Soliton Phenomenology

ISBN-13: 9789401074940 / Angielski / Miękka / 2014 / 452 str.

V. G. Makhankov

Opis

'Et moi, ..., si j'avait Sll comment en revemr, One service mathematics has rendered the je n'y serais point aIle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Spis treści

References.- I. Quantum Systems and Classical Behaviour.- 1. Some physical models and nonlinear differential equations.- 1. Magnetic chain (the Heisenberg model).- 2. Magnetic chain with magnon-phonon interaction.- 3. Nonlinearity of exchange integrals andphonon anharmonism in the Heisenberg model.- 4. Anisotropic magnetic chain in an external field breaking U(1)(XY) symmetry.- 5. Generalized Hubbard model.- 6. Low frequency wave interaction with a packet of h.f. waves in plasmas.- 7. The ?5Schrödinger equation as a model to describe collective motions in nuclei.- 8. ‘Colour’ generalization of a magnetic chain with magnon-phonon interaction.- 9. Multicolour Hubbard model.- 2. Physically interesting nonlinear differential equations.- 1. Equations with quadratic dispersion.- 2. Equations with ‘linear’ dispersion.- 3. Relativistically-invariant equations.- 4. Dynamical systems given by differential-difference equations.- References.- II. Some Exact Results in One-Dimensional Space.- 3. The Nonlinear Schrödinger equation and the Landau-Lifshitz equation.- 1. NSE associated with a symmetric space.- 2. The Sigma model representation of the NSE and the isotropic Landau-Lifshitz equation.- 3. Gauge connections of the LLE with uniaxial anisotropy and the NSE.- 4. Nonlinear Schrödinger equation with U(p,q) internal symmetry and the SG equation.- 1. Equations of motion and the internal symmetry group.- 2. U(p,q) NSE under trivial boundary conditions.- 3. The U(1,0) model.- 4. The U(0,1) model.- 5. The U(1,1) model.- 6. Quasi-classical quantization of the U(1,1) NSE.- 7. The SG equation.- References.- III. Noncompact Symmetries and Bose Gas.- 5. Dynamical symmetry and generalized coherent states.- 1. Bose gas and dynamical symmetry group.- 2. Quantum version (GCS).- 3. Quantum version. The representation in the form of a path integral over GCS.- 4. Quantum version. Some concrete models with dynamical symmetry.- 5. Weakly nonideal Bose gas. A classical approach.- 6. Bose gas, integrable NSE and Landau-Lifshitz models.- 1. Quantum models and nonlinear classical models corresponding to them. A new formulation of the reduction procedure.- 2. Nonlinear one-dimensional integrable models.- 3. The isotropic Landau-Lifshitz SU(1,1) model.- 4. Bose gas models and nonlinear sigma models. Summary.- 5. The third version — The sigma-model representation connected with the nonlinear Schrödinger equation.- 6. On the reduction procedure.- 7. ?6theory and Bose-drops.- 1. General relations and solitons — drops (particle-like solutions).- 2. Condensate states and their weak excitations.- 3. Localized soliton-like excitations of the condensate.- References.- IV. Soliton-Like Solutions in One-Dimension.- 8. The class of soliton solutions to the vector version of NSE with self-consistent potentials.- 1. Soliton solutions to the U(n) NSE. Linearization method.- 2. U(2) NSE. Dubrovin-Krichever technique.- 3. The self-consistent conditions.- 4. U(2) NSE. A modification of the Dubrovin-Krichever technique.- 5. U(n) system with the Boussinesq potential.- 9. The existence of soliton-like solutions.- 1. Virial relations.- 2. Mechanical analogy method.- 10. Soliton stability.- 1. Stability of hole-like excitations in the ?6model of nonlinear Schrödinger equation. The spectral analysis.- 2. Stability of drop-like solitons. Variational methods.- 3. Structural stability.- References.- V. Phenomenology of D = 1 Solitons.- 11. Dynamics of the formation and interaction of plane solitons.- 1. Computational procedures.- 2. KdV-like equations.- 3. NSE-like equations.- 4. Equation for induced processes.- 5. Relativistically invariant equations (RIE).- 6. Bound states of solitons (bions).- 7. Kink-antikink interactions in the ?4model.- 8. Kink-antikink collisions in the MSG model.- 9. Bions in the ?4-theory.- 10. Small-amplitude expansions.- 12. Structural stability and pinning of solitons.- 1. Static bound states.- 2. Bifurcational perturbation theory.- 3. Static states of the long Josephson junction with a single inhomogeneity.- 4. Passing region.- 13. Dynamical structure factors of soliton gas.- 1. General technique to calculate the dynamical formfactors of solitons.- 2. Dynamic structure factor scattering on a soliton gas. The SG model: phenomenological approach.- 3. CsNiF3and the SG model.- 4. The ideal gas phenomenology and the ?4-model.- 5. Soliton gas kinetics.- 6. Turbulence of a soliton gas.- References.- VI. Many-Dimensional Solitons.- 14. Existence and stability.- 1. Existence.- 2. Quasi-stationary solitons.- 3. Stability of many-dimensional stationary solitons.- 4. Static ring-shaped fluxons (the structure stability).- 15. Pulsons and Q-solitons.- 1. Collapse of circular and spherical bubbles.- 2. Properties of pulsons.- 3. Pulson stability.- 4. Pulson interaction.- 16. Interaction of Q-solitons.- 1. Nonrelativistic models.- 2. Relativistic models.- 3. Formfactors and DSF.- References.