Persistence of a unique periodic wave train in convecting shallow water fluid

1 Introduction

When modeling real-world problems, it is crucial to consider higher-order truncations or small corrections due to unavoidable external influences. In convective situations, the shallow water model [1], given by the equation

is more applicable than the classical KdV model

which was derived under ideal assumptions. In equation (1), ε is sufficiently small and appears alongside λ 2 , λ 4 , and λ 5 as specified in [1]. Equation (1) can model water motion in a variety of dissipative scenarios, such as a shallow liquid layer exposed to the air and heated from the air side, or a shallow water layer undergoing shear (modeling internal waves) [2]. The parameters λ i s can take a wide range of values, from positive to negative, and are generally considered as free parameters [3,4]. For example, when modeling shallow fluid layers under Marangoni-Bénard convection, both λ 4 and λ 5 are positive. The sign of λ 2 is negative if the instability threshold is greater than the diffusion, and positive otherwise. The sign of λ 3 is determined by the air-liquid surface tension. The signs of all coefficients may change when considering the buoyancy effect u u x , refer [3,4] and references therein. In the case of a long layer with much smaller surface tension, the coefficients can be approximated as λ 1 = λ 2 = λ 3 = λ 4 = 1 , and λ 5 = 0 , as demonstrated in Topper and Kawahara’s study [5] and related study on the existence of periodic waves [6,7]. Specifically, Aspe and Depassier [8] derived approximations for the parameters as follows:

where ε is a small parameter that expresses the excess of the Rayleigh number beyond its critical values as ε R 2 , while σ denotes the Prandtl number, and G denotes the Galileo number.

Equation (1) demonstrates numerous variations in various contexts, including delay dissipation [9], the elimination of one or two terms in (1) [10,11], and the extension of the model to incorporate a more potent nonlinear buoyancy term u 2 u x [12–14],

The fundamental study of equations (1) and (3) focuses on traveling waves, including traveling pulses (solitary waves) and periodic waves (wave trains). For equation (1), the existence of traveling pulses was rigorously proven in [3,15] through dynamical analysis, with numerical verification performed in [4] as a follow-up to [3]. For equation (3), the traveling pulse was constructed using the same method employed in [15], as detailed in [12,13]. Notably, the existence of periodic wave trains in both equations, as well as the coexistence of a periodic wave train and a pulse in equation (3) and their non-coexistence in equation (1), was rigorously derived in [2] through the analysis of a planar dynamical system on a normally hyperbolic manifold.

It is noteworthy that a stronger nonlinear buoyancy u 2 u x must induce a stronger dissipative effect ( u 2 u x ) x , since the perturbation in (1) and (3) is the first-order corrected approximation compared to the zero-order approximation (the unperturbed equation). Therefore, it is reliable to explore the periodic wave trains and pulse of the extended perturbed KdV equation,

The main purpose of this study is to provide complementary results to [2]. Specifically, we prove that (i) the mechanical balance on the wave coexistence of a traveling pulse and a periodic wave train is lost due to the stronger dissipation ( u 2 u x ) x . The convecting shallow model only possesses a unique periodic wave train within a fixed speed range or a unique pulse with a certain speed. (ii) We assess the impact of the dissipation on the wave speed and compare the new wave range for periodic wave trains with the previous ones by taking the same physical parameters as in [2].

The study is organized as follows: First, we briefly summarize the transformation of the partial differential equation (PDE) model into a planar dynamical system and prove that the Melnikov function for (3) has at most one zero, indicating the uniqueness of either a periodic wave train or a pulse in (4). In Section 3, we conduct a numerical study using the parameters employed in [2] to explore the periodic wave train and investigate its related wave speed. Finally, in Section 4, we present our conclusions.

2 Existence of a unique periodic wave train and pulse

In this section, we concisely illustrate the reduction of the PDE model to a planar dynamical system and study the zeros of the bifurcation function associated with the perturbed Hamiltonian system. We omit extensive computational analysis as our result is a complementary part to [2], and the detailed analysis is the same as the system reduction in [2], except for the terms with λ 5 . Substituting the wave profile z = x − c t into equation (4), direct integration yields a third-order ordinary differential equation (ODE),

where the boundary conditions are taken as d u d z , d 2 u d z 2 , d 3 u d z 3 → 0 as z → ∞ . Equation (5) is the standard form of singular perturbation as the higher-order term has a small parameter ε . To explore the associated critical manifold and its hyperbolic property, we can rewrite equation (5) into a first-order ODE system. This ODE system is a three-dimensional dynamical system, which has a projection on a normally hyperbolic manifold M ε to be a perturbed two-dimensional Hamiltonian system,

