Heterogeneous Elliptic BVPs with a Bifurcation-Continuation Parameter in the Nonlinear Mixed Boundary Conditions

This article ascertains the global structure of the diagram of positive solutions of a very general class of elliptic boundary value problems with spatial heterogeneities and nonlinear mixed boundary conditions, considering as bifurcation-continuation parameter a certain parameter γ that appears in the boundary conditions. In particular, in this work are obtained, in terms of such a parameter γ, the exact decay rate to zero and blow-up rate to infinity of the continuum of positive solutions of the problem, at the bifurcations from the trivial branch and from infinity. The new findings of this work complement, in some sense, those previously obtained for Robin linear boundary conditions by J. García-Melián, J. D. Rossi and J. C. Sabina de Lis in 2007. The main technical tools used to develop the mathematical analysis carried out in this paper are local and global bifurcation, continuation, comparison and monotonicity techniques and blow-up arguments.

Investigation of symmetry breaking in periodic gravity–capillary waves

In this paper, fully nonlinear non-symmetric periodic gravity–capillary waves propagating at the surface of an inviscid and incompressible fluid are investigated. This problem was pioneered analytically by Zufiria (J. Fluid Mech., vol. 184, 1987c, pp. 183–206) and numerically by Shimizu & Shōji (Japan J. Ind. Appl. Maths, vol. 29 (2), 2012, pp. 331–353). We use a numerical method based on conformal mapping and series truncation to search for new solutions other than those shown in Zufiria (1987c) and Shimizu & Shōji (2012). It is found that, in the case of infinite-depth, non-symmetric waves with two to seven peaks within one wavelength exist and they all appear via symmetry-breaking bifurcations. Fully exploring these waves by changing the parameters yields the discovery of new types of non-symmetric solutions which form isolated branches without symmetry-breaking points. The existence of non-symmetric waves in water of finite depth is also confirmed, by using the value of the streamfunction at the bottom as the continuation parameter.

Quantifying Poincare’s Continuation Method for Nonlinear Oscillators

In the sixties, Loud obtained interesting results of continuation on periodic solutions in driven nonlinear oscillators with small parameter (Loud, 1964). In this paper Loud’s results are extended out for periodically driven Duffing equations with odd symmetry quantifying the continuation parameter for a periodic odd solution which is elliptic and emanates from the equilibrium of the nonperturbed problem.

TRACING THE SOLUTION SURFACE WITH FOLDS OF A TWO-PARAMETER SYSTEM

We describe a special Gauss–Newton method for tracing solution manifolds with singularities of multiparameter systems. First we choose one of the parameters as the continuation parameter, and fix the others. Then we trace one-dimensional solution curves by using continuation methods. Singularities such as folds, simple and multiple bifurcations on each solution curve can be easily detected. Next, we choose an interval for the second continuation parameter, and trace one-dimensional solution curves for certain values in this interval. This constitutes a two-dimensional solution surface. The procedure can be generalized to trace a k-dimensional solution manifold. Numerical results in 1D, 2D and 3D second-order semilinear elliptic eigenvalue problems given by Lions [1982] are reported.