Multisoliton solutions, completely elastic collisions and non-elastic fusion phenomena of two PDEs

Solitary and periodic wave solutions to the family of nonlinear conformable fractional Boussinesq‐like equations

“…Case 1: When < 0 (hyperbolic function solutions) Substituting Equation (19) along with Equations (9) and (11) into Equation 18, we obtain a polynomial equation in ( ) and ( ). Setting each coefficient of the equation to zero, we obtain a system of algebraic equations (for minimalism, the equations are not displayed for this case and for subsequent other cases).…”

“…Substituting Equation (19) together with Equations (9) and (13) into Equation 18, we obtain a polynomial equation in ( ) and ( ). Setting each coefficient of the equation to zero, we have a system of algebraic equations, which on solving with the aid of the symbolic computation software Maple 17 yields the following solutions.…”

“…In this context, various symbolic computational packages, namely, Maple, Mathematica, and MATLAB, make much easier to create a platform for physicist, mathematicians, and engineers in evolving numerous analytical and numerical methods for seeking a variety of new exact solutions of nonlinear PDEs. The numerous evolution methods are the first integral method, 5 the modified tanh–coth method, 6 the extended Jacobi elliptic function expansion method, 6 the semi‐inverse variational principle, 7,10 the sine‐cosine method, 8 the modified trial equation method, 11 the extended rational trigonometric method, 9 the modified Kudryashov method, 1,12,13 the Adomian decomposition method, 14 the exponential rational function method, 2 the Exp‐function method, 3,10 the Darboux transform method, 15 the Hirota's bilinear method, 16 the Bäcklund transformation and inverse scattering method, 17 the multiple Exp‐function method, 18,19 the improved and ‐expansion methods, 20,21 the novel exponential rational function method, 22 the modified simple equation method, 23 the unified method, 24 the extended simple equation method, 25 the ‐expansion and its extension methods, 25,26 the sine‐Gordon expansion method, 27–29 the extended sinh‐Gordon expansion method, 30–33 the ‐expansion method, 34 the improved and ‐expansion methods, 35 the …”

Solitary and periodic wave solutions to the family of new 3D fractional WBBM equations in mathematical physics

“…Different symbolic computational sets, namely Mathematica, Maple, and MATLAB, make it far simpler for physicists, mathematicians, and engineers to build a forum to develop various numerical and analytical methods range of new precise nonlinear PDE solutions. The methods of numeral evolution are the first integral technique [7] , the modified Kudryashov technique [8] , [9] , [10] , the modified extended tanh-function method [11] , [12] , the improved simple equation technique [13] , the method of characteristics [14] , the novel exponential rational function technique [15] , the semi-inverse variational principle [16] , [17] , the multiple Exp-function system [18] , [19] , the sine-cosine method [20] , the Exp-function method [17] , [21] , the improved and -expansion methods [22] , [23] , the modified trial equation method [24] , the extended rational trigonometric method [25] , the unified method [26] , the Darboux transform method [27] , the Adomian decomposition method [28] , the exponential rational function method [29] , the Bäcklund transformation and inverse scattering method [30] , Hirota's bilinear method [31] , the advanced exp( )-expansion methods [32] , [33] , the extended simple equation method [32] , the extended sinh-Gordon expansion method [34] , [35] , [36] , [37] , the sine-Gordon expansion method [38] , [39] , [40] , the improved ( ) and ( …”

Existence of Solitary Waves in a Perturbed KdV-mKdV Equation

“…KdV-mKdV equation plays a most important role in soliton physics and appears in many practical scenarios such as thermal pulse and wave propagation of the bound particle [1]. Recently, many mathematicians are interested in studying traveling wave solutions for the KdV-mKdV equation, and there are many tools to find the traveling wave solutions, such as the inverse scattering method [2], the double subequation method [3], the sine-cosine method [4,5], the Darboux transformation [6], the Exp-expansion method [7,8], and the bifurcation method of dynamic systems [9]. KdV-mKdV equation is one of the most popular soliton equations; therefore, there were widely related investigations in [10][11][12][13][14].…”