Painlevé analysis and special solutions of generalized Broer–Kaup equations

Painlevé Integrability and N-Soliton Solution for the Whitham- Broer-Kaup Shallow Water Model Using Symbolic Computation

“…For N ≥ 3, analogous results can be obtained by choosing appropriate p i and q i with the constraint p i q i = 0. In this sense, the solutions (18) and (19) obtained in this paper are more general and abundant. Moreover, with suitable choices of the parameters, the solutions reflect various soliton surfaces and can so be used to describe many more realistic phenomena in shallow water waves.…”

“…A given PDE is said to possess the Painlevé property [18] or be a Painlevé integrable model when it passes the Painlevé PDE analysis, i.e., the solutions of this PDE are "single-valued" in the neighbourhood of noncharacteristic, movable singular manifolds [6]. The Painlevé integrability is a necessary condition for being Lax-or IST-integrable [19]. Sequentially, by truncating the Painlevé expansion at the constant level term, one can obtain the associated BT, the Lax pair, the Darboux transformation [20], and so on.…”

“…For N = 2, expressions (18) and (19) can be written as In order to better understand the mechanism of the WBK model, we draw some figures to analyze the behaviours of the solitons described by expressions (23) (27) - (31), we can obtain other two-soliton interaction modes. For the special case c = 0, expressions (27) -(31) could describe the coalescence phenomenon of two travelling waves [22].…”

Some Coherent Excitations for the (2+1)-Dimensional Generalized Broer-Kaup System

“…To determine u explicitly, one proceeds as follows: First, similar to the usual mapping approach, we can determine n by balancing the highest-order nonlinear term with the highest-order partial derivative term in (11). Second, substituting (12) with (4) and (5) into the given NPDE, collecting the coefficients of the polynomials of f i g j (i = 0, 1, ··· , j = 0, 1) and eliminating each of them, we can derive a set of partial differential equations for…”

“…where E, F, G are arbitrary constants, and it was recently derived from a typical (1+1)-dimensional BroerKaup (BK) system [12] by means of the Painlevé analysis [11]. Obviously, when E = F = G = 0, the GBK system degenerates to the celebrated (2+1)-dimensional BK system [13], which can be derived from an inner parameter dependent symmetry constraint of the Kadomtsev-Petviashvili model [14].…”