Nonintegrable reductions of the self-dual Yang–Mills equations in a metric of plane wave type

Abstract

Symmetry reductions of the self-dual Yang–Mills equations for SL(2,C) bundles with the background metric ds2=2 du dv−dx2+f2(u)dy2 are considered. One of the field components in the reduced equations can be cast into Jordan normal form after gauge transformations. The reduced equations for the two possible normal forms are equivalent, respectively, to certain generalizations of the Korteweg–de Vries (KdV) equation and the nonlinear Schrödinger (NLS) equation. It is shown that the generalized KdV and NLS equations fail the Painlevé test except when the metric is flat. The generalized KdV equation is transformed to a simple form in the case when f(u)=ua and it is shown that one may obtain either the KdV equation or the cylindrical KdV equation by this method only when the metric is flat.

Letter to the editor: Discussing some Korteweg-de Vries-directional contributions in fluid mechanics, atmospheric science, plasma physics and nonlinear optics concerning HFF 33, 3111 and 32, 1674

“…Vitalized by a sense of refreshing the KdV work from Wazwaz et al (2023) and Khan (2022), this Letter will consider the following generalized forced variable-coefficient KdV equation in fluid mechanics, atmospheric science, plasma physics or nonlinear optics (Brugarino, 1989; Kapadia, 2001; Zhang et al , 2013; Chen et al , 2020a, 2020b): with the subscripts denoting the partial derivatives, ϕ ( χ , τ ), the wave amplitude, being a real differentiable function as for the scaled time coordinate τ and scaled space coordinate χ , the external-force term f ( χ , τ ) and dissipative coefficient q ( χ , τ ) representing the differentiable functions as for χ and τ while ρ ( τ ), a ( τ ) and z ( τ ) corresponding to the effects of the nonlinearity, third-order dispersion and relaxation/perturbation, respectively (Brugarino, 1989; Kapadia, 2001; Zhang et al , 2013; Chen et al , 2020a, 2020b).…”

“…Painlevé property for equation (1) under some variable–coefficient constraints has been discussed (Brugarino, 1989; Kapadia, 2001). Brugarino (1989) has also obtained an auto-Bäcklund transformation and a Lax pair for equation (1) and shown that equation (1) is transformable to the KdV equation via some variable–coefficient constraints.…”

“…Vitalized by a sense of refreshing the KdV work from Wazwaz et al (2023) and Khan (2022), this Letter will consider the following generalized forced variable-coefficient KdV equation in fluid mechanics, atmospheric science, plasma physics or nonlinear optics (Brugarino, 1989;Kapadia, 2001;Zhang et al, 2013;Chen et al, 2020aChen et al, , 2020b:…”

“…(1) with the subscripts denoting the partial derivatives, f(x, t), the wave amplitude, being a real differentiable function as for the scaled time coordinate t and scaled space coordinate x, the external-force term f(x, t) and dissipative coefficient q(x, t) representing the differentiable functions as for x and t, while r(t), a(t) and z(t) corresponding to the effects of the nonlinearity, third-order dispersion and relaxation/perturbation, respectively (Brugarino, 1989;Kapadia, 2001;Zhang et al, 2013;Chen et al, 2020aChen et al, , 2020b. Painlev e property for equation ( 1) under some variable-coefficient constraints has been discussed (Brugarino, 1989;Kapadia, 2001). Brugarino (1989) has also obtained an auto-Bäcklund transformation and a Lax pair for equation (1) and shown that equation ( 1) is transformable to the KdV equation via some variable-coefficient constraints.…”

“…Chen et al (2020b) have constructed a reduction from equation (1) to another variable-coefficient KdV equation and gotten certain rational, periodic and mixed solutions for equation (1). Special cases of equation ( 1) have been seen in fluid mechanics, atmospheric science, plasma physics and nonlinear optics (Brugarino, 1989;Kapadia, 2001;Zhang et al, 2013;Chen et al, 2020aChen et al, , 2020b. By the way, other nonlinear models in fluid mechanics, atmospheric science, plasma physics and nonlinear optics have also been presented (Chen et al, 2024a(Chen et al, , 2024bPeng et al, 2024;Cheng et al, 2022Cheng et al, , 2023aCheng et al, , 2023bGao, 2023aGao, , 2024bGao, , 2024cGao et al, 2023aGao et al, , 2023bWu and Gao, 2023;Wu et al, 2023aWu et al, , 2023bShen et al, 2023aShen et al, , 2023bShen et al, , 2023cZhou and Tian, 2022;Zhou et al, 2023aZhou et al, , 2023bFeng et al, 2023).…”