Stair and Step Soliton Solutions of the Integrable (2+1) and (3+1)-Dimensional Boiti—Leon—Manna—Pempinelli Equations

Abstract

The multiple exp-function method is a new approach to obtain multiple wave solutions of nonlinear partial differential equations (NLPDEs). By this method one can obtain multi-soliton solutions of NLPDEs. In this paper, using computer algebra systems, we apply the multiple exp-function method to construct the exact multiple wave solutions of a (2+1)-dimensional Boiti—Leon—Manna—Pempinelli equation. Also, we extend the equation to a (3+1)-dimensional case and obtain some exact solutions for the new equation by applying the multiple exp-function method. By these applications, we obtain single-wave, double-wave and multi-wave solutions for these equations.

New soliton solutions of dissipative (2+1)-dimensional AKNS equation

“…x u x y t R ln f x y t (2) into the equation under discussion, where the auxiliary function f , for the single soliton solution, is given by…”

“…Also, the explicit formulas may provide physical information and help us to understand the mechanism of related physical models. In recent years, many kinds of powerful methods have been proposed to find solutions of nonlinear partial differential equations, numerically and/or analytically, e.g., Binary Bell Polynomials Method [1], the multiple exp-function method [2], the tanh method [3], the sine-cosine method [4], the Exp-function method [5], the homogeneous balance method [6] and the extended homoclinic test approach [7].…”

Soliton and Riemann theta function quasi-periodic wave solutions for a $$(2+1)$$ ( 2 + 1 ) -dimensional generalized shallow water wave equation

“…(2), some exact solutions were presented with an arbitrary function in y. In [52], the multi-soliton solutions for Eq. (2) were obtained by means of the multiple exp-function method.…”

On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

“…Equation (1) can also be used to describe the (2+1)-dimensional interaction of the Riemann wave propagating along the ξ-axis with a long wave propagating along the ζ-axis [10]. For ζ = ξ, (1) can be reduced to the KdV equation [6,7].…”

“…Higher-dimensional NLEEs are scientifically interesting: For example, some (3+1)-dimensional KdV-type equations can describe the dust-ion-acoustic waves in cosmic nonmagnetised dusty plasmas such as those in the supernova shells and Saturn's F-ring [28], and some (3+1)-dimensional NLEEs, with certain parameters, can reduce to the (2+1)-dimensional and (1+1)-dimensional NLEEs [29][30][31][32][33][34][35] where u is an analytic function depending on the scaled spatial coordinates (x, y, z) and temporal coordinate t. By virtue of the exp-function method [10] and bilinear method via the logarithm transformation [36], the soliton solutions of (2) have been discussed [10,36]. In the fluid and plasma dynamics, special cases of (2) are seen as follows: (a) When y = z, u(x, y, t) = (η, t) =  [x + χ(y), t] with as χ an analytic function of y, (2) degenerates into the KdV equation [37],…”