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In this paper, a ($2+1$)-dimensional generalized breaking soliton system, for the interactions of the Riemann wave with a long wave, is investigated. Via the Hirota method, bilinear forms different from those in the existing literatures are derived. $N$-soliton solutions are constructed via the Wronskian technique. Solitons with the crest curves being curvilineal are constructed, whose shape changes with the propagation. Parallel solitons have been obtained. Directions of the soliton propagation change, and speeds of the solitons are different: The higher the amplitude of the soliton is, the faster the soliton propagates. Breathers are constructed. Solutions consisting of a lump and two solitons are derived: Two solitons propagate in the same direction and the lump occurs in the region of the interaction between the two solitons.

Models for shallow water wave processes are routinely applied in coastal, estuarine and river engineering practice, to problems such as flood waves, tidal circulation, tsunami penetration, and storm tides. Evaluation and confirmation of application codes can be facilitated by analytical solutions that may represent this wide range of physical problems, without significantly compromising the contributing flow processes. The appropriate linearized shallow water wave equations are defined, together with quite general initial and boundary conditions that will represent the expected variety of shallow water wave problems. A completely general analytical solution for both water surface elevation and flow is established, in a manner suitable for the evaluation of shallow water wave codes. The contribution of both free and forced modes is identified, together with the role of friction.

A sequence of analytical solutions explore the spectrum of response patterns expected from numerical codes for flood and tide propagation in channels. Complete analytical details of the solutions are provided, together with specific suggestions for an associated set of analytical benchmark tests. Illustrations of predicted response patterns provide the basis for a discussion of many significant physical aspects and their representation in discrete numerical codes.

A periodical solution of linear long wave was obtained when considering the laminar or turbulent shear stress in the entire flow depth. The solution provides the variation of wave height due to bottom topography and bottom friction. The decay modulus of wave height and the phase differences between velocity, shear stress and water elevation obtained by the laminar flow solution are compared with those predicted by the laminar boundary layer theory. It is shown that the boundary layer approximation cannot be applied to the case where the water depth is small and the wave period is long. Using the turbulent flow solution, the variation of reflection coefficient due to bottom friction is examined. The reflection coefficient depends not only on the Iribarren number but also on the number of waves on a sloping beach and the bottom roughness. An empirical roughness coefficient n is obtained by equating the bottom shear stress evaluated by Manning's friction formula to that evaluated by the present theory. The proper value of n can be determined by considering bottom roughness only.

The force from a tsunami will damage buildings, such as houses, bridges, and many other coastal infrastructures. The impact and drag force of a tsunami depend on the scale and the characteristics of the tsunami as well as the infrastructures' characteristics.

This study was conducted using physical modeling to determine the effect of openings and protection on buildings to reduce the effects of a tsunami attack. The openings were symmetrical about the front and rear walls and were oriented in the direction of the tsunami. Barriers with dimensions that were the same width and three times higher than the building and of various lengths were used to simulate protection against a tsunami.

The results indicated that the force on the buildings depends on but is not linearly correlated to the opening. The barrier upstream of the building significantly reduced the force depending on its distance to the building and the surge Froude number; however, the openings still play an important role in reducing the force on a protected building. Simple equations for practical use are proposed to calculate the tsunami force on a building with openings with or without protection.

Solving the linear shallow water equations on a profile of four segments and matching the solutions on the turning points, runup of periodic long waves is obtained analytically. The resonant phenomena appear when periodic long waves propagate on piecewise topographies. According to the propagation path of tsunami waves, several profiles in the South China Sea are picked up. The topographies can be simplified to a four-segments profile which comprises a horizontal deep seabed, a continental foundation, a continental slope and a continental shelf from the trench to the shore. The runup amplification are calculated both analytically and numerically. Resonant phenomena appear and the values of runup are quite large at the resonant frequencies.

Some aspects of long surface waves are considered. Special attention is focused on the existence of conservation laws for this physical system and related problems.

In this article we explain the well-posedness of the initial value problem for water waves, and the shallow water and long wave approximations for water waves. Especially, we explain that the solution of the full water wave problem can be approximated by the solution of the shallow water equations, the Green–Naghdi equations, the KdV equation, the Kawahara equation, the forced KdV equation, and the Benjamin–Ono equation in some scaling regimes.