Similarity reductions and wave solutions for the 3D-Kudryashov–Sinelshchikov equation with variable-coefficients in gas bubbles for a liquid

Abundant Bounded and Unbounded Solitary, Periodic, Rogue‐Type Wave Solutions and Analysis of Parametric Effect on the Solutions to Nonlinear Klein–Gordon Model

“…Here are the solutions ( 7), ( 8), ( 9), ( 13), ( 14), ( 15), ( 22), (23), and (24) which arrived in the form of a trigonometric function, and the solutions (10), ( 11), ( 12), ( 17), ( 18), (19) t=−1 t=0 t=1…”

“…For ω 2 < β, the solution ( 22) is a complex valued function and it can be expressed as the trigonometric function in equations ( 22), (23), and (24). Solution (24) represents periodic rogue waves solution for ω � 0.5, α � 2, c � 1, β � 1 and has a nature like Figure 4.…”

“…Surveying literature, we realized ideas that many scientists worked to disclose innovative, efficient techniques for explaining internal behaviors of NLDEs with constant coefficients that are significant to elucidate different intricate problems such as a discrete algebraic framework [1], IRM-CG method [2], transformed rational function scheme [3], fractional residual method [4], new multistage technique [5], new analytical technique [6], extended tanh approach [7], Hirota-bilinear approach [8][9][10], multi exp-expansion method [11,12], Jacobi elliptic expansion method [13,14], Lie approach [15], Lie symmetry analysis techniques [16], generalized Kudryashov scheme [17,18], generalized exponential rational function scheme [19], MSE method [20][21][22], and many more. Such or similar schemes are also used to solve the model with variable coefficients to visualize various new nonlinear dynamics [23][24][25]. Recently, the nature of rogue waves and diverse dynamical interaction solutions has been studied in numerous fields.…”

“…Figures 6(b) and 6(d) for k � α � 1, c � 0.05, β � 1, ω � 0.5, taking the positive and negative signs of solutions, respectively.e profile in Figure7represents density, three dimensional control plot, and 2D plot of the imaginary chunk of (20) for c 1 � Ic 2 � 1, k � a 1 � α � 1, c � −1,β � −0.05. Figure 4For ω 2 < β, the solution (22) is a complex valued function and it can be expressed as the trigonometric function in equations (22),(23), and(24). Solution(24) represents periodic rogue waves solution for ω � 0.5, α � 2, c � 1, β � 1 and has a nature like Figure4.…”

Lie symmetry analysis, conservation laws and exact solutions for variable-coefficients (2+1)-dimensional dissipative long-wave system

“…According to the optimal system from the previous section. In this section, we are going to perform the similarity reduction for the equation (1.2) [28]. The partial differential equations containing two new variables σ and θ are obtained after the reduction, and it is given in table 3.…”