Multiple and exact soliton solutions of the perturbed Korteweg–de Vries equation of long surface waves in a convective fluid via Painlevé analysis, factorization, and simplest equation methods

In this research, the surface waves of a horizontal fluid layer open to air under gravity field and vertical temperature gradient effects are studied. The governing equations of this model are reformulated and converted to a nonlinear evolution equation, the perturbed Korteweg–de Vries (pKdV) equation. We investigate the latter equation, which includes dispersion, diffusion, and instability effects, in order to examine the evolution of long surface waves in a convective fluid. Dispersion relation of the pKdV equation and its properties are discussed. The Painlevé analysis is applied not only to check the integrability of the pKdV equation but also to establish the Bäcklund transformation form. In addition, traveling wave solutions and a general form of the multiple-soliton solutions of the pKdV equation are obtained via Bäcklund transformation, the simplest equation method using Bernoulli, Riccati, and Burgers' equations as simplest equations, and the factorization method.

Figure 1

Dispersion relation of the pKdV equation and its properties, Eqs. (14, 15), at $T=30{}^{\circ }\text{C}$. Blue (dark gray), dotted, dashed, and red (light gray) curves correspond to real and imaginary parts of dispersion relation, phase, and group velocities, respectively.

Figure 2

Exact wave solution of the pKdV equation [real part of solution $u\times {10}^{12}\left(\mathrm{mm}/\mathrm{s}\right)$] using the Bäcklund transformation with the dispersion relation (14) at $k=0.5,C_1=-1,C_2=1$, and $C_3=-1.$

Figure 3

Exact wave solution of the pKdV equation $[u\times {10}^{10}\left(\mathrm{mm}/\mathrm{s}\right)]$ using the Bäcklund transformation without the dispersion relation at $k=0.5,\omega =1,C_1=-1,C_2=1$, and $C_3=-1.$

Figure 4

Traveling wave solution (40) of the pKdV equation $[u_1\times {10}^{11}\left(\mathrm{mm}/\mathrm{s}\right)]$ using the Bernoulli equation at $k=0.05,q=1$, and ${\xi }_0=1.$

Figure 5

Traveling wave solution (A1) of the pKdV equation $[u_1\times {10}^{11}\left(\mathrm{mm}/\mathrm{s}\right)]$ using the Riccati equation at $k=0.05,p=0.5,q=2,$ and ${\xi }_0=-1.$

Figure 6

Traveling wave solution (A2) of the pKdV equation $[u_2\times {10}^{11}\left(\mathrm{mm}/\mathrm{s}\right)]$ using the Riccati equation at $k=0.05,p=0.5,q=2$, and ${\xi }_0=1.$

Figure 7

Traveling wave solution (42) of the pKdV equation $[u_1\times {10}^{11}\left(\mathrm{mm}/\mathrm{s}\right)]$ using the Burgers' equation at $k_i=0.05,\beta =1$, and ${\xi }_0=1.$

Figure 8

Traveling wave solution (43) of the pKdV equation $[u_2\times {10}^{11}\left(\mathrm{mm}/\mathrm{s}\right)]$ using the Burgers' equation at $k_i=0.05.$

Figure 9

Traveling wave solution (55) of the pKdV equation [real and imaginary parts of solution $u\left(\mathrm{mm}/\mathrm{s}\right)$] using the factorization method with the dispersion relation (14) at $k=0.05$ and ${\xi }_0=-1.$

Figure 10

Kink wave solution (55) of the pKdV equation $\left[u\right(\mathrm{mm}/\mathrm{s}\left)\right]$ using the factorization method without the dispersion relation (14) at $k=0.5,\omega =1$, and ${\xi }_0=-1.$

Figure 11

2D kink wave solution (55) of the pKdV equation using the factorization method with the dispersion relation (14) at $x=3\text{mm},k=0.05$, and ${\xi }_0=-1.$ Blue and dashed curves correspond to real and imaginary parts of solution $\left(u\right)$, respectively.