Letter to the Editor: Considering the Whitham-Broer-Kaup system in the shallow water inspired by HFF 33, 3801; 34, 1189 and 34, 2197

Wu, Xi-Hu; Gao, Yi-Tian

doi:10.1108/HFF-02-2025-950

Recently, Ramos and Garcia Lopez (2024) and Gao (2023a, 2023b) have conducted several comprehensive shallow-water studies in International Journal of Numerical Methods for Heat and Fluid Flow, that is, blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in the shallow water (Ramos and Garcia Lopez, 2024), novel hetero-Bäcklund transformations and similarity reductions for a (2 + 1)-dimensional generalized dispersive long-wave system that models certain dispersive and nonlinear long gravity waves on the oceanic shallow water (Gao, 2023a), and novel similarity reductions for a (2 + 1)-dimensional generalized modified dispersive water-wave system for the nonlinear and dispersive long gravity waves in the shallow water of uniform depth (Gao, 2023b).

Those studies (Gao, 2023a, 2023b; Ramos and Garcia Lopez, 2024) have sparked renewed interest in shallow-water research. Consequently, this Letter is dedicated to the Whitham-Broer-Kaup system for the dispersive long waves in the shallow oceanic water (Ahmad et al., 2015; Fan and Bao, 2024; Sabawi and Hamad, 2024; Fan and Bao, 2022; Gao et al., 2020; Cao et al., 2020; Arshad et al., 2017; Ren and Lin, 2018; Kuo, 2017; Fei et al., 2017; Zhou and Lu, 2017; Xu et al., 2017; Lin et al., 2011a, 2011b; Bhrawy et al., 2014; Shan et al., 2012; Wang et al., 2010, 2009; Zhang et al., 2008) (and references therein), i.e.:

\begin{array}{l} p_{t} + p p_{x} + q_{x} + β p_{x x} = 0, \\ q_{t} + {(p q)}_{x} + α p_{x x x} - β q_{x x} = 0, \end{array}

(1)

where x and t are the scaled space and time coordinates, the subscripts represent the partial derivatives, p(x, t) and q(x, t) are the real functions denoting the horizontal velocity field and height of the deviation from the equilibrium position, respectively, α is the real dispersion coefficient and β is the real diffusion coefficient.

For System (1), Bäcklund transformation, soliton and similar wave solutions have been studied (Fan and Bao, 2024); quartic B-spline collocation method has been used to obtain the numerical solutions (Sabawi and Hamad, 2024); Painlevé integrability, bilinear forms and Bäcklund transformations have been offered (Gao et al., 2020; Zhang et al., 2008; Fan and Bao, 2022); two different groups of variational principles have been constructed (Cao et al., 2020); travelling wave solutions in the form of solitons, dark solitons, bell and anti-bell periodic waves have been obtained (Ahmad et al., 2015; Arshad et al., 2017); nonlocal symmetry, consistent tanh-expansion solvability and power-series solutions have been studied (Ren and Lin, 2018; Zhou and Lu, 2017); modified simplest equation method has been used to derive some soliton solutions (Kuo, 2017); residual symmetries and interaction solutions have been investigated (Fei et al., 2017); double Wronskian solutions and soliton interactions have been studied via the Wronskian technique (Xu et al., 2017; Lin et al., 2011a, 2011b; Wang et al., 2009); solitons, cnoidal waves and snoidal waves have been presented (Bhrawy et al., 2014); as well as Darboux transformation and some soliton solutions have been investigated (Shan et al., 2012; Wang et al., 2010). Other contributions on System (1) have appeared as the references in the aforementioned papers.

However, to the best of our knowledge, for System (1), generalized Darboux transformation (GDT), which can be used to derive the rogue wave, multi-pole and mixed solutions, has not yet been reported. Using symbolic computation (Gao, 2023a, 2024), we aim to construct a GDT for System (1). With respect to the horizontal velocity field and the height of the deviation from the equilibrium position, the GDT supports the derivation of the multi-pole solitons, multi-pole breathers, higher-order rogue waves and mixed waves. We hope this study contributes to a deeper understanding of the physical dynamics of the nonlinear waves in shallow water environments.

