Letter to the Editor on HFF 32, 138 (2022); 32, 2282 (2022); 33, 965 (2023) and 34, 1189 (2024) for the recent shallow-water studies

Citation

Gao, X.-Y. (2024), "Letter to the Editor on HFF 32, 138 (2022); 32, 2282 (2022); 33, 965 (2023) and 34, 1189 (2024) for the recent shallow-water studies", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 34 No. 6, pp. 2197-2204. https://doi.org/10.1108/HFF-06-2024-944

Latterly, in International Journal of Numerical Methods for Heat and Fluid Flow, Wazwaz (2022), Li et al. (2022), Khuri (2023) and Ramos and Garcia Lopez (2024) have absorbingly studied the shallow water, i.e., on certain multiple soliton solutions and lump solutions for two integrable shallow water wave equations (Wazwaz, 2022), Gramian solutions and solitonic interactions for a (2+1)-dimensional Broer-Kaup-Kupershmidt system in the shallow water (Li et al., 2022), soliton solutions for a (2+1)-dimensional Korteweg-de Vries equation describing the shallow water waves (Khuri, 2023) as well as blowup in finite time of the numerical solutions for a (1+1)-dimensional, bidirectional, nonlinear wave model for the small-amplitude waves in shallow water, depending upon the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, etc. (Ramos and Garcia Lopez, 2024).

Enthusiasm for the shallow water has been warmed by Wazwaz (2022), Li et al. (2022), Khuri (2023) and Ramos and Garcia Lopez (2024), and consequently this shallow-water-oriented Letter aims to deal with a (2+1)-dimensional generalised modified dispersive water-wave system for the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth (Gao et al., 2023a; Liu et al., 2023):

(1a) uyt+αuxxy−2αvxx−βuuxy−βuxuy=0,

(1b) vt−αvxx−β(uv)x=0,

in which the real differentiable functions v(x, y, t) and u(x, y, t), respectively, imply the horizontal velocity of the water wave and the height of the water surface, α and β are the real non-zero constants, and the subscripts represent the partial derivatives concerning the scaled space variables x, y and time variable t. With symbolic computation (Anderson and Farazmand, 2024; Kovacs et al., 2024; Shen et al., 2023a, 2023b, 2023e, 2023f; Gao et al., 2023b; Wu and Gao, 2023; Wu et al., 2023a, 2023c, 2023d; Zhou and Tian, 2022) in planning, Gao et al. (2023a) have given a set of the hetero-Bäcklund transformations, a set of the scaling transformations and four sets of the similarity reductions for system (1), whereas Liu et al. (2023) have investigated certain Lie point symmetry generators, Lie symmetry groups and symmetry reductions for system (1) with some analytic solutions. Shallow-water special cases of system (1) have been seen in Ying and Lou (2000), Li and Zhang (2004), Ma et al. (2015), Zhao and Han (2015), Kassem and Rashed (2019), Yamgoué et al. (2019), Liang and Wang (2019), Ren et al. (2019), Cao et al. (2020), Li et al. (2022), Gao et al. (2023a) as well as Liu et al. (2023). Additionally, fluids from all over the Solar System have been observed and discussed (Lainey et al., 2024; Neish et al., 2024; Cheng et al., 2022, 2023a, 2023b, 2024; Feng et al., 2023; Gao, 2024a, 2024d; Gao et al., 2023c; Shen et al., 2023c, 2023d; Wu et al., 2023b; Zhou et al., 2023a, 2023b, 2024).

Key point: In accordance with Wazwaz (2022), Li et al. (2022), Khuri (2023) and Ramos and Garcia Lopez (2024), and using symbolic computation, this shallow-water-oriented Letter tries to construct two branches of the similarity reductions for system (1), different from those reported in Gao et al. (2023a) and Liu et al. (2023).

Towards system (1), similar to those in Clarkson and Kruskal (1989), Gao and Tian (2022, 2024) as well as Gao (2023, 2024b, 2024c), the assumptions

(2a) u(x,y,t)=θ(x,y,t)+ω(x,y,t)p[z(y,t)],

(2b) v(x,y,t)=γ(x,y,t)+κ(x,y,t)q[z(y,t)],

and choice of z_yz_t ≠ 0 develop into

(3a) ωzyztp″−2βωωxzypp′−β(ωxωy+ωωxy)p2−2ακxxq+ [(ωyzt+ωtzy+ωzyt)−β(ωθx+ωxθ)zy+αωxxzy]p′+ [ωyt−β(ωyθx+ωxθy+ωθxy+ωxyθ)+αωxxy]p+ [θyt−β(θxθy+θθxy)−2αγxx+αθxxy]=0,

(3b) κztq′−β(ωκx+ωxκ)pq+[κt−β(θκx+θxκ)−ακxx]q−β(ωγx+ωxγ)p+[γt−β(θγx+θxγ)−αγxx]=0,

with θ(x, y, t), ω(x, y, t), ≠ 0, γ(x, y, t), κ(x, y, t) ≠ 0 and z(y, t) ≠ 0 meaning certain real to-be-determined differentiable functions, p(z) and q(z) denoting two real differentiable functions while the “′” sign representing d/dz.