Taking the parameters α 1 = 3 c λ 1 , α 2 = c 3 λ 1 λ 3 , and k = λ 3 c , and introducing the scaling ε ˜ = 3 c ( λ 1 λ 4 − λ 3 λ 5 ) k ε λ 1 λ 3 2 , u = α 1 u ˜ , v = α 2 v ˜ , and z = k ξ , we have a simpler form by dropping the tildes on the symbols,

and

System (6) ε = 0 has a Hamiltonian function

The portrait of system (6) ε = 0 is depicted in Figure 1 in [2], and there is a double homoclinic loop. The right branches of this loop correspond to the light solitary wave of equation (3) ε = 0 , while the left branches correspond to the dark solitary wave. We investigate the right branch because the wave motion is investigated on a shallow water layer. Inside the right branch (satisfying H ( 0 , 0 ) = 0 ), the periodic orbits { Γ h } and the right homoclinic loop Γ 0 correspond to periodic wave trains and traveling pulses, respectively, for the shallow water model (3) ε = 0 . The center is defined by H ( 1 , 0 ) = − 1 4 . Under perturbation, most periodic orbits are spirally broken, only a certain number of Γ h persist as limit cycles of (6) or periodic wave trains of (4). Correspondingly, we take the Melnikov function as the main tool [16] to investigate the persisting periodic orbits and the homoclinic loop,

Figure 1

The speed of the unique persisting periodic wave train is an increasing function with respect to the amplitude. (a) c ( h ) is increasing on h ∈ ( − 1 4 , 0 ) and (b) c ( h ( u ) ) is increasing on u ∈ ( 1 , 2 ) .

3 Numerical study

We take the parameters used in [2] to perform the numerical study. In detail, we take Rayleigh number R = 1 0 4 , the water temperature equal to 2 0 ° C , and the layer length L = 1000 m . Then, the viscosity of water ν ≈ 1.0016 and the Prandtl number σ ≈ 7.56 , referring to Wikipedia. Furthermore, the Galilei number G = g L 3 ν 2 ≈ 9.768715103 × 1 0 9 , where g represents the gravitational acceleration. Substituting the above values into (2) and (8), the corresponding estimation

can be derived. The portraits of c ( h ) and c ( h ( u ) ) are depicted in Figure 1 and obtained by numerical computation. The numerical study reveals that the wave speed of the unique persistent periodic wave train increases as a function of h , which represents the amplitude of the periodic wave train. Notably, this wave speed is slower than the wave speed c * ranging between 32.90613608 and 46.06859050, as derived for equation (1) in Section 5 of [2]. Furthermore, we set c = 19 to analyze the zeros of M ( h ) and M ( h ( u ) ) . The results indicate that M ( h ) increases from ( − 1 4 , 0 ) to a maximum and then decreases towards the right endpoint, revealing a unique zero. Consequently, we simulate the oscillatory motion of equation (4), as illustrated in Figure 2, which shows the oscillator motion passing through the point u = 1.3328970104 .

Figure 2

Taking c = 19 to simulate the periodic wave and M : (a) A root of M ( h ) at h = − 0.0992175028 ; (b) a root u = 1.3328970104 of M ( h ( u ) ) ; (c) a time series indicates that an orbit initializing at u = 1.092 close to a periodic orbit crosses through u = 1.3328970104 ; and (d) a periodic wave crosses through u = 1.3328970104 .

4 Conclusion

In this study, we investigated a convecting shallow water model with a stronger nonlinear buoyancy and dissipation. We found that the mechanical balance of the coexistence of a traveling pulse and a periodic wave train is lost. The related wave speed of the unique persisting periodic wave train and the pulse becomes smaller than that of the original model (1). Another interesting problem is the spectral stability and nonlinear stability of the pulse when stronger buoyancy is considered, which will be explored in future study.

The weak dissipations ε ( u x x + u x x x x ) , always referred to as the Kuramoto-Sivashinsky perturbation, have been incorporated into the generalized BBM equation, as reported in [20,21,23], where only a unique periodic traveling wave was established. The Kuramoto-Sivashinsky perturbation has also been considered in the Camassa-Holm equation, as discussed in [18] and the Degasperis-Procesi equation in [24]; however, only a solitary wave was verified in the dissipative in the Camassa-Holm equation and the Degasperis-Procesi equation.

The methodology employed in this study can be adapted to other models beyond KdV-like equations, including the dissipative Camassa-Holm equation and the Degasperis-Procesi equation:

where α 1 and α 2 measure the linear and nonlinear viscosity, respectively. By utilizing a similar analytical approach as outlined above, we can determine which types of nonlinear dissipations affect or do not affect the balance of mechanisms underlying the existence of a unique periodic traveling wave. Moreover, there is ample scope for further research on these models. Beyond periodic waves emerging through Poincaré bifurcation, periodic peakon waves and peakon waves remain unexplored. The study of periodic traveling waves emerging near those periodic peakon waves and the peakon waves deserves further investigation. In this regard, the bifurcation theories and methods presented in [25–27] can serve as instrumental in studying periodic wave solutions.