Travelling reflectionless waves in shallow water channels with variable cross-section and current

As it is known, the problem of finding traveling waves in 1D nonlinear and dispersive media may be reduced to solving a system of ordinary differential equations. If the order of the system is large, then the internal wave structure can be very complicated and even random. If the medium is inhomogeneous, it is natural to expect the absence of solutions in the form of traveling waves due to the reflection and multiple reflection effects. If, however, the medium parameters change slowly (in comparison to the wavelength), and the reflection is weak, it becomes possible to construct an approximate solution in the form of a traveling wave with a variable amplitude and phase by using asymptotic methods (WKB, geometric optics or acoustics). For the media with a monotonic change in parameters, such solutions demonstrate the highest gain and the ability to transmit a signal over long distances without distortion.

It turns out to be possible to find exact solutions in the form of traveling waves with variable amplitude and phase in highly inhomogeneous media under certain assumptions on the medium parameters. Our paper reviews possible approaches to finding travelling reflectionless waves in the shallow water channels with variable cross-sections and currents. The basic equations are the classical 1D nonlinear shallow-water equations for water displacement and velocity averaged through the cross-channel. Mathematical procedure to get the solutions in the form of travelling reflectionless waves is based on the transformation of the original equations with variable coefficients to the constant-coefficient PDE. We first demonstrate this procedure using the example of the classical linear wave equation with variable coefficient when it can be reduced to the Klein-Gordon equation with constant coefficients. This gives rise to an ordinary second-order differential equation for finding a variable coefficient (the wave speed), so that traveling waves exist in a wide class of inhomogeneous water channels. The second procedure is the reducing of variable-coefficient 1D wave equation to the spherical symmetric wave equation in the odd-dimensional space, where waves traveling to and from the center are separated. More complicated procedure is developed for the channels with non-uniform current. In conclusion, we discuss the effectiveness of this procedure in the framework of Boussinesq systems.

The study is supported by grants RFBR (20-05-00162, 21-55-15008, 19-35-60022), President of the RF for the state support of Leading Scientific Schools of the RF (Grant No. NSH-70.2022.1.5).

Recent publications:

• Didenkulova I. and Pelinovsky E. On shallow water rogue wave formation in strongly inhomogeneous channels. Journal of Physics A: Mathematical and Theoretical, 2016, vol. 49, 194001.
• Pelinovsky E., Didenkulova I., Shurgalina E., and Aseeva N. Nonlinear wave dynamics in self-consistent water channels. J Phys. A, 2017, vol. 50, 505501.
• Pelinovsky E., Talipova T., Didenkulova I., Didenkulova E. Interfacial long traveling waves in a two-layer fluid with variable depth. Studies in Applied Mathematics, 2019, vol. 142, No. 4, 513–527.
• Churilov S.M., Stepanyants Yu.A. Reflectionless wave propagation on shallow water with variable bathymetry and current. J. Fluid Mech., 2022, vol. 931, A15.