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Received: 15 July 2020 Accepted: 5 November 2020

DOI: 10.1002/pamm.202000307

Study on the Interaction of Nonlinear Water Waves considering Random

Seas

Marten Hollm1,∗, Leo Dostal1, Hendrik Fischer1,and Robert Seifried1

1Institute of Mechanics and Ocean Engineering, Hamburg University of Technology, 21073 Hamburg, Germany

The nonlinear Schrödinger equation plays an important role in wave theory, nonlinear optics and Bose-Einstein condensation.

Depending on the background, different analytical solutions have been obtained. One of these solutions is the soliton solution.

In the real ocean sea, interactions of different water waves can be observed at the surface. Therefore the question arises, how

such nonlinear waves interact. Of particular interest is the interaction, also called collision, of solitons and solitary waves.

Using a spectral scheme for the numerical computation of solutions of the nonlinear Schrödinger equation, the nonlinear

wave interaction for the case of soliton collision is studied. Thereby, the inﬂuence of an initial random wave is studied, which

is generated using a Pierson-Moskowitz spectrum.

© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH

1 Description of extreme water waves

A closer look at the ocean sea reveals that the nature of water waves is random. Irregularities like wind, swell and currents lead

to a stochastic sea surface, which makes it hard to predict the amplitudes of the future incoming waves. Although most of the

time the waves are not high enough to damage ships or offshore structures, extreme waves can occur. These pose a huge risk

for these vessels and all persons present. Up to now, the origin of extreme waves is not fully understood and further studies

are needed. In recent years the inﬂuence of stochastic wind on water waves has been studied [1]. Also it was investigated how

waves change their behavior after a collision with other waves [2].

In a study of water waves, Dias et al. [3] have shown that in many cases it is enough to consider the Euler equation of ﬂuid

dynamics instead of the more complicated Navier-Stokes equations. But since solving the Euler equations is time-consuming

as well, a further problem reduction has be done in our study. By using the method of multiple scales and taking into account

terms up to order O(ε3), the Nonlinear Schrödinger equation (NLS) can be derived in deep water as [1]

iψτ=αψξξ +β|ψ|2ψ, (1)

whereby, α=ω

8k2,β=1

2ωk2and the scaled spatial and temporal coordinates are given by ξ=ε(x−cgt)and τ=ε2t.

Thereby, ψ(ξ, τ )∈Cdescribes the wave envelope, ε1is the wave steepness, xand tare the dimensional spatial and

temporal coordinate, kis the wave number, ωis the frequency of the carrier wave and cg=ω

2kis the deep water group velocity.

Is has to be noted that in comparison to [1], any wind pressure or viscosity effects are here neglected. The corresponding wave

period T, wavelength λand free surface elevation ηof a weakly nonlinear gravity wave are given by

T=2π

ω, λ =g

2πT2, η(x, t) = εiω

gψ(x, t) exp(i(kx −ωt)) + c.c. +O(ε2),(2)

whereby gis the gravity constant and c.c. denotes the complex conjugate.

One solution of Eq. (1) is the soliton solution. According to [4], a soliton solution which has an amplitude a0traveling in

space with velocity vis given by

ψ(ξ, τ ) = a0sech "a0rβ

2α(ξ−ξ0−vτ )#exp(i(cξ −wτ )) (3)

with a shift in space ξ0and the parameters c=v

2αand w=αc2−1

2βa2

0.

2 Collision of soliton solutions

In order to study the collision of different soliton solutions numerically, an initial condition must be used which contains

information about the colliding solutions. For this, a superposition of soliton solutions of the form

ψ0(ξ) =

N

X

i=1

ψi(ξ, 0) (4)

∗Corresponding author: e-mailmarten.hollm@tuhh.de, phone +00 49 40 42878 2308, fax +00 49 40 42878 2028

This is an open access article under the terms of the Creative Commons Attribution License, which permits use,

distribution and reproduction in any medium, provided the original work is properly cited.

PAMM ·Proc. Appl. Math. Mech. 2020;20:1 e202000307. www.gamm-proceedings.com 1 of 3

https://doi.org/10.1002/pamm.202000307 © 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH

2 of 3 Section 12: Waves and acoustics

is used, whereby ψi(ξ, 0) is given by Eq. (3) and characterized by own velocity vi, shift in space ξi

0and amplitude ai

0.

In order to model the sea surface as realistic as possible, an irregular sea surface is added to the colliding solitons. A well-

known model of random long-crested sea waves is given by the superposition of harmonic waves with frequencies ω, wave

numbers k(ω)and amplitude A, which depends on the underlying sea state given by the corresponding one-sided spectral

density S(ω). According to [2], the irregular sea surface is determined by

Z(ξ, τ ) = Z∞

0

cos (ωτ −κ(ω)ξ+ε(ω)) p2S(ω)dω, (5)

whereby the integral is not a Riemann integral but a summation rule over different ω. The perturbation of the undisturbed

solution can then be achieved by

e

ψ(ξ, τ ) = (1 + ΘZ(ξ, τ ))ψ(ξ, τ ),(6)

whereby Θregulates the amount of perturbation.

An initial condition of two soliton waves with the same amplitude a0= 1 m, speed v=±1m/s and frequency ω= 1 rad/s

is shown in Fig. 1 with and without perturbation of an irregular sea surface, respectively. The corresponding numerical

solutions are shown in Fig. 2. These results have been computed by using ﬁnite differences in time and a spectral scheme in

space as well as the Pierson-Moskowitz spectrum S(ω). Figure 2a illustrates that a signiﬁcant wave elevation can be observed

around the time of collision. After the collision the solitons regain their structure. An initial disturbance by irregular waves as

shown in Fig. 2b does not change this behavior much and only leads to perturbed solitons.

3 Conclusion

The obtained results indicate that soliton solutions can exist in irregular seas. An interaction leads to higher waves and the

soliton collision can appear also under realistic sea conditions.

Fig. 1: Initial condition for the soliton wave interaction with and without disturbance. Thereby, the Pierson-Moskowitz spectrum has been

used with signiﬁcant wave height Hs= 1.2m and the modal frequency ωm= 0.5rad/s. The scaling factor Θwas chosen as Θ = 0.2.

a) b)

Fig. 2: Soliton collision awithout and bwith perturbations by a random sea. In both cases, the initial condition from Fig. 1 has been used.

Acknowledgements Open access funding enabled and organized by Projekt DEAL.

References

[1] L. Dostal, M. Hollm and E. Kreuzer, Nonlinear Dyn. 99(3), 2319-2338 (2020).

© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH www.gamm-proceedings.com

PAMM ·Proc. Appl. Math. Mech. 20:1 (2020) 3 of 3

[2] H. Fischer, M. Hollm and L. Dostal, Soliton Collision in Random Seas, submitted, preprint available at arXiv.

[3] F. Dias, A. I. Dyachenko and V. E. Zakharov, Phys. Lett. A 372(8), 1297–1302 (2008).

[4] W. Bao, Q. Tang and Z. Xu, J. Comput. Phys. 235, 423–445 (2013).

www.gamm-proceedings.com © 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH