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November  2007, 8(4): 773-800. doi: 10.3934/dcdsb.2007.8.773

Two-pulse solutions in the fifth-order KdV equation: Rigorous theory and numerical approximations

Marina Chugunova 1, and  Dmitry Pelinovsky 1,

1. 

Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada, Canada

Received  May 2006 Revised  May 2007 Published  August 2007

We revisit existence and stability of two-pulse solutions in the fifth-order Korteweg–de Vries (KdV) equation with two new results. First, we modify the Petviashvili method of successive iterations for numerical (spectral) approximations of pulses and prove convergence of iterations in a neighborhood of two-pulse solutions. Second, we prove structural stability of embedded eigenvalues of negative Krein signature in a linearized KdV equation. Combined with stability analysis in Pontryagin spaces, this result completes the proof of spectral stability of the corresponding two-pulse solutions. Eigenvalues of the linearized problem are approximated numerically in exponentially weighted spaces where embedded eigenvalues are isolated from the continuous spectrum. Approximations of eigenvalues and full numerical simulations of the fifth-order KdV equation confirm stability of two-pulse solutions associated with the minima of the effective interaction potential and instability of two-pulse solutions associated with the maxima points.
Keywords: the Petviashvili method, persistence and stability of nonlinear waves., the fifth-order KdV equation, embedded eigenvalues, two-pulse solutions.
Mathematics Subject Classification: Primary: 35Q53, 34L16; Secondary: 65L20, 65M7.
Citation: Marina Chugunova, Dmitry Pelinovsky. Two-pulse solutions in the fifth-order KdV equation: Rigorous theory and numerical approximations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 773-800. doi: 10.3934/dcdsb.2007.8.773
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