Complex simplified Hirota’s forms and Lie symmetry analysis for multiple real and complex soliton solutions of the modified KdV–Sine-Gordon equation

Abstract

The present work consists of detailed exploration of modified KdV–Sine-Gordon equation in integrable form, owning to two-component nonlinear channel for modeling laser light propagation. For validating the behavior of this equation in the sense of integrability, we use the Painleve test. The simplified Hirota’s technique with new complex forms is developed suitably to construct multiple-soliton solutions with complex structure for considered equation. Moreover, Lie symmetry analysis has been implemented for perceiving symmetries of MKdV–SG equation and then culminating the invariant solitary wave solutions. The new findings obviously reveal that simplified Hirota’s technique with complex structure would be highly proficient for fabricating new multiple complex soliton solutions to other nonlinear equations with integrable properties from mathematical physics and dynamical systems community.