Intricate and multiple chirped waves geometric structures solutions of two-mode KdV equation, spectral and stability analysis

Abstract

In a fluid, when two modes are propagating simultaneously in the same direction with different velocities ([Formula: see text] and [Formula: see text]), they are regarded as first and second modes. The two-mode Korteweg de Vries equation (TMKdVE) has been currently studied in numerous works in the literature. However, the distinction between the two modes was not studied. Here, exact solutions of the TMKdVE are obtained by using the unified method. It is found that each solution depends either on [Formula: see text] or on [Formula: see text]. These solutions are taken to assign to the first mode (FM) and the second mode (SM), respectively. These results are completely new. In this work, attention is focused on investigating the different chirped wave structures. It is shown that intricate, complex and multiple chirped waves in the two modes occur. Furthermore, it is shown that each mode can be seen in two visions: right wave (RW) and left wave (LW). The stability of steady-state solution is analyzed, where it is found that it is stable against varying the velocity [Formula: see text]. While when varying the velocity [Formula: see text], it is unstable when [Formula: see text] and stable when [Formula: see text]. The properties of the waves, the characteristic wavelength, frequency, traveling speed and spectrum content are shown graphically.