Higher Order Approximation in the Reductive Perturbation Method. I. The Weakly Dispersive System

Normal Form and Solitons

“…The explicit form of the normal form is given for the perturbed KdV equation which contains the first three lowest weight approximate conserved quantities (Proposition 5.1). The normal form then admits one-soliton solution of the KdV equation, which confirms the result in [19]. We also discuss the Gardner-Miura transformation which is an invertible version of the Miura transformation, and show that the inverse Gardner-Miura transformation is nothing but the normal form transformation after removing the symmetries of the KdV equation (Proposition 5.2).…”

“…A series of lectures was carried out by the second author in the Euro Summer School 2001 held at the Isaac Newton Institute, Cambridge for August [12][13][14][15][16][17][18][19][20][21][22][23][24][25]2002. The lecture started with a brief summary of the Poincarè-Dulac normal form theory for a system of ordinary differential equations [3].…”

“…Then a natural question is to ask how the higher order corrections affect to the integrability of the leading order equations. In [19], the effect of the higher order corrections on one-soliton solution of the KdV equation was studied, and it was shown that the velocity of soliton is shifted by the secular terms in the higher order terms. Those secular terms or resonant terms are given by the symmetries of the KdV equation.…”

Boussinesq solitary-wave as a multiple-time solution of the Korteweg–de Vries hierarchy

“…In this case, a secular-free expansion can still be obtained and the process of returning to the laboratory coordinates can be made order-byorder at any higher order, implying in a successive solitary-wave velocity renormalization. 4,8 …”

Multiple - scale Expansion for Nonlinear Ion-acoustic Waves

“…To study the higher order terms in the perturbation expansion, the reductive perturbation method has been introduced by use of the stretched time and space variables (Taniuti [2]). However, in such an approach some secular terms appear which can be eliminated by introducing certain slow scale variables (Sugimoto and Kakutani [3]) or by a re-normalization procedure of the velocity of the KdV soliton (Kodama and Taniuti [4]). Nevertheless, this approach remains somewhat artificial, because there is no reasonable connection between the smallness parameters of the stretched variables and the one used in perturbation expansion of the field variables.…”