Generalized Euler, Smoluchowski and Schrodinger equations admitting self-similar solutions with a Tsallis invariant profile

.The damped isothermal Euler equations, the Smoluchowski equation and the damped logarithmic Schrodinger equation with a harmonic potential admit stationary and self-similar solutions with a Gaussian profile. They satisfy an H -theorem for a free energy functional involving the von Weizsacker functional and the Boltzmann functional. We derive generalized forms of these equations in order to obtain stationary and self-similar solutions with a Tsallis profile. In particular, we introduce a nonlinear Schrodinger equation involving a generalized kinetic term characterized by an index q and a power-law nonlinearity characterized by an index gamma . We derive an H -theorem satisfied by a generalized free energy functional involving a generalized von Weizsacker functional (associated with q and a Tsallis functional (associated with gamma . This leads to a notion of generalized quantum mechanics and generalized thermodynamics. When q=2 gamma-1 , our nonlinear Schrodinger equation admits an exact self-similar solution with a Tsallis invariant profile. Standard quantum mechanics (Schrodinger) and standard thermodynamics (Boltzmann) are recovered for q=gamma=1 .