Hyper-Riccati equations and integrable reductions permitting stationary solutions for complex hyperbolic field equations

摘要

We consider the hyper-Riccati differential equation y′2=P(y) where the degree of P is at least five. Unlike their degree-four counterparts, such equations are seldom exactly integrable. However, for appropriate parameter regimes, we can obtain perturbation solutions centered around solutions to lower-degree problems. We consider explicit perturbation solutions for the degree-five and six models. In the second half of the paper, we consider a class of stationary solutions for complex scalar fields. Such solutions are governed by hyper-Riccati differential equations of degree at least four. While the equation considered is sufficiently general to hold a number of models as special cases, particular attention is paid to examples related to (i) complex nonlinear self-interaction models (such as the φ4 model), and (ii) complex Rayleigh wave equations, where solutions may be expressed in terms of elliptic functions. For all examples considered, we provide a large number of exact stationary solutions of the form u(x,t)=e-ωtψ(x), including degenerate limits. In contrast to these models, the φ6 interaction model corresponds to a hyper-Riccati differential equation of degree six, and the qualitative behavior of solutions to this model depend greatly on the coefficient of the degree-six term in P(y). While this model is, in general, not integrable for ψ(x), we find many exactly solvable reductions. For this model, exact solutions are obtained when possible, and perturbation solutions are obtained for weak-coupling (which is an essential, physically relevant degenerate case). This weak coupling relates solutions of the φ4 model to those of the φ6 model. The mathematical results obtained are then related to the physics of these models. We are able to classify the physically significant solutions in terms of elliptic functions. We also demonstrate the existence of singular degenerate solutions. The specific class of solution obtained depends strongly on the spectrum ω, and for the φ6 model we are able to classify the existence of periodic solutions in terms of the spectrum and the sixth-order coupling parameter. Applications to the Goldstone boson and kink solutions for the φ6 model are discussed.