Wobbling Fractals for The Double Sine-Gordon Equation

This paper studies a perturbative approach for the double sine-Gordon equation. Following this path, we are able to obtain a system of differential equations that shows the amplitude and phase modulation of the approximate solution. In the case ? = 0, we get the well-known perturbation theory for the sine-Gordon equation. For a special value ? = -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. Using symmetry considerations, we can demonstrate the existence of envelope wobbling solitary waves, due to the critical observation the phase modulation depending on the solution amplitude and on the position. Because the double sine-Gordon equation has a very rich behavior, including wobbling chaotic and fractal solutions due to an arbitrary function in its solution, the main conclusion is that it is too reductive to focus only on coherent solutions.

Automated summary

This paper investigates the perturbative approach for the double sine-Gordon equation, which yields a set of differential equations that demonstrate the amplitude and phase modulation of the approximate solution. The well-known perturbation theory for the sine-Gordon equation is obtained when ? = 0. A phase-locked solution with the same frequency as the linear case is derived for a special value of ? = -1/8. Both coherent solutions (solitary waves, lumps, etc.) and fractal solutions are obtained in general. The presence of envelope wobbling solitary waves is demonstrated using symmetry considerations, highlighting the relationship between phase modulation, solution amplitude, and position. The main conclusion is that focusing solely on coherent solutions is overly simplistic due to the rich behavior of the double sine-Gordon equation, which includes wobbling chaotic and fractal solutions resulting from an arbitrary function in its solution.