Class diary of the course Nonlinear waves and solitons (Onde Non lineari e Solitoni, ONS), AA 2022-23. Tuesday 10:00-12:00 e friday 16:00-18:00, classroom Careri. Paolo Maria Santini ================== 24/02/2023 (2) Overview of the subjects of the course: 1) Linear dispersive waves. 2) Nonlinear hyperbolic waves, gradient catastrophe and dissipative regularization. 3) Equations of physics; multiscale methods in weakly nonlinear regime, and nonlinear model equations of mathematical physics. 4) Inverse spectral transform method and solitons; finite gap method and periodic anomalous waves. Chapter 1. Definition of linear dispersive PDE, Cauchy problem on the line, Fourier transform (FT) method, and Fourier integral representation of the solution. In the first order approximation of very localized FT, the carrier wave travels with the phase velocity, and the envelope travels with the group velocity. ========================= 28/02/2023 (4) Fundamental solution for delta function initial condition; if omega(k)=k^n, it becomes a similarity solution. Explicit example of the similarity solution for the linear Schroedinger equation. Fresnell integral. The FT of real functions. Heuristic derivation of the longtime asymptotics t>>1, x/t=O(1)) of the solution through the stationary phase method. Case of a single stationary point on the real axis. Slowly varying wave train. Amplitude modulation and carrier wave. The wave number, satisfying a nonlinear hyperbolic PDE, and the angular frequency travel with the group velocity; the phase propagates with the phase velocity. ========================================== 03/03/2023 (6) The energy of the wave packet propagates with the group velocity; dispersion of the wave packet. The simplest example: the Schroedinger equation for a free particle and the longtime solution. A second example: the linear Korteweg de Vries (KdV) equation. Fundamental solution through the Airy function. Longtime solution for x/t=O(1)<0: the slowly varying wave train travels to the right. For x/t>0, the stationary points lie on the imaginary axis of the complex k plane and the stationary phase method cannot be used. Digression on asymptotic methods: i) integration by parts; ii) Laplace method, heuristic derivation of the leading order term, and Stirling formula. ======================== 07/03/2023 (8) Rigorous derivation of the formulae of the Laplace method, in both cases t_0\in (a,b), and t_0=a or t_0=b, with the estimate of the error. The saddle point method. ================================== 10/03/2023 Sala Touschek 11:00-12:00 (9) Applications of the saddle point method: 1) study of the longtime behavior of the solutions of the Cauchy problem for the linear Schroedinger equation. 2) the integral of 1), integrated in the real interval (a,b), ax/2t. 3) the integral of 1), integrated in the real interval (a,b), a real such that ax/2t and Im(b)>0; in this last case the main contribution comes from integration by parts. ================================== 10/03/2023 Sala Touschek 13:00-14:00 (10) 4) longtime behavior of solutions of the linear KdV when x/t>0. Chapter 2. The Riemann equation u_t+c(u)u_x=0 as basic example of first order quasi-linear PDE; derivation from the continuity equation. ================================ 10/03/2023 (12) The method of characteristics: the Riemann equation is equivalent to a system of two ODEs (great simplification). Integration through the inversion of a algebro-transcendental equation. Deformation of the profile and wave breaking. Intersection of the characteristic curves. Characterization of the beaking point. Example: the Hopf equation u_t+u u_x=0 with a gaussian initial condition. The solution in terms of elementary functions in the case of rarefaction and compression waves, and calculation of the breaking point in the compression case. Homework: i) rarefaction wave with discontinuous initial condition for the Hopf equation u_t+u u_x=0; ii) rarefaction wave with discontinuous initial condition for the Riemann equation u_t+u^2 u_x=0. ================================ 14/03/2023 10:00-12:00 (14) Perturbative study, in a neighborhood of the breaking point, of the evolution of a single bump initial condition according to the Hopf equation. The equation of the characteristics is well approximated by a cubic equation, and one solves it using