Algebraic operator method for the construction of solitary solutions to nonlinear differential equations

Solutions of the KdV equation are derived by the algebraic operator method based on generalized operators of differentiation. The algebraic operator method based on the generalized operator of differentiation is exploited for the derivation of analytic solutions to the KdV equation. The structure of solitary solutions and explicit conditions of existence of these solutions in the subspace of initial conditions are derived. It is shown that special solitary solutions exist only on a line in the parameter plane of initial and boundary conditions. This new theoretical result may lead to important findings in a variety of practical applications.