Multi-waves, breather wave and lump–stripe interaction solutions in a (2 $$+$$ 1)-dimensional variable-coefficient Korteweg–de Vries equation

A numerical study of third-order equation with time-dependent coefficients: KdVB equation

“…On the other hand, by introducing variable coefficients α(t) > 0 and β(t) > 0, the KdVB equation (2.1) is useful to describe solitonic propagation in fluids [38], a variety of cosmic plasma phenomena [15,26,36,16], among others. Motivated by those applications and as mentioned, as far as we know, an exhaustive numerical study for (2.1) has not been reported.…”

“…As mentioned, the KdVB equation is considered to investigate the impact of bottom configurations on the free surface waves and describe a wide variety of phenomena arise in plasma physics, among others. Motivated by those applications and using as starting point the references [15,26,16,22,20,21], in this subsection we develop three parametric configurations among the coefficients α and β. Although several constant physics have been simplified in our analysis, all profiles below are consistent with the previous sections and the references above mentioned.…”

“…As mentioned at the beginning of this section, the chosen temporal profiles are based upon the papers found in the literature. From [26] and [22] and by simplicity, some physical data have been modified. More precisely, we analyze the following two cases:…”

Soliton solutions and their degenerations in the (2+1)-dimensional Hirota–Satsuma–Ito equations with time-dependent linear phase speed

“…In addition, the structures of the solutions in the nonlinear equations with variable coefficients are more diverse. Many authors pay attention to a variety of variable-coefficient nonlinear equations, such as Hirota equation [24] and the nonlinear Schrödinger equation [25] in inhomogeneous optical fibers, the Kadomtsev-Petviashvili (KP) equation in a fluid [26], the KdV equation [27,28], and the Boiti-Leon-Manna-Pempinelli (BLMP) equation [10,29,30]. The bright one and two soliton solutions of Kundu-Eckhaus equation with variable coefficients were constructed by the bilinear method and the corresponding movements and collisions were illustrated in [31].…”

Lump periodic wave, soliton periodic wave, and breather periodic wave solutions for third-order (2+1)-dimensional equation

“…Ilhan et al [23] have derived the lump solution and thier interaction with the exponential soliton and the hyperbolic soliton to a variable-coefficient Kadomtsev-Petviashvili equation. Liu et al [24,25] have derived interaction solutions between a lump and two solitary waves, using the Hirota bilinear method. The same Authors have investigated also multi-waves solutions by using the three waves method and breather wave solutions, using the homoclinic breather approach.Beside the lump, soliton and rogue wave solutions, that are the rational solutions, the lump periodic solutions, solitary wave periodic solutions and the breather periodic wave of these equations have importance physical significance to observe the oscillatory behaviour in applied sciences and engineering, especially in elastic media, fluid dynamics, biotechnology, shallow water wave theory, to name just a few.…”