Letter to the Editor: Singular-manifold view on a (3+1)-dimensional fourth-order nonlinear equation in a fluid via HFF 32, 1664 (2022)

Citation

Gao, X.-Y. (2023), "Letter to the Editor: Singular-manifold view on a (3+1)-dimensional fourth-order nonlinear equation in a fluid via HFF 32, 1664 (2022)", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 33 No. 11, pp. 3561-3563. https://doi.org/10.1108/HFF-11-2023-938

Recently, Wazwaz (2022) and Meng et al. (2023) have made some outstanding contributions to a (3 + 1)-dimensional integrable fourth-order nonlinear equation in a fluid, which is:

(1) vtt−vxxxt−3(vxvt)x+αvxt+βvyt+γvzt=0,

with γ, β and α being the real nonzero constants, v(x, y, z, t) denoting a real differentiable function of the independent variables x, y, z and t, while the subscripts representing the partial derivatives (Meng et al., 2023). For equation (1), Wazwaz (2022) has investigated the Painlevé integrability, lump and multiple soliton solutions, while Meng et al. (2023) has presented the special cases in fluid dynamics, bilinear auto-Bäcklund transformations, breather and mixed lump-kink solutions.

This Letter, based on the work in Wazwaz (2022) and Meng et al. (2023), aims to seek an auto-Bäcklund transformation for equation (1), which is different from those in Meng et al. (2023).

In equation (1) let us put the truncated Painlevé expansion, in a generalized Laurent series (Zhou and Tian, 2022; Zhou et al., 2023; Gao, 2023a, 2023b, 2023c), around a noncharacteristic movable singular manifold conferred by an analytic function ψ(x, y, z, t) = 0, as:

(2) v (x, y, z, t)=ψ−K(x, y, z, t)∑k=0K vk (x, y, z, t) ψk (x, y, z, t),

where v_k(x, y, z, t) ’s also represent the analytic functions, with v₀(x, y, z, t) ≠ 0, ψ_x(x, y, z, t) ≠ 0 and ψ_t(x, y, z, t) ≠ 0, and if the powers of ψ at the lowest orders cancel out, the positive integer:

(3) K=1.

Using symbolic computation (Wu et al., 2022a, 2022b; Shen et al., 2022, 2023; Gao and Tian, 2022; Gao et al., 2021, 2022) and substituting formulae (2) and (3) into equation (1), we recommend that the coefficients of like powers of ψ fade away, to obtain the Painlevé-Bäcklund equations:

(4) ψ−5:  v0=2ψx,

(5) ψ−4:  (satisfied)ψ−3:  αψxψt+βψyψt+γψzψt−3ψxψtv1,x−3ψx2v1,t+ψt2−ψxxxψt+  3ψxtψxx−3ψxψxxt=0,

(6) ψ−2:  2αψxtψx+αψtψxx+βψytψx+βψyψxt+βψtψxy+γψztψx+ γψzψxt+γψtψxz−3ψx2v1,xt−6ψxtψxv1,x−9ψxxψxv1,t−  3ψtψxv1,xx−3ψtψxxv1,x+ψttψx−4ψxxxtψx+2ψtψxt+ 2ψxtψxxx−ψtψxxxx=0,

(7) ψ−1:  αψxxt+βψxyt+γψxzt−3ψxxv1,xt−3ψxtv1,xx−3ψxxtv1,x− 3ψxxxv1,t+ψxtt−ψxxxxt=0,

(8) ψ0:  v1,tt−v1,xxxt−3(v1,xv1,t)x+αv1,xt+βv1,yt+γv1,zt=0.

Mutually consistent or as noticed below, explicitly solvable with respect to ψ(x, y, z, t), v₀(x, y, z, t) and v₁(x, y, z, t), equations (2)–(8) fashion an auto-Bäcklund transformation for equation (1).

Next, the assumptions:

ψ(x,y,z,t)=e η1x+η2y+η3z+η4t+η5+1,v1(x,y,z,t)=η6x+η7y+η8z+η9t+η10,

are substituted into auto-Bäcklund transformation (2)–(8) via symbolic computation, leading to:

η9=η4(αη1+βη2−η13−3η1η6+η3γ+η4)3η12

and the following explicit soliton solutions for equation (1):

(9) v(x,y,z,t)=η1tanh⁡(η1x+η2y+η3z+η4t+η52)+η6x+η7y+η8z+η4(αη1+βη2−η13−3η1η6+η3γ+η4)3η12t+η1+η10,

where η_{1 …} η₁₀ are the real constants with η₁ ≠ 0 and η₄ ≠ 0.

Our results are linked to γ, β and α, the coefficients in equation (1).