Analytical insights into three models: Exact solutions and nonlinear vibrations

Abstract

Three models are investigated in this paper, including the generalized nonlinear Schr€ odinger equation with distributed coefficients, the time-dependent-coefficient nonlinear Whitham-Broer-Kaup system and the fractional nonlinear vibration governing equation of an embedded single-wall carbon nanotube. With an analytical method, the aim of this paper is to exactly solve these models. As a result, some explicit and exact solutions which include hyperbolic function solutions, trigonometric function solutions and rational solutions are obtained. To gain more insights into the obtained exact solutions, dynamical evolutions with nonlinear vibrations of the amplitudes are simulated by selecting oscillation functions, noises, coefficient functions and fractional orders. It is graphically shown in the dynamical evolutions that the nonlinear vibrations of the amplitudes are influenced not only by the coefficient functions but also by the oscillation functions, noises and fractional orders.

Exact Solutions of Nonlinear Equations in Mathematical Physics via Negative Power Expansion Method

“…Theorem 2.1 shows that the NPE method for determining the value of n in (2.2) is different from that of the auxiliary equation methods [23,25,27,[33][34][35]38], in which the value of n is related to the auxiliary equations. The main reason is that the expansions of the ansatz solutions for these two methods are different.…”

“…Constructing exact solutions of nonlinear mathematical physical equations is of theoretical and practical significance. Since the famous Korteweg-de Vries (KdV) equation was solved in 1967 [7], a large number of exact solutions like [2][3][4][5][6][7]10,11,[13][14][15][16][18][19][20][22][23][24][25][26][27][28]31,33,34,36,38,41,43] of nonlinear partial differential equations (PDEs) have been found. The exp-function method [10] proposed by He and Wu has been widely used for constructing exact solutions of nonlinear PDEs.…”

Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

“…x a ϕ(ω, t)(dω) α (0 < α ≤ 1) (6) for any nondifferentiable functions φ(x, t) and ϕ(x, t) defined on a fractal set , respectively. The concept of local fractional derivative was first proposed by Kolwankar and Gangal [17], and it has received continuous developments and extensive applications like [18][19][20][21][22][23][24][25]. In addition, this article will show some obtained fractal solutions for more insights into novel nonlinearities hidden behind the fractional order models.…”

Variational iteration method for two fractional systems with boundary conditions

“…Since the concept of local fractional derivative was presented by Kolwankar and Gangal [9] in 1996, it was widely studied, and much achievement was obtained [10][11][12][13][14][15][16][17][18][19][20]. One of the graceful properties of the local fractional calculus is that it can be used to describe the non-differential problems in science and engineering.…”