Exact breathing soliton solutions in combined time-dependent harmonic-lattice potential

We investigate the explicit novel localized nonlinear matter waves of the cubic-quintic nonlinear Schr¨odinger equation with spatiotemporal modulation of the nonlinearities and the harmonic-lattice potential using a modified similarity transformation. We also find that when the modulus of the Jacobian elliptic function in the limit closes to 1, the shapes of the breathing solitons may exhibit some interesting features, i.e., one breathing soliton dividing into two in the ground state.The stability of the exact solutions is investigated numerically such that some stable breathing soliton solutions are found.