Modified Bianchi Equation with Nonlinear Right-Hand Side

We study a nonlinear Bianchi equation which contains power nonlinearities in unknown function and its first derivatives. The method of separation of variables is used for investigation. An exponential solution is found in the case of an autonomous equation with linearly homogeneous right-hand side. It is shown that the equation has a solution in the form of a polylinear function in the case when the right-hand side of the equation contains the product of power functions of independent variables. Also, we have found solutions in the form of linear combination of exponents and in the form of generalized polynomials. The conditions on the parameters of the equation are given under which the above-mentioned solutions exist. Theorems determining the possibility of decreasing the dimension of the equation are proved. In particular, the initial equation is reduced to an ordinary differential equation for the solutions in the form of one- dimensional traveling waves, and it is reduced to a partial differential equation of lesser dimension for the solutions in the form of multi-dimensional traveling waves. In the latter case, a solution is found in the form of a generalized polynomial in linear combinations of independent variables.