Via the transformations:

\begin{array}{l} p = 2 \sqrt{β^{2} + α} P, q = 4 (β^{2} + α) Q - 2 \sqrt{β^{2} + α} (\sqrt{β^{2} + α} + β) P_{x}, \end{array}

(2)

we transform System (1) into:

\begin{array}{l} P_{t} + \sqrt{β^{2} + α} (2 Q_{x} + 2 P P_{x} - P_{x x}) = 0, \\ Q_{t} + \sqrt{β^{2} + α} [Q_{x x} + 2 {(P Q)}_{x}] = 0. \end{array}

(3)

Wang et al. (2010) has offered a Lax pair for System (3):

\begin{array}{l} Φ_{x} = U Φ, U = \frac{1}{2} (\begin{matrix} λ - P & - 2 Q \\ 2 & P - λ \end{matrix}), \\ Φ_{t} = V Φ, V = \frac{\sqrt{β^{2} + α}}{2} (\begin{matrix} P^{2} - P_{x} - λ^{2} & 2 Q (P + λ) + 2 Q_{x} \\ - 2 (P + λ) & - (P^{2} - P_{x} - λ^{2}) \end{matrix}), \end{array}

(4)

where Φ = (ϕ, ψ)^T is the eigenfunction, λ is a real spectral parameter and ϕ₁ and ϕ₂ are the complex functions of x and t. The compatibility condition Φ_xt = Φ_tx leads to System (3).

We consider an n-fold gauge transformation that transforms Lax Pair (4) into:

\begin{array}{l} Φ^{[N]} = Δ^{[N]} Φ, \\ Φ_{x}^{[N]} = U^{[N]} Φ^{[N]}, \\ Φ_{t}^{[N]} = V^{[N]} Φ^{[N]}, \end{array}

(5)

where [N] represents the N-fold solutions/matrices, Φ^[^N^] = (ϕ^[^N^], ψ^[^N^])^T, Δ^[^N^] is the 2 × 2 gauge transformation matrix and N is a positive integer. U^[^N^] and V^[^N^] own the same forms as U and V except that the initial potentials P and Q are replaced with their N-fold versions P^[^N^] and Q^[^N^]. Combining Gauge Transformation (5) with Lax Pair (4), we derive:

\begin{array}{l} {(Δ^{[N]})}_{x} + Δ^{[N]} U - U^{[N]} Δ^{[N]} = 0, \\ {(Δ^{[N]})}_{t} + Δ^{[N]} V - V^{[N]} Δ^{[N]} = 0. \end{array}

(6)

Motivated by Wang et al. (2010), the n-fold gauge matrix Δ^[n] can be set as:

\begin{array}{l} Δ^{[N]} = (\begin{matrix} ω (λ^{N} + \sum_{j = 0}^{N - 1} a_{j} λ^{j}) & ω (\sum_{j = 0}^{N - 1} b_{j} λ^{j}) \\ ω^{- 1} (\sum_{j = 0}^{N - 1} c_{j} λ^{j}) & ω^{- 1} (λ^{N} + \sum_{j = 0}^{N - 1} d_{j} λ^{j}) \end{matrix}), \end{array}

(7)

where j = 0, 1, 2,…, N – 1, a_j’s, b_j’s, c_j’s and d_j’s are 4 N functions of x and t, and ω² = 1 + c_N_–1. To determine 4N unknown functions a_j’s, b_j’s, c_j’s and d_j’s, we suppose that λ₁, λ₂, …, λ₂_n are the 2n roots of Det(Δ^[^N^]), i.e.:

\begin{array}{l} Det (Δ^{[N]}) = \prod_{k = 1}^{n} {(λ - λ_{2 k - 1})}^{m_{k} + 1} {(λ - λ_{2 k})}^{m_{k} + 1}, \end{array}

(8)

where Det denotes the determinant, k = 1, 2, …, n, n ≤ N, m_k is a nonnegative integer and $n + \sum_{k = 1}^{n} m_{k} = N$ ⁠. We assume that Φ₂_k_–1 = (ϕ₂_k_–1, ψ₂_k_–1)^T and Φ₂_k = (ϕ₂_k, ψ₂_k)^T respectively are the eigenfunction of Lax Pair (4) at λ = λ₂_k_–1, and λ = λ₂_k, and the superscript T denotes the transpose of the matrix.

