Physics

CLAUDIO CONTI
CLAUDIO CONTI

Formative targets:

The objectives of the course are to bring the student to a deep knowledge and understanding of the basic mathematical properties i) of the nonlinear wave propagation with or without dispersion or dissipation; ii) of the construction of nonlinear mathematical models of physical interest, through the multiscale method, like the soliton equations, and of the mathematical techniques to solve them, arriving to the introduction of current research topics in the theory of solitons and anomalous waves. At the end of the course the student must be able i) to apply the acquired methods to problems in nonlinear physics even different from those studied in the course, in fluid dynamics, nonlinear optics, theory of gravitation, etc .., solving typical problems of the nonlinear dynamics; ii) to integrate in autonomy the acquired knowledges through the suggested literature, to solve also problems of interest for him/her, and not investigated in the course. The student will have the ability to consult supplementary material, interesting scientific papers, having acquired the right knowledges and critical skill to evaluate their content and their potential benefits to his/her scientific interests. At last the student must be able to conceive and develop a research project in autonomy. In order to achieve these goals, we plan to involve the student, during the theoretical lectures and exercises, through general and specific questions related to the subject; or through the presentation in depth of some specific subject agreed with the teacher.

Basic knowledge of nonlinear wave propagation, shock waves, solitons, related experiments in various fields such as fluid dynamics and optics, and the construction of mathematical models.
Basic knowledge of perturbative techniques, in particular the method of multiple scales.
Ability to solve first-order partial differential equations using the method of characteristics, including numerical techniques for the integration of ordinary differential equations.
Ability to solve the Sturm–Liouville problem for the Schrödinger equation with a potential, formulated as an eigenvalue problem.
Basic knowledge of numerical techniques for the direct spectral problem, including the computation of the continuous spectrum and the discrete spectrum.
Solution of the inverse scattering problem using Riemann–Hilbert methods.
Basic knowledge of conservation laws and of the Hamiltonian structure of integrable equations, with particular reference to the Korteweg–De Vries equation and the nonlinear Schrödinger equation.

This course should i) provide students with a thorough knowledge and understanding of the topics covered, and ii) enable them to successfully apply this knowledge to various areas of physics. To achieve these goals, and in order for students to develop the ability i) to communicate what they have learned, and ii) to continue studying independently, they will be engaged during lectures and exercises through general and specific questions related to the topics covered, or through classroom presentations of in-depth analyses agreed upon with the instructor—particularly examples of nonlinear waves in optics, biophysics, fluid dynamics, gravitation, and quantum processes.

Courses for undergraduate students

Introduction to nonlinear waves in physics
Examples in optics, fluid dynamics, gravitation, quantum solitons
Wave and non-linear propagation
Linear dispersive waves
Hyperbolic waves
Shock waves and wave regularization
Rogue waves
Multiple scale method
Model equations in 1+1, 2+1 and 3+1
Universality and integrability
Inverse scattering and soliton theory
Solitons in non-integrable systems
Advanced topics:
Non-local solitons, applications in biophysics, and dark matter
Nonlinear waves and lasers
Nonlinear waves and solitons in general relativity
Nonlinear waves and machine learning
Statistics and thermodynamics of solitons, replica symmetry breaking
Quantum Solitons
Experimental investigations
Numerical methods for nonlinear waves

Specific notes distributed during the classrooms
Theory of Solitons, S. Novikov, S.V. Manakov, L. P. Pitaevskii, V. E. Zakharov
Drazin and Johnson, Solitons: an introduction, Cambridge University Press
Ablowitz, Nonlinear Dispersive Waves
M. J. Ablowitz and A. S. Fokas, Complex Variables
An Introduction to Partial Differential Equations, Pinchover and Rubinstein

In addition to the adopted books, the student may also consider,

Whitham, Linear and Nonlinear Waves, Wiley
M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and Inverse Scattering, London Math. Society Lecture Note Series,vol. 194, Cambridge University Press, Cambridge (1991)

Lectures and excercises in presence

In presence

Oral with optional study project

Multiple scale method, examples
Nonlinear Fourier transform
Solitons in the nonlinear Schroedinger equation

• Topic 1: introduction to nonlinear waves;  experiments Vs theory (first lesson, presentation of the course, 2 hours);
  • Books: Notes from the teacher


• Topic 1: Fermi-Pasta-Ulam-Tsingou recurrence and the concept of solitons (first week)
  • Books: Ablowitz 2011, Chap 1


• Topic 1: The Kortweg-De-vries equation, the solitary-wave solution via elliptic integrals (second week)
  • Books: Drazin and Johson


• Topic 1 : Numerical solution of the KdV, differences between solitary waves and solitons
  • Books: Course notes


• Topic 1: The Burgers equations, difference between dissipation and dispersion
  • Books: Course notes


• Topic 1:  The Nonlinear Schroedinger equation, the WKB method, links to shock waves; universality and experiment in different fields
  • Books: Course Notes


• Topic 2: Shock waves and the Hopf equation, numerical methods
  • Books: Course Notes


• Topic 2: Theory of Characteristic, Solutions of the First Order PDE
  • Books: Course Notes


• Topic 3: Linear waves and dispersion, the Fourier transform
  • Books: Ablowitz, 2011


• Topic 3: Numerical solutions of linear equations
  • Books: Course Notes


• Topic 3: The method of multiple scales, derivation of universal equations
  • Books: Course Notes


• Topic 4: Advanced Mathematical Methods: From the Fourier transform to the Lax Pair; introduction to the Riemann Hilbert Boundary value problem
  • Books: Ablowitz and Fokas; Novikov et al.; Course Notes


• Topic 4: The Lax Pair for the KdV
  • Books: Novikov et al.; Course Notes


• Topic 4: Spectral theory for the KdV
  • Books: Novikov et al; Course Notes


• Topic 4: Numerical solutions of the spectral problem
  • Books: Course Notes


• Topic 5: Solitons and Hamiltonians; Solitons and Darboux transform
  • Books: Course Notes


• Topic 5: Hamiltonian structure and conserved quantities in nonlinear PDE
  • Books: Course Notes


• Topic 5: Generalizations of Lax Pair, AKNS
  • Books: Ablowitz 2011


• Topic 6: Applications and Experiments (various topics, also following the student interests)
  • Books: Course Notes