prorogation velocity and w is the wave frequency and fl and f2 are the field (electromagnetic) distributions of core and cladding, respectively.
In this example, relation (28) gives continuity conditions at t a. Here A and B are the 2 2 identity matrices, n rn d 2 and d'= 0. Hence, by Theorem 3 and Remark 2, there exists a unique IFS solution for IFSIVP associated with (26)-(28). Also, by Theorem 4 and Remark 2, there exist exactly two linearly independent IFS (continuous) solutions of (26)-(28). Example 3 One-dimensional scattering in quantum theorem [3]: In quantum theory, the one-dimensional time-independent scattering problem with the delta function scattering potential is represented by the problem: together with the interface conditions given by where k 2-2mE/h 2, v o is a constant, and the functions fl and f2 are associated with the flux density of the particle of the two regions, respectively. Here, m denotes the mass of the particle, E denotes its total energy, and h denotes the Planck constant divided by 277. In this example, where k I and k 2 are constants. Problems (34)-(37) can be thought of as the transverse vibrations of a string stretched between a and b, fixed at a and b, with different uniform linear densities (in the portion) between a and c and between c and b, and plucked at the point t c.
In this example, there is only one condition at the interface (i.e., the continuity condition), and no definite relation between the derivatives is available. Therefore, we may take fl)(c)-fl)(c) o, c e N, We note that relation (38) fl)(c) 0 1 fl)(c) A=B=the (2x2) identity matrix, n m d 2, and d'= 1. Therefore, by Theorem 3, there exist one or two linearly independent IFS solutions of the IFSIVP associated with problems (34)-(36) depending on whether the initial data is zero or nonzero. Also, by Theorem 4, there exist three linearly independent IFS solutions of problems (34)-(36).
Remark 3: The results of this paper are used in studying the deficiency indices and selfadjoint boundary value problems associated with (L1,L2) satisfying interface-spatial conditions which we shall establish elsewhere.