According to $(Δ^{[N]} Φ) |_{λ = λ_{2 k - 1}} = 0$ and $(Δ^{[N]} Φ) |_{λ = λ_{2 k}} = 0$ ⁠, we give the following 4n equations:

\begin{array}{l} ϕ_{2 k - 1} \sum_{j = 0}^{N - 1} a_{j} λ_{2 k - 1}^{j} + ψ_{2 k - 1} \sum_{j = 0}^{N - 1} b_{j} λ_{2 k - 1}^{j} = - λ_{2 k - 1}^{N} ϕ_{2 k - 1}, \\ ϕ_{2 k} \sum_{j = 0}^{N - 1} a_{j} λ_{2 k}^{j} + ψ_{2 k} \sum_{j = 0}^{N - 1} b_{j} λ_{2 k}^{j} = - λ_{2 k}^{N} ϕ_{2 k}, \\ ϕ_{2 k - 1} \sum_{j = 0}^{N - 1} c_{j} λ_{2 k - 1}^{j} + ψ_{2 k - 1} \sum_{j = 0}^{N - 1} d_{j} λ_{2 k - 1}^{j} = - λ_{2 k - 1}^{N} ψ_{2 k - 1}, \\ ϕ_{2 k} \sum_{j = 0}^{N - 1} c_{j} λ_{2 k}^{j} + ψ_{2 k} \sum_{j = 0}^{N - 1} d_{j} λ_{2 k}^{j} = - λ_{2 k}^{N} ψ_{2 k} . \end{array}

(9)

Nonetheless, the 4n equations presented in equations (9) are inadequate for uniquely determining the 4N unknown functions A_j’s, B_j’s, C_j’s and D_j’s. We consider the Taylor expansions $(Δ^{[N]} Φ) |_{λ = λ_{2 k - 1} + ϵ} = 0$ and $(Δ^{[N]} Φ) |_{λ = λ_{2 k} + ϵ} = 0$ at ϵ = 0:

(Δ^{[N]} Φ) |_{λ = λ_{2 k - 1} + ε, λ_{2 k} + ε} = \sum_{ζ = 0}^{m_{k}} \sum_{ι = 0}^{ζ} Δ^{[N] (ι)} |_{λ = λ_{2 k - 1}, λ_{2 k}} Φ^{(ζ - ι)} |_{λ = λ_{2 k - 1}, λ_{2 k}} ε^{ζ} + O (ε^{m_{k}}),

\begin{array}{l} Δ^{[N] (ζ)} |_{λ = λ_{2 k - 1}} = (\begin{matrix} C_{N}^{ζ} λ_{2 k - 1}^{N - ζ} + \sum_{j = ζ}^{N - 1} C_{j}^{ζ} a_{j} λ_{2 k - 1}^{j - ζ} & \sum_{j = ζ}^{N - 1} C_{j}^{ζ} b_{j} λ_{2 k - 1}^{j - ζ} \\ \sum_{j = ζ}^{N - 1} C_{j}^{ζ} c_{j} λ_{2 k - 1}^{j - ζ} & C_{N}^{ζ} λ_{2 k - 1}^{N - ζ} + \sum_{j = ζ}^{N - 1} C_{j}^{ζ} d_{j} λ_{2 k - 1}^{j - ζ} \end{matrix}), \\ Δ^{[N] (ζ)} |_{λ = λ_{2 k}} = (\begin{matrix} C_{N}^{ζ} λ_{2 k}^{N - ζ} + \sum_{j = ζ}^{N - 1} C_{j}^{ζ} a_{j} λ_{2 k}^{j - ζ} & \sum_{j = ζ}^{N - 1} C_{j}^{ζ} b_{j} λ_{2 k}^{j - ζ} \\ \sum_{j = ζ}^{N - 1} C_{j}^{ζ} c_{j} λ_{2 k}^{j - ζ} & C_{N}^{ζ} λ_{2 k}^{N - ζ} + \sum_{j = ζ}^{N - 1} C_{j}^{ζ} d_{j} λ_{2 k - 1}^{j - ζ} \end{matrix}), \\ Φ^{(ζ)} |_{λ = λ_{2 k - 1}} = (\begin{matrix} φ_{2 k - 1}^{(ζ)} \\ ψ_{2 k - 1}^{(ζ)} \end{matrix}) = \frac{1}{ζ!} \frac{\partial^{ζ}}{\partial λ_{2 k - 1}^{ζ}} Φ |_{λ = λ_{2 k - 1}} = (\begin{matrix} \frac{1}{ζ!} \frac{\partial^{ζ}}{\partial λ_{2 k - 1}^{ζ}} φ_{2 k - 1} \\ \frac{1}{ζ!} \frac{\partial^{ζ}}{\partial λ_{2 k - 1}^{ζ}} ψ_{2 k - 1} \end{matrix}), \\ Φ^{(ζ)} |_{λ = λ_{2 k}} = (\begin{matrix} φ_{2 k}^{(ζ)} \\ ψ_{2 k}^{(ζ)} \end{matrix}) = \frac{1}{ζ!} \frac{\partial^{ζ}}{\partial λ_{2 k}^{ζ}} Φ |_{λ = λ_{2 k}} = (\begin{matrix} \frac{1}{ζ!} \frac{\partial^{ζ}}{\partial λ_{2 k}^{ζ}} φ_{2 k} \\ \frac{1}{ζ!} \frac{\partial^{ζ}}{\partial λ_{2 k}^{ζ}} ψ_{2 k} \end{matrix}), \end{array}

