Hazards Earth Syst. Asymptotic integrability of water waves. Hydrodynamics of the Coastal Zone. Elsevier , Amsterdam , Bore formation , evolution and disintegration into solitons in shallow inhomogeneous channels.

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Tsunami ascending in rivers as an undular bore. Natural Hazards , , 4 , — Analytical and numerical study of nonlinear effects at tsunami modelling. Runup of nonlinearly deformed waves on a coast. Earth Sci. Steepness and spectrum of a nonlinearly deformed wave on shallow waters. Izvestiya Atmos. Runup of nonlinear asymmetric waves on a plane beach. Springer , , — Nonlinear deformation of a large-amplitude wave on shallow water. Doklady Physics , , 56 , — Theoretical Foundations of Nonlinear Acoustics.

Consultants Bureau , New York , Manchester University Press , Manchester , Nonlinearity and Disorder: Theory and Applications pp Cite as. The scattering of a wavepacket by a random nonlinear medium is analyzed. In the linear limit strong localization occurs, which means that the transmission coefficient decays exponentially with a characteristic localization length. In some nonlinear homogeneous media solitons propagate without changes in their shape or velocity. Solitons are therefore candidates to test the robustness of the exponential localization in random nonlinear media.

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## Random-Phase Surface-Wave Solitons in Nonlocal Nonlinear Media

Lifshitz, I. Google Scholar. Knapp, R. Devillard, P.

Doucot, B. CrossRef Google Scholar. Kivshar, Y.

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For solving the problem 2 , we can use a numerical method and for illustration, two examples of the wave maker movement are proposed. The plan of this paper is as follows. The last section presents numerical applications for illustrating the theoretical model. We consider a fixed Oxy reference system, where the y -axis is vertically ascendant and the x -axis coincides with the initial free surface.

The new coordinates X and Y are introduced as follows:. In this part, we transform the Eqs. Afterwards we use the approximate solution at fourth order and the inverse scattering method for more details, see Aktosun in order to obtain Sturm—Liouville spectral problem. The Eqs.

The double distortion is introduced as follows:. Using new distortion variables in Eqs. Substituting 22 and 23 in 17 — 19 and approximating at fourth order, we choose, among the various approximations, the following. Now in order to show that the KdV equation admits as particular solution a solitary wave, we give the proposition below. The KdV equation 1 admits as particular solution a solitary wave soliton :. Multiply the Eq. It is easy to verify that the function f is a solution of the KdV equation 1.

The solution of the KdV equation 1 corresponding to the reflections potential can be asymptotically represented as a superposition of N single-soliton solutions propagating to the right and ordered in space by their speeds. For this, we give the proposition below. The function 36 is asymptotically represented by a linear superposition. As we said previously, the solution of the KdV equation can be transformed to the Sturm—Liouville linear ordinary differential equation for more details, see Alquran and Al-Khaled ; Temperville ; add to this the boundary conditions, the problem becomes:.

The upper bound for the number N of solitons-solutions can be estimated by the formula see, Grimshaw The spectrum comprises a continuous and discrete spectrum. To solve 39 , three cases are to be considered.

## OSA | Random-Phase Surface-Wave Solitons in Nonlocal Nonlinear Media

By using 45 , we can calculate. These yield. Note that the Eq. Let the solution 44 and the condition 45 , then we have. According to 44 , we have:. We rewrite 60 as a system of first order equations. We take the following data Table 1. Comment : this example shows the propagation of two solitons: the soliton of high amplitude 1.

A soliton propagates more quickly than its amplitude is large and if two solitons of different amplitudes are created, there is a collision that does not change the shape of the waves. We take the following data Table 3. We give the results of two solitons see Fig.

Now, we modify the data as follows Table 4 :.