Symplectic structures, their B?acklund transformation and hereditary symmetries

A time division communication system wherein a plurality of time slots occurs in repetitive frames includes a plurality of lines, an incoming time division bus and an outgoing time division bus. Each line has an associated hybrid circuit that comprises balancing apparatus for minimizing the portion of the incoming bus signal returned to the outgoing bus through the hybrid circuit. Adjustment of the balance for a selected hybrid is done during the plurality of time frames in a predetermined time interval. The polarities of the incoming signal to the hybrid and the outgoing signal from the hybrid are detected in each frame. A first type signal is generated when the detected polarities are the same and a second type signal is generated when the detected polarities are different. Responsive to the sequence of first and second type signals, a control signal is produced which is applied to the balancing apparatus of the selected hybrid circuit.

[1]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[2]  David J. Kaup,et al.  On the Inverse Scattering Problem for Cubic Eigenvalue Problems of the Class ψxxx + 6Qψx + 6Rψ = λψ , 1980 .

[3]  T. Benjamin.  Internal waves of permanent form in fluids of great depth , 1967, Journal of Fluid Mechanics.

[4]  F. Bayen,et al.  Deformation theory and quantization. I. Deformations of symplectic structures , 1978 .

[5]  Athanassios S. Fokas,et al.  A symmetry approach to exactly solvable evolution equations , 1980 .

[6]  John D. Gibbon,et al.  A new hierarchy of Korteweg–de Vries equations , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[7]  Benno Fuchssteiner,et al.  Pure soliton solutions of some nonlinear partial differential equations , 1977 .

[8]  A. Fokas,et al.  Bäcklund transformations for hereditary symmetries , 1981 .

[9]  Franco Magri,et al.  A Simple model of the integrable Hamiltonian equation , 1978 .

[10]  David Robinson,et al.  Local Jet Bundle Formulation of Bäcklund Transformations , 1979 .

[11]  Peter J. Olver,et al.  On the Hamiltonian structure of evolution equations , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  I. Gel'fand,et al.  Fractional powers of operators and Hamiltonian systems , 1976 .

[13]  A. Lichnerowicz,et al.  Les variétés de Poisson et leurs algèbres de Lie associées , 1977 .

[14]  I. Gel'fand,et al.  A Lie algebra structure in a formal variational calculation , 1976 .

[15]  Benno Fuchssteiner,et al.  Application of hereditary symmetries to nonlinear evolution equations , 1979 .

[16]  Peter J. Olver,et al.  Evolution equations possessing infinitely many symmetries , 1977 .

[17]  Peter D. Lax,et al.  Almost Periodic Solutions of the KdV Equation , 1976 .

[18]  Hiroaki Ono.  Algebraic Solitary Waves in Stratified Fluids , 1975 .

[19]  Athanassios S. Fokas,et al.  Group theoretical nature of Bäcklund transformations , 1979 .

[20]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

[21]  M. Ablowitz,et al.  The Inverse scattering transform fourier analysis for nonlinear problems , 1974 .

[22]  Athanassios S. Fokas,et al.  On the structure of symplectic operators and hereditary symmetries , 1980 .