the Ferro-Cardano-Tartaglia formula for the roots. The solution before breaking, at breaking, and after breaking, in terms of elementary functions. Geometric meaning of a first order scalar quasi linear PDE, in an arbitrary number of independent variables, and the equivalent system of ODEs. Some explicit examples: general and particular solutions. Homework: find the general and particular solutions of other explicit examples. ================================ 17/03/2023 Sala Touschek 11:00-12:00 (15) Solution of the homework. Systems of N real, quasi-linear, first order PDEs are hyperbolic if the relevant matrix has real eigenvalues (not necessarily distinct) and N independent eigenvectors. ============================= 17/03/2023 Sala Touschek 13:00-14:00 (16) Example: the gas dynamics equations are hyperbolic; derivation of their characteristic form. One can simplify further the system if it is possible to introduce the Riemann variables (invariants), for which the differential part of the system decouples. The Riemann variables always exist if N=1,2. For N>= 3 they exist in special cases only. ============================ 17/03/2023 (18) Example: The gas dynamics equations in the iso-entropic case can be written in Riemann form; construction of the Riemann invariants. Their form in the case of a polytropic transformation. The problem of the regularization of a gradient catastrophe through the existence of weak (discontinuous) solutions (shock waves). Characterization of the shock condition (the Rankine-Hugoniot law) and matching with the solution coming from the method of characteristics. The shock front cuts equal area lobi; analytic formula for the Hopf equation. The confluence of the characteristics curves on the shock trajectory. The loss of information in the interval (eta_1,eta_2), and the growth of entropy. ============================ 21/03/2023 (20) Example of regularization: the shock of the compression wave. The piston problem for a polytropic gas with homogeneous initial conditions. Entropy and r_ are constant everywhere. The solution in the compression and rarefaction cases; multivaluedness in the compression case. Entropy and the Riemann invariant r_ are constant everywhere. The general solution with the characterization of the multivalued solution regions. The explicit case of piston of constant speed V, positive or negative, and wave breaking at t=0 for V positive. =================== 24/03/2023 Sala Touschek 11:00-12:00 (21) Shock regularization of the piston problem, in general and for piston with constant positive speed V. Dissipative regularization of the Riemann and Hopf hyperbolic PDEs; conservation of mass and loss of energy. ==================== 24/03/2023 Sala Touschek 13:00-14:00 (22) Burgers equation and its solution schema through the Hopf-Cole transformation and the heat (diffusion) equation. Solution of the Cauchy problem for the heat equation and its fundamental solution. Examples: the solution for the delta function and step function initial conditions. Solution of the Cauchy problem for the Burgers equation through the solution of the Cauchy problem for the heat equation and the Hopf-Cole transformation. The general solution of the heat equation, and the particular solutions for the delta function and the step function initial conditions (1st part). ================= 24/03/2023 (24) Solution of the Cauchy problem for the Burgers equation (2nd part). Use of the Laplace method in the small dissipation problem for the Burgers equation showing the following. i) In the case of a single critical point, the solution reduces to that of the Hopf equation obtained through the method of characteristics, while ii) in the case of three critical points (two relative maxima and one minimum) the solution reduces to the shock wave solution of the Hopf equation. Construction of the exact shock wave solution describing the structure of the wave front of the regularized shock wave; shock strength and shock layer. Two scale problem. ================================== 28/03/2023 (26) The multiscale method applied to the simple pendulum equation, in the case of a weak non linearity. Expansion in power series of the small parameter epsilon, and secularity with linear divergence in time, incompatible with the hamiltonian character of the dynamics. Introduction of the slow time