New integrable (2+1) -dimensional sine -Gordon equations with constant and time -dependent coefficients: Multiple optical kink wave solutions

This work studies new integrable (2+1)-dimensional sine-Gordon equations, the first is char- acterized with constant coefficients, while the other is defined with time-dependent coefficients. We show that the first equation passes the Painleve test for complete integrability. We also study the compatibility conditions that ensure the integrability of the second model with time-dependent coefficients. We show that the dispersion relations of the two equations are distinct, whereas the phase shifts are identical. We apply the simplified Hirota's method where multiple optical kink wave solutions will be reported for each equation.