Solitons in Mathematics and Physics


Nonlinear localized objects, solitons, play a very important role in as different areas of physics as nonlinear optics, hydrodynamics, plasma theory, superfluivity, and magnetism. They are important for the theory of general relativity . the black holes are solitons. It is remarkable that this broad variety of physical phenomena, from microscopic to astronomical scale, can be described by unified mathematical apparatus, which was intensively developed during last four decades. The mathematical theory of solitons, also known as the Inverse Scattering Method (ISM) is closely connected to the spectral theory of differential operators and to the classical theory of integrable Hamiltonian systems.

In this course we will discuss the basic elements of both physics and mathematics of solitons. We will develop ISM for the Shrodinger and Dirac operators and study periodic and quasiperiodic solutions of the solitonic equation. This part of the course requires explanation of some elements of classical theory of integrable systems, such as the theory of Riemann surfaces and their Jacoby manifolds. Then we elucidate a connection between theory of solitons and classical theory of integrable systems, including the free motion of a rigid body in n-dimensional space and the Kovalevskaya top. The course will be organized as follow.

  1. Solitons in Physics (3 weeks)
    1. Waves in shallow water. Solitons in the ocean. Theory of tsunami.
    2. Derivation of the Korteg-de Vries (KdV) equation.
    3. From KdV to the Nonlinear Shrodinger equation. Solitons in fiber optics. Bright and dark solutions.
    4. Solitons in thin vortex lines and elastic rods. Solitons in one-dimensional magnetics. Hasimoto transformation.
    5. Solitons in the sine-Gordon equation. Self-induced transparency.
    6. Solitons in plasmas. Solitons in magnetospheres of the Earth.
  2. Inverse scattering method (4 weeks)
    1. Direct scattering theory for the Shrodinger operator.
    2. Inverse scattering theory for the Shrodinger operator.
    3. Lax representation for the KdV equation.
    4. Bargmann.s potential and N-soliton solutions. Solitonic gas.
    5. Direct and Inverse scattering theory for the Dirac equation.
    6. Lax representations for the NSLE. Focusing and defocusing.
    7. Interaction of solitons in optic fibers.
    8. Lax representation for the sine-Gordon equation. Kink, breathers and their interaction.
  3. Dressing method (2 weeks)
    1. Dressing method for construction of solitonic equation and their solutions. Kadomtsev-Petviashvily (KP) equation and its physical applications.
    2. The 3-wave equation and its generalizations. Nonelastic interaction of solitons.
    3. Solitons in KP equation as an example of .integrable. solitons in higher dimensions.
  4. Nonintegrable solitons in higher dimensions (1 week)
    1. Self-focusing and Bose-condensation.
    2. Solitons in rotating plasmas. The Red spot of Jupiter.
  5. Application of classical algebraic geometry to the solitonic equation.
    1. Riemann surfaces. Elliptic functions.
    2. Compact Riemann surfaces of higher and Abelian integrals and their periods.
    3. Abelian differentials. Riemann-Roch theorem.
    4. Abel's map. Jacoby variety. Theta-functions.
    5. Periodic and quasiperiodic solutions of solitonic equations.
  6. Solitonic equations as integrable systems (2 weeks)
    1. Complete integrability of KdV equation.
    2. Complete integrability of NLSE.
    3. Classical integrable systems as stationary point for solitonic equations. Free motion of the rigid body in n-dimensional space. Kovalevskaya map.

The course is naturally separated into easy part (chapters 1-4) and more hard part (chapters 5-6). The easy part can be taken by undergraduate students. They could skip the hard part without harm to their grades.