Dimension breaking of nonlinear elliptic PDEs: satisfying the spectral condition geometrically
نویسندگان
چکیده
Dimension breaking occurs when the solution of a nonlinear partial differential equation (PDE) depending on n independent variables bifurcates to one depending on n + 1. A central hypothesis in the theory of dimension breaking is that a certain operator should have a non-zero purely imaginary eigenvalue. This hypothesis is difficult to verify in general. We present a geometric theory for verifying this hypothesis. Moreover, for a large class of partial differential equations, namely multi-symplectic Hamiltonian PDEs, we show that the verification of this hypothesis is encoded in the basic state. The theory is demonstrated by obtaining new results on dimension breaking of localized states for three examples: the (2+1)-Boussinesq equation, the Zakharov-Kuznetsov equation and the Kadomtsev-Petviashvili equation.
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