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) $$v_{tt}-v_{xxxt}-3{\left(v_x{v}_t\right)}_x+\alpha v_{xt}+\beta v_{yt}+\gamma v_{zt}=0,$$

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) $$v(x,y,z,t)={\psi }^{-K}(x,y,z,t){\displaystyle \sum _{k=0}^K{v}_k}(x,y,z,t){\psi }^k(x,y,z,t),$$

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) $${\psi }^{-5}:v_0=2{\psi }_x,$$

(5) $$\begin{array}{l}{\psi }^{-4}:\left(\text{satisfied}\right){\psi }^{-3}:\alpha {\psi }_x{\psi }_t+\beta {\psi }_y{\psi }_t+\gamma {\psi }_z{\psi }_t-3{\psi }_x{\psi }_t{v}_{1,x}-3{\psi }_x^2{v}_{1,t}+{\psi }_t^2-{\psi }_{xxx}{\psi }_t+3{\psi }_{xt}{\psi }_{xx}-3{\psi }_x{\psi }_{xxt}=0,\end{array}$$

(6) $$\begin{array}{l}{\psi }^{-2}:2\alpha {\psi }_{xt}{\psi }_x+\alpha {\psi }_t{\psi }_{xx}+\beta {\psi }_{yt}{\psi }_x+\beta {\psi }_y{\psi }_{xt}+\beta {\psi }_t{\psi }_{xy}+\gamma {\psi }_{zt}{\psi }_x+\gamma {\psi }_z{\psi }_{xt}+\gamma {\psi }_t{\psi }_{xz}-3{\psi }_x^2{v}_{1,xt}-6{\psi }_{xt}{\psi }_x{v}_{1,x}-9{\psi }_{xx}{\psi }_x{v}_{1,t}-3{\psi }_t{\psi }_x{v}_{1,xx}-3{\psi }_t{\psi }_{xx}{v}_{1,x}+{\psi }_{tt}{\psi }_x-4{\psi }_{xxxt}{\psi }_x+2{\psi }_t{\psi }_{xt}+2{\psi }_{xt}{\psi }_{xxx}-{\psi }_t{\psi }_{xxxx}=0,\end{array}$$

(7) $$\begin{array}{l}{\psi }^{-1}:\alpha {\psi }_{xxt}+\beta {\psi }_{xyt}+\gamma {\psi }_{xzt}-3{\psi }_{xx}{v}_{1,xt}-3{\psi }_{xt}{v}_{1,xx}-3{\psi }_{xxt}{v}_{1,x}-3{\psi }_{xxx}{v}_{1,t}+{\psi }_{xtt}-{\psi }_{xxxxt}=0,\end{array}$$

(8) $${\psi }^0:v_{1,tt}-v_{1,xxxt}-3{\left(v_{1,x}{v}_{1,t}\right)}_x+\alpha v_{1,xt}+\beta v_{1,yt}+\gamma v_{1,zt}=0.$$

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:

$$\begin{array}{l}\psi (x,y,z,t)=e^{{\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}z+{\eta }_{4}t+{\eta }_5}+1,\\ v_1(x,y,z,t)={\eta }_{6}x+{\eta }_{7}y+{\eta }_{8}z+{\eta }_{9}t+{\eta }_{10},\end{array}$$

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

$${\eta }_9=\frac{{\eta }_4\left(\alpha {\eta }_1+\beta {\eta }_2-{\eta }_1^3-3{\eta }_1{\eta }_6+{\eta }_3\gamma +{\eta }_4\right)}{3{\eta }_1^2}$$

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

(9) $$v(x,y,z,t)=η1tanh⁡ (η1x+η2y+η3z+η4t+η52) + η6 x + η7 y + η8 z + η4(αη1+βη2−η13−3η1η6+η3γ+η4)3η12 t + η1 + η10 ,$$

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).