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

Citation

Gao, X.-Y., Guo, Y.-J. and Shan, W.-R. (2024), "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", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 34 No. 5, pp. 1929-1936. https://doi.org/10.1108/HFF-05-2024-943

Recently, in International Journal of Numerical Methods for Heat & Fluid Flow, Wazwaz et al. (2023) and Khan (2022) have made some interesting Korteweg-de Vries (KdV)-directional contributions, respectively, on a linear structure of components of the modified KdV hierarchy including the third-order, fifth-order and seventh-order modified KdV equations with the solitons and breathers, as well as on a Hausdorff-fractal isotropically extended Lax–KdV equation for an incompressible fluid.

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):

(1) ϕτ+ρ(τ)ϕϕχ+a(τ)ϕχχχ+[q(χ,τ)ϕ]χ+z(τ)ϕ+f(χ,τ)=0  ,

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. Zhang et al. (2013) have worked out certain bilinear forms, Bäcklund transformations, Lax pair, infinite conservation laws and soliton solutions for equation (1). Chen et al. (2020a) have obtained a bilinear Bäcklund transformation, a non-isospectral Ablowitz-Kaup-Newell-Segur system and some infinite conservation laws for equation (1). 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., 2020a, 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, 2024b; Peng et al., 2024; Cheng et al., 2022, 2023a, 2023b; Gao, 2023a, 2024b; Gao et al., 2023a, 2023b; Wu and Gao, 2023; Wu et al., 2023a, 2023b; Shen et al., 2023a, 2023b, 2023c; Zhou and Tian, 2022; Zhou et al., 2023a, 2023b; Feng et al., 2023).

However, to our knowledge, reductions for equation (1) beyond those in Brugarino (1989) and Chen et al. (2020b) have not been published as yet. This Letter will find out two sets of the similarity reductions for equation (1) with symbolic computation (Gao et al., 2024; Wu et al., 2023c; Shen et al., 2023d, 2023e, 2023f; Zhou et al., 2023c), which are different from those in Brugarino (1989) and Chen et al. (2020b).

For the sake of constructing certain similarity reductions for equation (1), similar to those in Clarkson and Kruskal (1989); Gao (2023b, 2024a); and Gao and Tian (2024), the form of ϕ(χ, τ) can be assumed as:

(2) ϕ(χ,τ)=α(χ,τ)+β(χ,τ)r[w(χ,τ)],

of which α(χ, τ), β(χ, τ) ≠ 0 and w(χ, τ) ≠ 0 represent some real differentiable functions to be determined while r(w) denotes a real differentiable function as for w only.

Making use of symbolic computation and inserting assumption (2) into equation (1), we find that:

(3) m0r″′+m1r″+m2rr′+m3r2+m4r′+m5r+m6=0,

with m_i’s (i = 0, …, 6) meaning the real differentiable functions of χ and τ:

(4a) m0=a(τ)βwχ3 ,

(4b) m1=3a(τ)wχ(βχwχ+βwχχ) ,

(4c) m2=ρ(τ)β2wχ ,

(4d) m3=ρ(τ)ββχ ,

(4e) m4=3a(τ)(βχwχχ+βχχwχ)+β[wτ+q(χ,τ)wχ+ρ(τ)αwχ+a(τ)wχχχ],

(4f) m5=z(τ)β+βτ+[q(χ,τ)β]χ+ρ(τ)(βα)χ+a(τ)βχχχ,

(4g) m6=f(χ,τ)+z(τ)α+ατ+[q(χ,τ)α]χ+ρ(τ)ααχ+a(τ)αχχχ,

while the prime sign denoting the differentiation, hereby with respect to w.

Because equation (3) should be required as an ordinary differential equation (ODE) about r(w), we order that the ratios of different derivatives and powers of r(w) denote the functions as for w only. Hence, each set of α(χ, τ), β(χ, τ) ≠ 0 and w(χ, τ) = 0, which we plan to work out, can bring about a similarity reduction.

Case 1: w_χ ≠ 0.

As a result of m₀ ≠ 0, we obtain:

(5) mi=Ωi(w)m0,

on which Ω_i(w)’s imply some real to-be-determined functions with respect to w only.

The second freedom in remark 2 in Clarkson and Kruskal (1989) simplifies equations (5) with i = 2 into:

(6) β(χ,τ)=a(τ)ρ(τ)wχ2  , Ω2(w)=1.

Based on the first freedom in remark 2 in Clarkson and Kruskal (1989), equations (5) with i = 3 can result in:

(7) w(χ,τ)=σ1(τ)χ+σ2(τ)  , Ω3(w)=0,

and then equations (5) with i = 1 may come to:

(8) Ω1(w)=0,

with σ₁(τ) ≠ 0 and σ₂(τ) as two real differentiable functions of τ.

