Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Wobbling Fractals for the Double–Sine–Gordon Equation

Version 1 : Received: 5 February 2023 / Approved: 7 February 2023 / Online: 7 February 2023 (08:25:03 CET)

A peer-reviewed article of this Preprint also exists.

Maccari, A. Wobbling Fractals for The Double Sine–Gordon Equation. Symmetry 2023, 15, 639. Maccari, A. Wobbling Fractals for The Double Sine–Gordon Equation. Symmetry 2023, 15, 639.


This paper presents the perturbation theory for the double–sine–Gordon equation. We obtain a system of differential equations that shows the amplitude and phase modulation of the approximate solution.. In the particular case λ = 0 we get the well-known perturbation theory for the sine–Gordon equation. For a special value λ=-1/8, we derive a phase-locked solution with the same frequency of the linear case. In general we obtain both coherent (solitary waves, lumps and so on) solutions as well as fractal solutions. We can demonstrate the existence of envelope wobbling solitary waves, because of the phase modulation depending on the solution amplitude and on the position. The main conclusion is that it is too reductive focus only on coherent solutions for the double sine-Gordon equation, because of the very rich behavior for the DSG nonlinear equation, including wobbling chaotic and fractal solutions.


Double–sine–Gordon equation; perturbation theory; soliton


Physical Sciences, Mathematical Physics

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