Table of contents

CONTENTS Introduction Chapter I. Integrable equations with attractors § 1. Algebraic construction of differential equations with attractors § 2. Dynamical systems with attractors § 3. 1+1-dimensional integrable equations Chapter II. Breaking solitons in 2+1-dimensional integrable equations § 1. 2+1-dimensional integrable equation § 2. Basic lemma § 3. Breaking solitons and N-soliton solutions § 4. The second 2+1-dimensional integrable equation § 5. Connection with the Kadomtsev-Petviashvili equation § 6. Dynamics of the poles of meromorphic solutions § 7. Integrable two-dimensionalization of the Burgers equation and dynamics of singularities § 8. 3+1-dimensional integrable equation § 9. The third 2+1-dimensional integrable equation § 10. Application of the Painlevé method Chapter III. 2+1-dimensional modified integrable equation § 1. 2+1-dimensional modified equation § 2. Countable set of conservation laws § 3. Lax representation for the 2+1-dimensional modified equation (1.5) § 4. Lax representation for 2+1-dimensional equations (1.5) and (1.6) § 5. Lax representation with a Hermitian operator L § 6. Breaking solitons § 7. Evolution of scattering data § 8. Integrable extension of the KdV equation with the fourth-order Lax operator L § 9. 3+1-dimensional complex integrable equation § 10. Integrable complexifications of the KdV and MKdV equations Chapter IV. Breaking solutions in continual limits of dynamical systems § 1. Continual limits of the Toda lattice and its two-dimensionalization § 2. Continual limits of the Fermi-Pasta-Ulam systems and their two-dimensionalization References

CONTENTS Introduction § 1. Generalized functions and Gaussian integrals on finite-dimensional non-Archimedean spaces § 2. Non-Archimedean Hilbert spaces § 3. Spaces of functions square-integrable with respect to Gauss and Lebesgue distributions on non-Archimedean spaces § 4. The Schrödinger and Bargmann-Fock representations in non-Archimedean quantum mechanics § 5. Non-Archimedean statistical quantum mechanics § 6. Existence and uniqueness theorems for the solution of linear partial differential equations on non-Archimedean spaces § 7. The solubility of Schrödinger's, Heisenberg's and Liouville's equations in non-Archimedean mechanics § 8. Trotter's formula in non-Archimedean commutative Banach algebras, and the chronological representation of the solution of Schrödinger's equation with a potential § 9. Generalized functions and Gaussian integrals on infinite-dimensional non-Archimedean spaces § 10. Existence and uniqueness theorems for the solution of linear partial differential equations on infinite-dimensional non-Archimedean spaces § 11. The path integral for a non-Archimedean scalar bosonic field § 12. Non-Archimedean bosonic strings in a photonic-like gauge § 13. Non-Archimedean bosonic string fields in a photonic-like gauge § 14. Singular functions of a non-Archimedean scalar bosonic field References