(10)

where ϵ is a small parameter, ι = 0, 1, …, ζ, ζ = 0, 1, …, m_k and $C_{N}^{ζ} = \frac{N!}{ζ! (N - ζ)!}$ ⁠.

We truncate $(Δ^{[N]} Φ) |_{λ = λ_{2 k - 1} + ϵ} = 0$ and $(Δ^{[N]} Φ) |_{λ = λ_{2 k} + ϵ} = 0$ to the m_k-th order and obtain:

\lim_{ϵ \to 0} (Δ^{[N]} Φ) |_{λ = λ_{2 k - 1} + ϵ, λ_{2 k} + ϵ} ϵ^{- ζ} = 0,

i.e.:

\begin{array}{l} \sum_{ι = 0}^{ζ} Δ^{[N] (ι)} |_{λ = λ_{2 k - 1}} Φ^{(ζ - ι)} |_{λ = λ_{2 k - 1}} = 0, \sum_{ι = 0}^{ζ} Δ^{[N] (ι)} |_{λ = λ_{2 k}} Φ^{(ζ - ι)} |_{λ = λ_{2 k}} = 0. \end{array}

(11)

Combining Transformations (2), equations (6), N-Fold DT Matrix (7) and equations (11), we calculate out the Nth-order solutions for System (1) as:

\begin{array}{l} p^{[N]} = 2 \sqrt{β^{2} + α} P^{[N]}, q^{[N]} = 4 (β^{2} + α) Q^{[N]} - 2 \sqrt{β^{2} + α} (\sqrt{β^{2} + α} + β) {(P^{[N]})}_{x}, \\ P^{[N]} = P^{[0]} - {[ln (1 + c_{N - 1})]}_{x}, Q^{[N]} = (1 + c_{N - 1}) (Q^{[0]} + b_{N - 1}), \\ b_{N - 1} = \frac{Υ_{b}}{Υ}, c_{N - 1} = \frac{Υ_{c}}{Υ}, \\ Υ = Det {(\begin{matrix} Υ_{1}^{T}, & Υ_{2}^{T}, & \dots, & Υ_{n}^{T} \end{matrix})}^{T}, \end{array}

