Separation of variables for nonlinear equations of hyperbolic and Korteweg–de Vries type

Invariant solutions of a system of Euler equations that satisfy the Rankine–Hugoniot conditions

“…Invariant solutions of system (5), (6) and Rankine-Hugoniot conditions. In this section, we find solutions of system (5), (6) in the case n = 1 that are compatible with the Rankine-Hugoniot conditions.…”

“…Namely, the two first equations in system (17) appear only when n > 1. As was demonstrated in [9], for the equation of state of the form 3 3 M p = ρ , which describes an ideal polytropic gas, system (5), (6) under n = 1, admits an infinite group. Due to this fact, the general solution was obtained for system (5), (6) in this case [9].…”

“…There are many examples of explicitly solved problems of fluid mechanics in the literature. All known solutions and multiparametric families of new particular solutions appear to be obtainable by means of group-theoretical methods [1][2][3][4][5][6][7][8]. Moreover, these methods are useful for finding the particular solutions of nonlinear differential equations that satisfy certain prescribed initial or boundary conditions.…”

On exact solutions of the nonlinear heat equation

“…This approach gives a description of a class of equations of the forms (1) and (2) that have solutions of the form (3), as well as an effective technique for constructing such solutions. An ansatz of the form (3) is used in [6,7] for constructing exact solutions of nonlinear wave equations and Korteweg-de Vries equations.…”

“…are linearly dependent. The functions ϕ ′ ϕ , 1 ′ ϕ are linearly independent, so all other functions (6) should obey the condition that they are representable as a linear combination of the functions ϕ ′ ϕ , 1 ′ ϕ . We have…”