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Exactly Solvable Nonhomogeneous Burgers Equations with Variable Coefficients

We consider a nonhomogeneous Burgers equation with time variable coefficients, and obtain an explicit solution of the general initial value problem in terms of solution to a corresponding linear ODE. Special exact solutions such as generalized shock and multi-shock solitary waves, triangular wave, N-wave and rational type solutions are found and discussed. As exactly solvable models, we study forced Burgers equations with constant damping and an exponentially decaying diffusion coefficient. Different type of exact solutions are obtained for the critical, over and under damping cases, and their behavior is illustrated explicitly. In particular, the existence of inelastic type of collisions is observed by constructing multi-shock solitary wave solutions, and for the rational type solutions the motion of the pole singularities is described.