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The initial value problem for the general coupled Hirota system with nonzeroboundary conditions at infinity is solved by reporting a rigorous theory of theinverse scattering transform. With the help of a suitable uniformizationvariable, both the inverse and the direct problems are analyzed which allows usto develop the inverse scattering transform on the complex $z$-plane. Firstly,analyticity of the scattering eigenfunctions and scattering data, properties ofthe discrete spectrum, symmetries, and asymptotics are discussed in detail.Moreover, the inverse problem is posed as a Riemann-Hilbert problem for theeigenfunctions, and the reconstruction formula of the potential in terms ofeigenfunctions and scattering data is presented. Finally, the maincharacteristics of these obtained soliton solutions are graphically discussedin the $2\times2$ self-focusing case. This family of solutions contains novelAkhmediev breather and Kuznetsov-Ma soliton. These results would be of muchimportance in understanding and enriching breather wave phenomena arising innonlinear and complex systems, especially in Bose-Einstein condensates.

Inverse scattering transform of the general coupled Hirota system with nonzero boundary conditions