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## Classification of 2nd order PDEs

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**An important case:**the heat equation in 1D Homogeneous state and its stability**the heat equation in 1D**(Fourier approach)**The heat equation does not have**travelling wave solutions**Reaction-diffusion equations**in 1D, for a scalar field**The simplest case:**the KISS model for plankton blooms**Need for a saturation mechanism:**the Fisher equation**The equation is now nonlinear…**Fourier decomposition does not help now**Fisher equation: homogeneous solution**(the logistic equation)**dissipative and conservative**dynamical systems**types of bifurcation:**normal form**types of bifurcation:**normal form**Fisher equation: travelling wave**c ≥ 2 z http://commons.wikimedia.org/wiki/File:Travelling_wave_for_Fisher_equation.svg**Fisher equation: travelling wave**general initial condition**Fisher equation: travelling wave**asymptotic form stability of the travelling wave**A different form of nonlinearity…**Again, Fourier decomposition does not help**Heuristic, just to understand…**This is an hyperbolic equation Singularity in finite time**The Burgers equation**Shock solutions**The Burgers equation:**shock structure general solution and confluence of shocks test for numerical methods stochastically-forced Burgers eq.**The Korteweg-de Vries equation**Travelling wave solutions: “solitary waves” on the periodic: “cnoidal waves”**The inverse scattering transform**for the Korteweg-de Vries equation