New solitary and optical wave structures to the (1 + 1)-dimensional combined KdV–mKdV equation

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

“…Nonlinear evolution equations are often used to describe some physical aspects that arise in the various fields of nonlinear sciences, such as plasma physics, quantum mechanics, biological sciences, chemistry, chemical physics, and so forth. Various powerful techniques have been formulated and used by different scholars to find the solutions of some NLEEs, such as the sine-Gordon expansion method [1][2][3], the generalized Kudryashov method [4,5], the extended tanh method [6,7], the new generalized and improved (G /G)-expansion method [8], the Jacobi elliptic function method [9,10], the improved Bernoulli subequation function method [11], the tanh method [12,13], the sine-cosine method [14], the Lie group analysis method [15][16][17], the homogeneous balance method [18], the modified simple equation method [19,20], the meshless method of radial basis functions [21], He's variational iteration method [22], the explicit multistep Galerkin finite element method [23], the differential quadrature based numerical method [24], the partitioned second-order method [25], the adaptive pseudo-transient-continuation-Galerkin methods [26]. In general, various efficient techniques have been implemented to explore the search for the solutions of the different kind of NLEEs [27][28][29][30][31][32][33][34][35][36][37][38].…”

Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation

“…For solving the nonlinear partial differential equations, there have been many schemes applied such as the Kudryashov method by M. Foroutan et al in [15] and K. K. Ali et al in [16]; the modified Kudryashov method by K. Hosseini et al in [17,18], D. Kumar et al in [19], A. K. Joardar et al in [20] and A.R. Seadawy et al in [21]; the generalized Kudryashov method by F. Mahmud et al in [22], S. T. Demiray et al in [23] and S. Bibi et al in [24]; the sine-cosine method by K. R. Raslan et al in [25]; the sine-Gordon method by H. Bulut et al in [26]; the sinh-Gordon equation expansion method by H. M. Baskonus et al in [27], Y. Xian-Lin et al in [28] and A. Esen et al in [29]; the extended trial equation method by K. A. Gepreel in [30], Y. Pandir et al in [31] and Y. Gurefe et al in [32]; the Exp-function method by L.K. Ravi et al in [33], A. R. Seadawy et al in [34] and M. Nur Alam et al in [35]; the Jacobi elliptic function method by S. Liu et al in [36]; the F-expansion method by A. Ebaid et al in [37]; and the extended G G method by E. M. E. Zayed and S. Al-Joudi et al in [38].…”

New soliton properties to the ill-posed Boussinesq equation arising in nonlinear physical science

“…Nonlinear evolution equation are often used to describe complex aspects of various models arising in the field of nonlinear sciences such as mathematical physics, chemical physics, chemistry, biological sciences etc. Attention from different researchers has been paid to this area in searching for new solutions to the different class of NLEEs where various powerful method are formulated such as the generalized and improved   / GG  -expansion method [1], the Jacobi elliptic-function method [2], the modified simple equation method [3,4], the sine-Gordon expansion method [5][6][7], the extended tanh method [8], the improved Bernoulli sub-equation function method [9], the rational sine-cosine method [10], the RicattiBernoulli sub-ODE method [11], the Homotopy perturbation method [12] and so on. However, in this work we aim at investigating solution of the ill-posed Boussinesq equation [13] by using the modified exp…”