  A new design paradigm of topology has recently emerged to manipulate the flow
of phonons. At its heart lies a topological transition to a nontrivial state
with exotic properties. This framework has been limited to linear lattice
dynamics so far. Here we show a topological transition in a nonlinear regime
and its implication in emerging nonlinear solutions. We employ nonlinearity
management such that the system consists of masses connected with two types of
nonlinear springs, "stiffening" and "softening" types, alternating along the
length. We show, analytically and numerically, that the lattice makes a
topological transition simply by changing the excitation amplitude and invoking
nonlinear dynamics. Consequently, we witness the emergence of a new family of
finite-frequency edge modes, not observed in linear phononic systems. We also
report the existence of kink solitons at the topological transition point.
These correspond to heteroclinic orbits that form a closed curve in the phase
portrait separating the two topologically-distinct regimes. These findings
suggest that nonlinearity can be used as a strategic tuning knob to alter
topological characteristics of phononic crystals. These also provide fresh
perspectives towards understanding a new family of nonlinear solutions in light
of topology.
