Numerical Computations of a Nearly Singular Nonlinear Equation: Weakly Nonlocal Bound States of Solitons for the Fifth-Order Korteweg–deVries Equation

✍ Scribed by John P. Boyd

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 427 KB
• Volume: 124
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.0044

✦ Synopsis

is small. The next order in the linear dispersion relation (fifth derivative) is then as important as the third We numerically calculate bions, which are bound states of two solitary waves which travel together as a single coherent structure derivative. It is then necessary to replace the KdV with a fixed peak-to-peak separation, for the fifth-order Kortewegequation by its generalization; rescaling the variables deVries equation. R. H. J. Grimshaw and B. A. Malomed (J. Phys. gives the canonical form (1.1). There is one crucial A 26 (1993), 4087-4091) predicted such bions using perturbation distinction which cannot be scaled away, which is the theory. We find that the nearly singular quasi-translational eigenrelative sign of the third and fifth derivative terms. When mode which is the heart of the theory is also numerically important in the sense that later iterations are approximately proportional to the signs are opposite, the solitary waves decay to zero this eigenmode. However, the near-singularity does not create any as ͉x͉ ⇒ 0. When the signs are the same, as here, the serious problems for our Fourier pseudospectral/Newtonsolitary waves are ''weakly nonlocal'' in the sense that Kantorovich/pseudoarclength continuation algorithms. This type of the soliton is a fusion of a single large peak which travels, theory for weakly overlapping solitary waves has been previously locked in phase, with a small amplitude oscillation which developed by Gorshkov, Ostrovskii, Papko, and others. However, Grimshaw and Malomed's work and our own are the first on bions fills all of space. ''Nonlocal'' means that the solitary which are ''weakly nonlocal,'' that is, decay for large ͉x͉ to small wave has nonzero amplitude even infinitely far from the amplitude oscillations rather than to zero. Our numerical calculalarge amplitude ''core'' of the soliton; ''weakly'' means tions confirm the main assertions of Grimshaw and Malomed. Howthat the amplitude of these ''far field oscillations'' or ever, there are other features, such as a complicated branch strucoscillatory ''wings'' is exponentially small in 1/, where ture with multiple turning points and the existence of bions with the parameter measures the amplitude of the core narrow peak-to-peak separation, which are not predicted by the theory.