Class diary of the course: Nonlinear waves and solitons (Onde Non lineari e Solitoni, ONS), AA 2023-24. Tuesday 10:00-12:00 and friday 16:00-18:00 classroom Careri. Teacher: Paolo Maria Santini =========================== 27/02/2024 (2) Introduction on linear and nonlinear physical theories. Chapter 1. Definition of linear dispersive PDE, Cauchy problem on the line, the Fourier transform (FT) method, and the Fourier integral representation of the solution. Fundamental solution for delta function initial condition. If omega(k)=k^n, it becomes a similarity solution. The FT of real functions. 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. Amplitude modulation. ========================= 01/03/2024 (4) 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 (the crests) propagates with the phase velocity. 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 behavior of the solution. Connection with the fundamental solution. ========================================== 05/03/2024 (6) A second example: the linear Korteweg de Vries (KdV) equation. Fundamental solution through the Airy function Ai(x). Longtime solution for x/t=O(1)<0: the slowly varying wave train travels to the left. For x/t>0, the two 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. ============================ 08/03/2024 (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. ============== 12/03/2024 (10) Applications of the saddle point method to the following problems. 1) The solutions of the Cauchy problem for the linear Schroedinger equation on the real axis. 2) The integrand of 1), integrated on the real interval (a,b), ax/2t. 3) The integrand of 1), integrated on the 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. 4) The longtime behavior of solutions of the linear KdV when x/t>0 on the real axis. ============= 15/03/2024 (12) The Riemann equation u_t+c(u)u_x=0 as basic example of first order quasi-linear PDE; derivation from the continuity equation. The method of characteristics: the Riemann equation is equivalent to a system of two ODEs (great simplification). Integration through the inversion of an 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. ====================== 19/03/2024 (14) Perturbative study, through the Hopf equation, of a localized initial condition in a neighborhood of the breaking point. The solution before breaking, at breaking, and after breaking, in terms of elementary functions, via the Ferro-Cardano-Tartaglia formulas. 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. ================== 22/03/2024 (16) 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. 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. 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. General solution and particular solutions of some Riemann equations in 1+1 and n+1 dimensions. ================== 26/03/2024 (18) The regularization of the gradient catastrophe through the existence of a weak (discontinuous) solution (shock wave). 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. Example of regularization: the shock of the compression wave. The piston problem for a polytropic gas with homogeneous initial conditions is an iso-entropic problem. ================= 05/04/2024 (20) Continuation of the piston problem. also the Riemann invariant r_ is constant everywhere. The solution of the piston problem in the rarefaction and compression cases; multivaluedness in the compression case. The explicit case of piston of constant speed V, positive or negative, and wave breaking at t=0 for V positive. Shock regularization of the piston problem, in general and for a piston with constant positive speed V. =================== 09/04/2024 (22) Regularization of the gradient catastrophe through the introduction of a dissipation (or diffusion) term. The Burgers equation; conservation of mass and dissipation of energy. Burgers equation and the solution scheme of its Cauchy problem via the Hopf-Cole transformation + the solution of the Cauchy problem for the heat equation. The solution of the Cauchy problem for the heat equation with delta function and step function initial conditions. The solution of the Cauchy problem for the Burgers equation in the small dissipation limit, via the Laplace method. In the case of a single stationary point, the solution reduces to that of the Hopf equation obtained through the method of characteristics, and no breaking takes place. In the case of three stationary points (two relative maxima and one minimum) the solution reduces to the shock wave solution of the Hopf equation. ================== 12/04/2024 (24) 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 problems. 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. =============== 16/04/2024 (26) The multiscale method applied to PDEs. Example: application of the multiscale method to the sine-Gordon equation, a nonlinear Klein Gordon equatio, in the weakly nonlinear and quasi-monochromatic regime. 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 equations in d+1 dimensions; the slow variables again through the dispersion relation. The Hopf equation as model equation for weakly nonlinear and hyperbolic PDEs (1st part). =========================== 19/04/2024 (28) The Hopf equation as model equation for weakly nonlinear and hyperbolic PDEs (2nd part). 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. The Korteweg - de Vries (KdV) equation as model equation for weakly nonlinear and weakly dispersive PDEs. Derivation of the Euler and Navier-Stokes equations from fluid dynamics. =========================== 23/04/2024 (30) Hydrodynamics in the irrotational regime. The surface water wave equations, and their linear (weak field) limit. 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 (long wave) regime of the Euler equation in 1+1 dimensions and the Korteweg-de Vries equation. =========================== 26/04/2024 (32) The weakly nonlinear and weakly dispersive (long wave) regime of the Euler equation in 2+1 dimensions, when the wave is longer in the y direction, and the Kadomtsev-Petviashvili 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+d n(I). Paraxial approximation and NLS equation with saturation potential. Weak intensity, cubic nonlinearity, and the elliptic NLS equation in 2+1 dimensions. =========================== 30/04/2024 (34) The cubic focusing and defocusing nonlinear Schroedinger (NLS) equations in 1+1 dimensions as the integrability condition X_t-T_x+[X,T]=0 of the Zakharov - Shabat vector Lax pair psi_x=X psi, psi_t=T psi. The equation psi_x=X psi as a vector 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. A basis of 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. =========================== 03/05/2024 (36) 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. 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. =========================== 07/05/2024 (38) The spectral operator 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 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_0 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 and S_{11}(lamba)->1 for |\lambda|>>1, 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