Integrability and soliton-like solutions for the coupled higher-order nonlinear Schrödinger equations with variable coefficients in inhomogeneous optical fibers

• We studied the dynamics of femtosecond soliton pulse in inhomogeneous optical fibers.
• Infinitely-many conservation laws are obtained based on the Lax pair.
• Bilinear Bäcklund transformation are derived through the Hirota method.
• Propagation and interactions of some special shaped pulses are analyzed.
• We get the bound states with periodic attraction and repulsion between two solitons.

Under investigation in this paper are the coupled higher-order nonlinear Schrödinger equations with variable coefficients, which represent the propagation of femtosecond soliton pulses comprising of two fields with the left and right polarization in the inhomogeneous optical fiber media. Infinitely-many conservation laws are obtained based on the Lax pair. Via the Hirota method and symbolic computation, bilinear forms, bilinear Bäcklund transformations, one- and two-soliton-like solutions are also derived. With different coefficients, bell-shaped, periodic-changing, quadratic-varying, exponential-decreasing and exponential-increasing soliton-like profiles are seen, to describe the propagation and interactions of the femtosecond soliton pulses. Head-on and overtaking elastic interactions are shown, which are decided by the directions of the velocities. We also get the bound states with periodic attraction and repulsion between two solitons.