Perturbative Analysis of Wave Interaction in Nonlinear Systems

Abstract

This work proposes a new way for handling obstacles to asymptotic integrability in perturbed nonlinear PDE's within the method of Normal Forms (NF) for the case of multi-wave solutions. Instead of including the whole obstacle in the NF, only its resonant part (if one exists) is included in the NF, and the remainder is assigned to the homological equation. This leaves the NF integrable and its solutions retain the character of the solutions of the unperturbed equation.We exploit the freedom in the expansion to construct canonical obstacles which are confined to the interaction region of the waves. For soliton solutions (e.g., in the KdV equation), the interaction region is a finite domain around the origin; the canonical obstacles then do not generate secular terms in the homological equation. When the interaction region is infinite (or semi-infinite, e.g. in wavefront solutions of the Burgers equation), the obstacles may contain resonant terms.The obstacles generate waves of a new type which cannot be written as functionals of the solutions of the NF. When the obstacle contributes a resonant term to the NF, this leads to a non-standard update of the wave velocity.

Wave interactions and the analysis of the perturbed Burgers equation

“…WE have studied the case of the perturbed KdV equation. Some of the results are published in [19,20]. …”

Ion Acoustic Waves in Weakly Relativistic Plasma—Separation of Relativistic and Electron-Temperature Effects

“…The effect of these new driving terms on the solution cannot be written in closed form as differential polynomials in the zero-order approximation (polynomials in the solution and its spatial derivatives). They have been called obstacles to asymptotic integrability [21][22][23][24][25][26][27][28][29][30][31][32][33]. The previous studies of weak relativistic effects [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] have missed these effects as they did not perform a second order analysis.…”

“…The analysis of Eq. (27) will be carried out through O(µ 2 ) because obstacles to asymptotic integrability emerge in the multi-soliton case only from O(µ 2 ) and onwards [21][22][23][24][25][26][27][28][29][30][31][32][33].…”

“…The equation for U (2) contains a driving term, the contribution of which toU (2) cannot be described by the most general differential polynomial in U (0) consistent with the expansion procedure. Such a driving term has been called an obstacle to asymptotic integrability [21][22][23][24][25][26][27][28][29][30][31][32][33]. It generates dispersive waves that run away from the soliton trajectories (See Section 8.2.).…”

“…In Eqs. (33) and (34), 3 0 4 0 1 1 1 1 3 2 2 2 5,1 0 5, 2 0 1 5,3 0 1 1 1 3 2 3 2 6,1 0 1 6, 2 0 1 6,3 0 1 1 1 1 , ,…”