On the solitary wave solutions for nonlinear Euler equations

Exact solutions to two higher order nonlinear Schrödinger equations

“…(2) can be applied to describe the propagation of femtosecond pulses in optical fibers, too. In recent years, various powerful methods have been presented to derive the nonlinear transformations and exact solutions to the nonlinear PDEs in mathematical physics, such as: the homogeneous balance principle [3][4][5][6][7][8][9][10][11][12][13][14][15][16], F-expansion method [17][18][19][20][21][22][23][24][25], tanh method [26][27][28][29], and so on. In this letter, we use the homogeneous balance principle and Fexpansion method to explore the kink type solitary waves, bell type solitary waves and sinusoidal waves to Eqs.…”

The ( $\frac{G'}{G})$ -expansion method and its applications to some nonlinear evolution equations in the mathematical physics

“…Many powerful methods have been presented such as the inverse scattering transform [3], the Backlund transform [15,17], the generalized Riccati equation [18,25], the Jacobi elliptic function expansion [7,12,26,30,33], the extended tanh-function method [1,8,31,32,39], the F-expansion method [2,[19][20][21]36], the exp-function expansion method [5,9,28,37,38], the sub-ODE method [13,22], the extended sinhcosh and sine-cosine methods [23], the complex hyperbolic function method [34], the truncated Painlevé expansion [35] and others.…”

Numerical Solutions for Ito Coupled System

“…Let us consider an one-parameter Lie group of infinitesimal transformations [7,8] of the form: 1 (x, t, u, v, w), 2 (x, t, u, v, w),…”

“…8, which calculated by Richardson extrapolation error [2]. 6 Comparison between analytical solution of system (8) and two numerical solutions at several step size Fig. 7 Comparison between analytical solution of system (8) and two numerical solutions at several step size …”