\begin{array}{l} {(Υ_{k})}_{ς, τ} = {\begin{cases} \sum_{ρ = 0}^{ς - 1} C_{N - τ}^{ρ} λ_{2 k - 1}^{N - τ - ρ} φ_{2 k - 1}^{(ς - 1 - ρ)}, 1 \leq ς \leq m_{k} + 1, 1 \leq τ \leq N, \\ \sum_{ρ = 0}^{ς - 1} C_{2 N - τ}^{ρ} λ_{2 k - 1}^{2 N - τ - ρ} ψ_{2 k - 1}^{(ς - 1 - ρ)}, 1 \leq ς \leq m_{k} + 1, N + 1 \leq τ \leq 2 N, \\ \sum_{ρ = 0}^{ς - m_{k} - 2} C_{N - τ}^{ρ} λ_{2 k}^{N - τ - ρ} φ_{2 k}^{(ς - 2 - m_{k} - ρ)}, m_{k} + 2 \leq ς \leq 2 (m_{k} + 1), 1 \leq τ \leq N, \\ \sum_{ρ = 0}^{ς - m_{k} - 2} C_{2 N - τ}^{ρ} λ_{2 k}^{2 N - τ - ρ} ψ_{2 k}^{(ς - 2 - m_{k} - ρ)}, m_{k} + 2 \leq ς \leq 2 (m_{k} + 1), N + 1 \leq τ \leq 2 N, \end{cases} \\ {(υ)}_{2 N \times 1} = {(υ_{1}, υ_{2}, \dots, υ_{n})}^{T}, {(μ)}_{2 N \times 1} = {(μ_{1}, μ_{2}, \dots, μ_{n})}^{T}, \\ {(υ_{k})}_{1, ς} = {\begin{cases} - \sum_{ρ = 0}^{ς - 1} C_{N}^{ρ} λ_{2 k - 1}^{N - ρ} φ_{2 k - 1}^{(ς - 1 - ρ)}, 1 \leq ς \leq m_{k} + 1, \\ - \sum_{ρ = 0}^{ς - m_{k} - 2} C_{N}^{ρ} λ_{2 k}^{N - ρ} φ_{2 k}^{(ς - m_{k} - 2 - ρ)}, m_{k} + 2 \leq ς \leq 2 (m_{k} + 1), \end{cases} \\ {(μ_{k})}_{1, ς} = {\begin{cases} - \sum_{ρ = 0}^{ς - 1} C_{N}^{ρ} λ_{2 k - 1}^{N - ρ} ψ_{2 k - 1}^{(ς - 1 - ρ)}, 1 \leq ς \leq m_{k} + 1, \\ - \sum_{ρ = 0}^{ς - m_{k} - 2} C_{N}^{ρ} λ_{2 k}^{N - ρ} ψ_{2 k}^{(ς - m_{k} - 2 - ρ)}, m_{k} + 2 \leq ς \leq 2 (m_{k} + 1), \end{cases} \end{array}

(12)

where Y_k is a 2(m_k + 1) × 2N matrix, 1 ≤ ς ≤ 2(m_k + 1), 1 ≤ τ ≤ 2N, (Y_k)_ς_,_τ denotes the element in the ςth row and τth column of Y_k, Y_b can be obtained via the modified Y with its (N + 1) th column replaced by the vector (υ)₂_N_×1, Y_c can be obtained via the modified Y with its 1st column replaced by the vector (µ)₂_N_×1.

In conclusion, N-Fold DT Matrix (7) and The Nth-Order Solutions (12) together establish an N-fold GDT for System (1). In The Nth-Order Solutions (12), the variable n indicates the number of sets of spectral parameters utilized, the variable m_k represents that the kth set of spectral parameters is iterated m_k times (a total of m_k + 1 times) and the value of N represents the total order of The Nth-Order Solutions (12). Notably, when n = N, i.e. m_k = 0, each of the n spectral parameters is iterated only once. In this case, the resulting GDT simplifies to the N-fold DT proposed by Shan et al. (2012) and Wang et al. (2010).

In relation to the horizontal velocity field and height of the deviation from the equilibrium position, the previously obtained GDT allows for multiple iterations of the spectral parameters, leading to the generation of the multi-pole solitons, multi-pole breathers, higher-order rogue waves and mixed waves. This study might assist in exploring the complex and dynamic natural mechanisms underlying real-world shallow water waves.

Statements and Declarations

Funding: This work has been supported by the National Natural Science Foundation of China under Grant No.11772017, and the National Scholarship for Doctoral Students under Grant BSY202306241.

Conflict of interest: The authors declare that they have no conflict of interest.

Data availability: Data sharing not applicable to this article, as no datasets were generated or analysed during the current study.

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2024

Emerald Publishing Limited

Letter to the Editor: Considering the Whitham-Broer-Kaup system in the shallow water inspired by HFF 33, 3801; 34, 1189 and 34, 2197

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Letter to the Editor: Considering the Whitham-Broer-Kaup system in the shallow water inspired by HFF 33, 3801; 34, 1189 and 34, 2197