variables of the multiscale expansion to eliminate the secularities of the naive expansion. The multiscale method applied to PDEs. Example: application of the multiscale method to the sine-Gordon equation, a nonlinear Klein Gordon equation arising as continuous limit of the discrete Scott model, in the weakly nonlinear and quasi-monochromatic regime. ========================== 31/03/2023 (28) Weakly nonlinear and quasi-monochromatic waves in nature, and the nonlinear Schroedinger (NLS) equation in 1+1 dimensions through the multiscale method. Generality of the result. Definition of the slow variables through the dispersion relation. NLS focusing and defocusing. ================ 04/04/2023 (30) The NLS equation in d+1 dimensions; the slow variables again through the dispersion relation. The Hopf equation as model equation for weakly nonlinear and hyperbolic PDEs. The Burgers equation as model equation for i) weakly nonlinear and weakly dissipative PDEs, and ii) weakly nonlinear, dissipative PDEs in the long wave regime. Part 1. ===================== 14/04/2023 Sala Touschek 11:00-12:00 (31) The Korteweg - de Vries (KdV) equation as model equation for weakly nonlinear and weakly dispersive PDEs. Part 2. Derivation of the Euler and Navier-Stokes equations from fluid dynamics. ==================== 14/04/2023 Sala Touschek 13:00-14:00 (32) Hydrodynamics in the irrotational regime. The surface water wave equations, their linear (weak field) limit, and the surface water wave dispersion relation. ===================== 14/04/2023 (34) Solution of the linearized equations and dispersion relation. Shallow water and deep water limits; the tsunami and the wind waves as examples. The weakly nonlinear and weakly dispersive regime of the Euler equation and the Korteweg-de Vries equation. The weakly nonlinear and quasi-monochromatic regime in deep water, and the hyperbolic nonlinear Schroedinger equation in 2+1 dimensions. Nonlinear optics in non magnetic media, in the absence of external charges and currents. Refraction index as a slowly varying function of the light intensity I=|A|^2: n=n_0+dn(I). Paraxial approximation and NLS equation with saturation potential. Weak intensity, cubic nonlinearity, and the NLS model in 2+1 dimensions. ==================== 18/04/2023 (36) The Lax pair: psi_x=X psi, psi_t=T psi, its integrability condition X_t-T_x+[X,T]=0 for every lambda,and the nonlinear Schroedinger (NLS) equation of focusing and defocusing type. The equation psi_x=X psi as eigenvalue equation. General properties: i) The reality symmetry. ii) (det Psi)_x=(det Psi) (tr X)=0, (det Psi)_t=(det Psi) (tr T)=0 from Jacobi's formula. Jost solutions, scattering equations and scattering matrix S(lambda). Scattering picture. det S(lambda)=1. The reality symmetry for the Jost eigenfunctions and for the scattering matrix. ====================== 21/04/2023 Sala Touschek 10:00-11:00 (37) Volterra integral equations for the Jost eigenfunctions. Existence of the Jost solutions through the total convergence of the corresponding Neumann series. In addition all the terms of the Neumann series are analytic in the upper (or lower) part of the complex lambda plane: and since the convergence is total, it follows that these analyticity properties are transfered to the sum of the series, the Jost eigenfunctions. ================= 21/04/2023 Sala Touschek 10:00-11:00 (38) Analyticity properties of the diagonal elements of the scattering matrix. The reflection coefficient does not have, in general, analyticity properties. The potential and its modulus square in terms of the eigenfunctions. ================== 21/04/2023 (40) The operator of the eigenvalue equation is self-adjoint in the defocusing case; then its spectrum is real. phi^(1) and psi^(2) are analytic in the upper part of the complex lambda plane, together with the S_{11}. If S_{11}(lamba_0)=0 for Im(lambda_0)>0, then phi^(1)(lambda_0) is proportional to psi^(2)(lambda_0): phi^(1)(lambda_0)=b psi^(2)(lambda_0), and it is exponentially localized for |x|-> infty. Therefore lambda_0 belongs to the discrete spectrum. Since S_{11}(lamba) is analytic for Im(lambda)>0, its zeroes (the eigenvalues) are isolated points. If lambda_0 is real we have no discrete spectrum. Therefore, in the defocusing NLS equation, there is no discrete spectrum. In the focusing case, if ||u||_1