On account of the first freedom in remark 2 in Clarkson and Kruskal (1989), equation (5) with i = 4 should give rise to:

(9) α(χ,τ)=−q(χ,τ)ρ(τ)−σ1′(τ)χ+σ2′(τ)ρ(τ)σ1(τ), Ω4(w)=0,

and equation (5) with i = 5 would make for:

(10) q(χ,τ)=q1(τ)χ+q2(τ), a(τ)=μ0ρ(τ), σ1(τ)=λ1e−∫z(τ)dτ  , Ω5(w)=0,

as a result that the first freedom in remark 2 in Clarkson and Kruskal (1989) helps us transform equation (5) with i = 6 into:

(11a) σ2(τ)=λ2  , q2(τ)=μ2ρ(τ)  , z(τ)=μ1ρ(τ)+q1(τ),

(11b) f(χ,τ)=−μ1[2μ1ρ(τ)+3q1(τ)]χ+μ2[μ1ρ(τ)+2q1(τ)]  , Ω6(w)=0,

with λ₁ ≠ 0, λ₂, μ₀ ≠ 0, μ₁ and μ₂ as five real constants.

Thus, making use of symbolic computation, we are able to transform equation (3) into the following ODE:

(12) r″′+rr′=0,

which could be integrated once with respect to w, i.e.:

(13) r″+12r2+η1=0,

with η₁ as a real constant of integration.

In sum, under the following variable–coefficient constraints:

(14a) ρ(τ)≠0  , q(χ,τ)=q1(τ)χ+μ2ρ(τ)  , z(τ)=μ1ρ(τ)+q1(τ)≠0,

(14b) a(τ)=μ0ρ(τ)  , f(χ,τ)=−μ1[2μ1ρ(τ)+3q1(τ)]χ+μ2[μ1ρ(τ)+2q1(τ)],

we find out a set of the similarity reductions for equation (1), i.e.:

(15a) ϕ(χ,τ)=(μ1χ−μ2)+μ0λ12e−2[μ1∫ρ(τ)dτ+∫q1(τ)dτ]r[w(χ,τ)],

(15b) w(χ,τ)=λ1e−∫z(τ)dτχ+λ2,

(15c) r″+12r2+η1=0.

ODE (15c), a known ODE, has been investigated in Ince (1956).

In fluid mechanics, atmospheric science, plasma physics and nonlinear optics, with respect to the wave amplitude, under variable–coefficient constraints (14), similarity reductions (15) rely on the nonlinearity coefficient ρ(τ) and relaxation/perturbation coefficient z(τ) for equation (1).

Case 2: w_χ = 0.

On account of m₀ = m₁ = m₂ = 0 and m₄ ≠ 0, we find that:

(16a) m3=∑1(w)m4,

(16b) m5=∑2(w)m4,

(16c) m6=∑3(w)m4,

of which ∑_h(w)’s (h = 1, 2, 3) mean some real to-be-determined functions as for w only.

Because the second freedom in remark 2 in Clarkson and Kruskal (1989) makes us simplify equation (16a) into:

(17) β(χ,τ)=w′(τ)ρ(τ)[χ+θ0(τ)], ∑1(w)=1,

according to the first freedom in remark 2 in Clarkson and Kruskal (1989), equations (16b) and (16c) bring about:

(18a) w(τ)=θ1∫ρ(τ)dτ, θ0(τ)=θ2, α(χ,τ)=−12(δ1χ+δ2),

(18b) f(χ,τ)=14δ12ρ(τ)χ+14[δ1δ2ρ(τ)+(δ2−δ1θ2)z(τ)],

(18c) q(χ,τ)=12[δ1ρ(τ)−z(τ)]χ+12[δ2ρ(τ)−θ2z(τ)], ∑2(w)=∑3(w)=0,

in which θ₁ ≠ 0, θ₂, δ₁ and δ₂ denote four real constants, whereas θ₀(τ) represents a real differentiable function of τ.

In short, under the following variable–coefficient constraints:

(19a) ρ(τ)≠0,  q(χ,τ)=12[δ1ρ(τ)−z(τ)]χ+12[δ2ρ(τ)−θ2z(τ)],

(19b) f(χ,τ)=14δ12ρ(τ)χ+14[δ1δ2ρ(τ)+(δ2−δ1θ2)z(τ)],

we conclude with another set of the similarity reductions for equation (1), i.e.:

(20a) ϕ(χ,τ)=−12(δ1χ+δ2)+θ1(χ+θ2)r[w(τ)],

(20b) w(τ)=θ1∫ρ(τ)dτ,

(20c) r′+r2=0.

ODE (20c), a known ODE, has been reported in Zwillinger and Dobrushkin (2022), and is hereby solved out as:

r(w)=1w−ϒ,

where ϒ is a constant.

In fluid mechanics, atmospheric science, plasma physics and nonlinear optics, with respect to the wave amplitude, under variable–coefficient constraints (19), similarity reductions (20) rely on the nonlinearity coefficient ρ(τ) for equation (1).