Branching of Solutions of Nonlinear Equations

The article is concerned with aspects of the equation Au = u, where is a real parameter, u is an element of a real Banach space B, and A is a nonlinear operator from B into itself such that A0 = 0. Unlike the linear case, when the only nontrivial solutions are linear manifolds at particular values of, the nonlinear problem may have a very complicated solution set that always includes, however, the solution u = 0. Branching theory studies nontrivial solutions u () tending to zero as tends to some particular value ₀, a so-called branch-point. After some examples and definitions, the connection between branch-points and the eigenvalues of the linearized problem is established through theorems of Leray–Schauder and Krasnoselskii. The construction of branches is described using the methods of Lyapunov–Schmidt and Poincar–Keller. Three physical examples are then examined in some detail: the Taylor instability for rotating viscous fluids, the nonexistence of buckled states for certain nonlinear rods, and the criticality condition for a simple nuclear reactor.