Applications

  • D. H. Sattinger
  • O. L. Weaver
Part of the Applied Mathematical Sciences book series (AMS, volume 61)

Abstract

The equations
$$\begin{array}{*{20}{l}} {{u_t} + {u_{xxx}} + u{u_x} = 0}&{Korteweg - deVries} \\ {{u_{xt}} = \sin u}&{Sine - Gordon} \\ {i{u_t} + {u_{xx}} + 2|u{|^2}u = 0}&{Nonlinear Schrodinger} \end{array}$$
have several remarkable features in common, among them
  1. (i)

    a Hamiltonian structure.

     
  2. (ii)

    an infinite number of conservation laws; all of them Hamiltonians in involution.

     
  3. (iii)

    an associated spectral problem, invariant under the flow.

     
  4. (iv)

    solvability by inverse scattering method.

     
In addition to the above three examples there are entire hierarchies of such “completely integrable Hamiltonian systems” based on any semi-simple Lie algebra. In this section we shall explain these ideas and give some further examples.

Keywords

Convection Manifold Covariance Hexagonal 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • D. H. Sattinger
    • 1
  • O. L. Weaver
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of PhysicsKansas State UniversityManhattanUSA

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