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We consider evolution of wave pulses with formation of dispersive shock waves in framework of fully nonlinear shallow-water equations. Situations of initial elevations or initial dips on the water surface are treated and motion of the dispersive shock edges is studied within the Whitham theory of modulations. Simple analytical formulas are obtained for asymptotic stage of evolution of initially localized pulses. Analytical results are confirmed by exact numerical solutions of the fully nonlinear shallow-water equations.
We derive an asymptotic formula for the amplitude distribution in a fully nonlinear shallow-water solitary wave train which is formed as the long-time outcome of the initial-value problem for the Su-Gardner (or one-dimensional Green-Naghdi) system. O
When a $(1+1)$-dimensional nonlinear PDE in real function $eta(x,t)$ admits localized traveling solutions we can consider $L$ to be the average width of the envelope, $A$ the average value of the amplitude of the envelope, and $V$ the group velocity
We consider the one-dimensional dynamics of nonlinear non-dispersive waves. The problem can be mapped onto a linear one by means of the hodograph transform. We propose an approximate scheme for solving the corresponding Euler-Poisson equation which i
Process of the nonlinear deformation of the surface wave in shallow water is studied. Main attention is paid to the relation between the Fourier-spectrum and wave steepness. It is shown that the spectral harmonics of the initially sine wave can be ex
Process of the nonlinear deformation of the shallow water wave in a basin of constant depth is studied. The characteristics of the first breaking are analyzed in details. The Fourier spectrum and steepness of the nonlinear wave is calculated. It is s