We prefer to consider the following ordinary differential equations (ODEs): By reason that equations (3) are designed as a set of the ODEs with respect to p(z) and q(z), we make the ratios of different derivatives and powers of p(z) and q(z) become the functions of z only. Consequently,

(4a) Ω1(z)ωzyzt=−2βωωxzy,

(4b) Ω2(z)ωzyzt=−β(ωxωy+ωωxy),

(4c) Ω3(z)ωzyzt=−2ακxx,

(4d) Ω4(z)ωzyzt=(ωyzt+ωtzy+ωzyt)−β(ωθx+ωxθ)zy+αωxxzy,

(4e) Ω5(z)ωzyzt=ωyt−β(ωyθx+ωxθy+ωθxy+ωxyθ)+αωxxy,

(4f) Ω6(z)ωzyzt=θyt−β(θxθy+θθxy)−2αγxx+αθxxy,

(4g) Γ1(z)κzt=−β(ωκx+ωxκ),

(4h) Γ2(z)κzt=κt−β(θκx+θxκ)−ακxx,

(4i) Γ3(z)κzt=−β(ωγx+ωxγ),

(4j) Γ4(z)κzt=γt−β(θγx+θxγ)−αγxx,

with Ω_i(z)’s (i = 1, …, 6) and Γ_j(z)’s (j = 1, …, 4) standing for the real to-be-determined functions as for z only.

Accordingly, any solutions for θ(x, y, t), ω(x, y, t) ≠ 0, γ(x, y, t), κ(x, y, t) ≠ 0 and z(y, t) ≠ 0 can bring about, at least, a similarity reduction.

Let us keep the similarity-reduction issue in mind and turn to Clarkson and Kruskal (1989).

On the score of the second freedom in remark 3 in Clarkson and Kruskal (1989), equations (4a) and (4c) bring about

(5) ω(x,y,t)=−zt2βx,  κ(x,y,t)=zyzt224αβx3,  Ω1(z)=Ω3(z)=1.

In the light of the first freedom in remark 3 in Clarkson and Kruskal (1989), equation (4b) comes to

(6) z(y,t)=λ1y+λ2t+λ3,  Ω2(z)=0,

as a result that equation (4g) turns into

(7) Γ1(z)=2,

with λ₁ and λ₂ implying two real non-zero constants, whereas λ₃ meaning a real constant.

So far, two branches of θ(x, y, t) can help us simplify Γ₂(z) to a constant:

Branch 1:

In this branch, based on the first freedom in remark 3 in Clarkson and Kruskal (1989), equations (4h) and (4i) give rise to

(8) θ(x,y,t)=−3αβx−1,  γ(x,y,t)=0,  Γ2(z)=Γ3(z)=0.

Then, equations (4d), (4e), (4f) and (4j) bloom into

(9) Ω4(z)=Ω5(z)=Ω6(z)=Γ4(z)=0.

We are in a position to transform system (1) into two ODEs, i.e.,

(10a) p″+pp′+q=0,

(10b) q′+2pq=0,

and then to simplify ODEs (10) to a single ODE, i.e.,

(11) p‴+3pp″+p′2+2p2p′=0,

with the choice of

(12) q=−(p″+pp′).

In short, we build up the following branch of the similarity reductions for system (1):

(13a) u(x,y,t)=−3αβx−1−λ22βxp[z(y,t)],

(13b) v(x,y,t)=−λ1λ2224αβx3{p″[z(y,t)]+p[z(y,t)]p′[z(y,t)]},

(13c) z(y,t)=λ1y+λ2t+λ3,

(13d) p‴+3pp″+p′2+2p2p′=0.

ODE (13d) means a known ODE, the information of which has been given in Zwillinger and Dobrushkin (2022).

In relation to the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, as for u(x, y, t), the height of the water surface, and v(x, y, t), the horizontal velocity of the water wave, similarity reductions (13) are dependent on α and β, the shallow-water coefficients in system (1).

Branch 2:

In this branch, on account of the second freedom in remark 3 in Clarkson and Kruskal (1989), equation (4h) becomes

(14) θ(x,y,t)=−λ24βx−3αβx−1,  Γ2(z)=1.

On the ground of the first freedom in remark 3 in Clarkson and Kruskal (1989), equation (4i) makes for

(15) γ(x,y,t)=0,  Γ3(z)=0,

so that equations (4d), (4e), (4f) and (4j) issue in

(16) Ω4(z)=12,  Ω5(z)=Ω6(z)=Γ4(z)=0.

We are now capable of transforming system (1) into the following two ODEs:

(17a) p″+pp′+q+12p′=0,

(17b) q′+2pq+q=0,

and then of simplifying ODEs (17) into the following single ODE:

(18) p‴+3pp″+32p″+p′2+2p2p′+2pp′+12p′=0,

with the choice of

(19) q=−(p″+pp′+12p′).

In brief, we construct another branch of the similarity reductions for system (1), i.e.,

(20a) u(x,y,t)=−(λ24βx+3αβx−1)−λ22βxp[z(y,t)],

(20b) v(x,y,t)=−λ1λ2224αβx3{p″[z(y,t)]+p[z(y,t)]p′[z(y,t)]+12p′[z(y,t)]},

(20c) z(y,t)=λ1y+λ2t+λ3,

(20d) p‴+3pp″+32p″+p′2+2p2p′+2pp′+12p′=0.

ODE (20d) implies a known ODE, the information of which has been presented in Zwillinger and Dobrushkin (2022).

In relation to the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, as for u(x, y, t), the height of the water surface, and v(x, y, t), the horizontal velocity of the water wave, similarity reductions (20) depend on α and β, the shallow-water coefficients in system (1).