Painlevé Analysis, Soliton Solutions and Bäcklund Transformation for Extended (2 + 1)-Dimensional Konopelchenko—Dubrovsky Equations in Fluid Mechanics via Symbolic Computation

Abstract

This paper is to investigate the extended (2+1)-dimensional Konopelchenko-Dubrovsky equations, which can be applied to describing certain phenomena in the stratified shear flow, the internal and shallow-water waves, plasmas and other fields. Painlevé analysis is passed through via symbolic computation. Bilinear-form equations are constructed and soliton solutions are derived. Soliton solutions and interactions are illustrated. Bilinear-form Bäcklund transformation and a type of solutions are obtained.

Transverse spectral instabilities in Konopelchenko–Dubrovsky equation

“…Lemma 3. The operator  𝑎,𝜏 (𝜇, 𝛾) is not invertible in 𝐿 2 (𝕋) for some 𝜇 ∈ ℂ and 𝜏 ≠ 0 if and only if 𝜇 ∈ spec 𝐿 2 (𝕋) ( 𝑎 (𝛾, 𝜏)), 𝐿 2 (𝕋)-spectrum of the operators,  𝑎 (𝛾, 𝜏) ∶= 𝑘𝑐(𝜕 𝑧 + 𝑖𝜏) + 𝑘 3 (𝜕 𝑧 + 𝑖𝜏) 3…”

“…We consider the (2+1)‐dimensional Konopelchenko–Dubrovsky (KD) equation 1,2 $$$$\begin{equation} {\left\lbrace \begin{aligned} u_{t}-u_{xxx}-6 \rho u u_{x} +\dfrac{3}{2}\phi ^{2} u^{2} u_{x}-3 v_{y}+3 \phi u_{x} v &= 0, \\ u_{y} &= v_{x}, \end{aligned}\right.} \end{equation}$$$$where, $$u=u(x,y,t)$$, $$v=v(x,y,t)$$, the subscripts denote partial differentiation, ρ and ϕ are real parameters, defining the magnitude of nonlinearity in wave propagation, modeled for stratified shear flow, the internal and shallow‐water waves, and the plasmas, 3 can also be regarded as combined KP and modified KP equation, 4 or generalized (2+1)D Gardner equation 5 …”

“…It is among the few nonlinear evolution equations which are completely integrable in different settings. Notably, the considered KD equation is also integrable in the Painlevé sense and solvable through IST 1,3,5 …”

“…where, 𝑢 = 𝑢(𝑥, 𝑦, 𝑡), 𝑣 = 𝑣(𝑥, 𝑦, 𝑡), the subscripts denote partial differentiation, 𝜌 and 𝜙 are real parameters, defining the magnitude of nonlinearity in wave propagation, modeled for stratified shear flow, the internal and shallow-water waves, and the plasmas, 3 can also be regarded as combined KP and modified KP equation, 4 or generalized (2+1)D Gardner equation. 5…”

Invariance of Painlevé property for some reduced (1+1)-dimensional equations

“…which have been proposed by Konopelchenko and Dubrovsky [23] with the inverse scattering transformation method. Xu et al [24] studied the Painlevé integrability of KD equation. Many researchers have investigated Eq.…”