*3.1. Korteweg-de Vries Equation Theory*

The interfacial waves in an arbitrarily stratified fluid can be expressed by the classical Korteweg–de Vries (KdV) equation

$$
\frac{\partial \eta}{\partial t} + \mathcal{C}\_0 \frac{\partial \eta}{\partial x} + \alpha \eta \frac{\partial \eta}{\partial x} + \beta \frac{\partial^3 \eta}{\partial x^3} = 0 \tag{1}
$$

where *η* characterizes the vertical displacement of the isopycnal surface [15,29]. *C*<sup>0</sup> is a linear speed. The nonlinear parameter *α* and dispersion parameter *β* are considered environmental parameters and make contributions to density stratification and sheer currents [10].

In the two-layer fluid model, the upper layer has a thickness *h*<sup>1</sup> and a density *ρ*1, and the lower layer has a thickness *h*<sup>2</sup> and a density *ρ*2, and *ρ*<sup>2</sup> > *ρ*<sup>1</sup> [15]. This models a typical pycnocline. In this case, the linear speed is

$$\mathcal{C}\_0 = \sqrt{\frac{gh\_1h\_2\delta\rho}{\rho\_{av}(h\_1+h\_2)}},\tag{2}$$

where *ρav* = (*ρ*<sup>1</sup> + *ρ*2)/2 is the mean density, *δρ* = *ρ*<sup>2</sup> − *ρ*<sup>1</sup> is the density difference.

To illustrate the dynamics of a single wave packet of ISW in Figure 2, the solitary solution of KdV equation is

$$
\eta = \eta\_0 \text{sech}^2 \frac{\chi - \mathcal{C}\_p t}{l},
\tag{3}
$$

where *η*<sup>0</sup> is the amplitude of the displacement, *x* is the horizontal position of an onedimensional interfacial wave, and the nonlinear velocity *C<sup>p</sup>* and soliton width *l* [15,29].

$$\mathbf{C}\_{p} = \mathbf{C}\_{0} + \frac{\alpha \eta\_{0}}{3}, \ l = \sqrt{\frac{12\beta}{\alpha \eta\_{0}}} \tag{4}$$

The nonlinear and dispersion parameters (*α* and *β*) of the model are

$$\mathfrak{a} = \frac{\mathfrak{NC}\_0 (h\_1 - h\_2)}{2h\_1 h\_2}, \mathfrak{f} = \frac{\mathfrak{C}\_0 h\_1 h\_2}{6}. \tag{5}$$

Using the explicit form of environmental parameters *α* and *β*, Equation (4) becomes

$$\mathcal{C}\_{p} = \left[1 + \frac{\eta\_{0}(h\_{1} - h\_{2})}{2h\_{1}h\_{2}}\right], l = \frac{2h\_{1}h\_{2}}{\sqrt{3\eta\_{0}|h\_{1} - h\_{2}|}}. \tag{6}$$

This solitary solution is the typical characteristic of ISW in many areas of the ocean [15].
