# The Behavior of Nonlinear Tsunami Waves Running on the Shelf
###### Abstract  
The problem of creating methods for calculating tsunami parameters and predicting these dangerous events is currently being solved by integrating the equations of the theory of water waves. Both numerical methods and powerful computers are used, as well as analytical solutions. The essential stage is the stage of the tsunami reaching the shelf and shallow coastal waters. The tsunami amplitude increases here, and nonlinear effects become important. Nonlinearity excludes the solution's unicity and the superposition principle's fulfillment. The nonlinear theory can have many solutions, depending on various external conditions; there could be nontrivial ones. In this article, we explore the properties of several nonlinear solutions. With their help, we can find the maximum possible amplitude of tsunami waves when approaching the coast and estimate the seismological parameters of an earthquake-generating tsunami.  
nonlinear theory of shallow water; shelf tsunami; dissipative solitons; switching wave; earthquake hazard +
Footnote †: journal: Medical  
increase significantly [20; 21]. Sometimes, tsunamis move in a phase in which the water level ahead is lowered. Such waves are called N-waves [22; 23].  
Selcluded waves were discovered by the English engineer [PERSON] in 1834. He measured the wave resistance of barges moving in channels [18]. The linear theory of water waves was developed in the first half of the 19 th century by [PERSON] and [PERSON] [15; 16]. It was based on the equations of a heavy, nonviscous (ideal) fluid in the Earth's gravitational field. This theory is still used in studying tsunamis near sources [4]. The corresponding models are called nonhydrostatic since they allow large vertical velocities and do not obey the hydrostatic equation. Additional consideration of nonlinearity in the equations of hydrodynamics allowed [PERSON], [PERSON], [PERSON], and [PERSON] to construct a theory of solitary waves [17; 18; 24; 25]. This is used to examine tsunami waves [15; 16; 19] but needs to consider turbulent friction, which becomes essential during tsunami propagation along the shelf and coastal regions.  
The problems of periodic, surface-gravity waves propagating in one horizontal dimension on the water of variable depth were considered by [PERSON] et al. [26]. [PERSON] et al. [27] and [PERSON] [28] analyzed nonlinear packets of surface gravity waves over rugged topography governed by a nonlinear Schrodinger equation with variable coefficients.  
The effect of dissipation on low-amplitude gravitational-capillary waves was considered in [29]. However, waves of large amplitudes, such as tsunamis running along the shelf, are conveniently studied using hydrodynamic equations integrated over the vertical. One theory shows that eliminating the vertical velocities and using the hydrostatic equation is possible in this case. Therefore, these models are called hydrostatic models or shallow water theory models.  
In this paper, we study the nonlinear equations of the shallow water theory and find new solutions that describe solitary tsunami waves traveling along the shelf. It should be noted that these solutions differ from Russell solitons, which arise in an ideal fluid without friction due to the balance of nonlinearity and dispersion typical for waves on water of tremendous or medium depth. There is no dispersion in shallow water; solitary waves can exist here with a balance of nonlinearity and energy dissipation due to friction. These waves can be called dissipative solitons; they also arise, for example, when studying storm surges in oceans and seas [30].