Numerical computations of a nearly singular nonlinear equation: Weakly nonlocal bound states of solitons for the fifth-order Korteweg-deVries equation

We numerically calculate bions, which are bound states of two solitary waves which travel together as a single coherent structure with a fixed peak-to-peak separation, for the fifth-order Korteweg-deVries equation. R. H. J. Grimshaw and B. A. Malomed (J. Phys. A 26 (1993), 4087-4091) predicted such bions using perturbation theory. We find that the nearly singular quasi-translational eigenmode which is the heart of the theory is also numerically important in the sense that later iterations are approximately proportional to this eigenmode. However, the near-singularity does not create any serious problems for our Fourier pseudospectral/Newlon-Kantorovich/pseudoarclength continuation algorithms. This type of theory for weakly overlapping solitary waves has been previously developed by Gorshkov, Ostrovskii, Papko, and others. However, Grimshaw and Malomed's work and our own are the first on bions which are ''weakly nonlocal,'' that is, decay for large \x\ to small amplitude oscillations rather than to zero. Our numerical calculations confirm the main assertions of Grimshaw and Malomed. However, there are other features, such as a complicated branch structure with multiple turning points and the existence of bions with narrow peak-to-peak separation, which are not predicted by the theory. (C) 1996 Academic Press, Inc.