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M. Chugunova and D. Pelinovsky

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

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Discrete and Continuous Dynamical Systems, Series B 8, 773-800 (2007)

**Abstract:**

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 related to the minima of the effective
interaction potential and instability of two-pulse solutions
related to the maxima points.

**Keywords**:

Fifth-order Korteweg-de Vries equation, Solitary waves, Multi-pulse solutions,
Lyapunov-Schmidt reduction methods, Petviasvhili iteration method, Embedded and isolated
eigenvalues, Exponentially weighted spaces