A Reformulated Krein Matrix for Star-Even Polynomial Operators with Applications

Kapitula, Todd; Parker, Ross; Sandstede, Björn

doi:10.1137/19m124246x

Cited by 3 publications

(1 citation statement)

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“…A much more difficult problem concerns the spectral stability of multi-pulses in Hamiltonian PDEs in the case where the essential spectrum consists of the entire imaginary axis. (If the essential spectrum is imaginary but bounded away from the origin, the spectral problem is considerably easier; see, for example, an application to a fourth-order beam equation in [24,Secton 6]). As a concrete example, consider double pulse solutions to the fifth-order Korteweg-de Vries equation (KdV5)…”

Section: Introductionmentioning

confidence: 99%

Periodic multi-pulses and spectral stability in Hamiltonian PDEs with symmetry

Parker,

Sandstede

2020

Preprint

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We consider the existence and spectral stability of periodic multi-pulse solutions in Hamiltonian systems which are translation invariant and reversible, for which the fifth-order Korteweg-de Vries equation is a prototypical example. We use Lin's method to construct multi-pulses on a periodic domain, and in particular demonstrate a pitchfork bifurcation structure for periodic double pulses. We also use Lin's method to reduce the spectral problem for periodic multi-pulses to computing the determinant of a block matrix, which encodes both eigenvalues resulting from interactions between neighboring pulses and eigenvalues associated with the essential spectrum. We then use this matrix to compute the spectrum associated with periodic single and double pulses. Most notably, we prove that brief instability bubbles form when eigenvalues collide on the imaginary axis as the periodic domain size is altered. These analytical results are all in good agreement with numerical computations.

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Section: Introductionmentioning

confidence: 99%