Modulation instability and wavenumber bandgap breathers in a time layered phononic lattice

We provide the first experimental realization of wavenumber bandgap <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mo>(</a:mo><a:mi>q</a:mi><a:mtext>−</a:mtext><a:mtext>gap</a:mtext><a:mo>)</a:mo></a:math> breathers. Experiments are obtained in the setting of a time-periodic phononic lattice where the model and experiment exhibit good qualitative agreement. <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>q</b:mi><b:mtext>−</b:mtext><b:mtext>gap</b:mtext></b:math> breathers are localized in time and periodic in space, and are the counterparts to the classical breathers found in space-periodic systems. We derive an exact condition for modulation instability that leads to the opening of wavenumber bandgaps in which the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>q</c:mi><c:mtext>−</c:mtext><c:mtext>gap</c:mtext></c:math> breathers can arise. The <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>q</d:mi><d:mtext>−</d:mtext><d:mtext>gap</d:mtext></d:math> breathers become more narrow and larger in amplitude as the wavenumber goes further into the bandgap. In the presence of damping, these structures acquire a nonzero, oscillating tail. The controllable temporal localization that <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>q</e:mi><e:mtext>−</e:mtext><e:mtext>gap</e:mtext></e:math> breathers make possible has potential applications in the creation of phononic frequency combs, energy harvesting or acoustic signal processing.