The scientific team had to deal with the following two categories of equations:






From this category it is going to be examined the equations applied to shallow waters. More precisely it is going to be examined the following:

a.             Classical equations otherwise the equations for which it is known the theoretical solution. The research has to improve some of the existing numerical methods and to propose new methods for their solution. The equations which are going to be examined are

i.              Kortweg & de Vries



and its extension in two dimensions the Kadomtsev-Petviashvili equation given by

uxt+3(u^2)x+uxxx -3σuyy=0


where |σ|=1 is a parameter.


ii.          Davey-Stewartson


                                            i u t+η1uxx+η2uyy=λ1 u|u|^2+λ2 uv  

                                                 ρ1 vxx+ρ2 vyy=μ(|u^2)xx


where u=u(x,y,t), v=v(x,y,t) and η1, η2 ρ1, ρ2, μ, λ1, λ2 are parameters.

iii.         Boussinesq


utt=uxx+q uxxxx+(u^2)xx



and its extensions in two dimensions.


iv             nonlinear cubic Schrödinger equation


i u t+uxx+q|u|^2 u=0


Where i is the imaginary unity and q is a real parameter, as well as the coupled Schrödinger equation given by the following system




where u1, u2 are the wave amplitudes and e, δ are parameters.


b.             Equations describing real physical phenomena


We consider for study the following problem describing the wave propagation in swallow water under true natural conditions, which is an application of the Boussinesq equation under real natural circumstances. The problem is briefly described as follows:

The system of wave equations for a wave approaching a coast with variable deepness is defined by the following equations


                                                                           ζ t+px+qy=0                                                                           (2.1)

(continuity equation)



pt +(p^2/h)x+(pq/h)y+gh ζx+(gp/c^2h^2)(p^2+q^2)^{1/2}-E(pxx+pyy)=Dh(pxxt+qxyt)/3                        (2.2)

(impulse preservation equation in the x direction)


qt +(q^2/h)y+(pq/h)x+gh ζy+(gp/c^2h^2)(p^2+q^2)^{1/2}-E(qyy+qyy)=Dh(qxxt+pxyt)/3                        (2.3)

 (impulse preservation equation in the y direction)



where ζ(x,y,t) is the elevation of the free surface of the sea above a reference level, p(x,y,t) is the density of the flow volume (m^3/s/m) in the x direction, q(x,y,t) is the density of the flow volume (m^3/s/m) in the y direction, h(x,y,t) is the deepness from the bottom of the sea, D(x,y) is the deepness from the bottom of the sea in still surface, E(x,y) is the coefficient of noisy flow, C is a constant, H is the wave height, L is the wave length etc.

These equations include non-linear terms as well as frequency distribution. They can be applied in the study of wave potential at sea shelters, ports and small coast territories. The wave disturbance at ports is considerable for the run aground of the boats as well as for the cargo loading and unloading. The disturbance within the port basins is one of the most important factors for the engineers, who are called to choose the construction place for the port and to determine the highest efficiency of the port. The wave movement from the open sea into a port or a sea shelter is a process that basically involves the phenomena of decrease of the depth of the sea, the diffraction, the reflection and the refraction. At this point we must point out that according to the conclusions from extended studies the solution of the above equations is accomplished by the National Center for Marine Researches (NCMR) with the help of credible programs available in the free market. The access to these programs is impossible since their availability depends on the owner companies. As a result, the scientific personnel of the NCMR has not the ability to apply and use productively their specific knowledge on these programs. They are merely called to use them by entering some data.




                                                              u t=k uxx+f(u)                                                                                                  (3.1)


where u=u(x,t), k the contactivity and f(u) a nonlinear function.






a.             Classical equations

i.              Kortweg & de Vries


Freeman, N.C., Soliton solutions of Non-linear Evolution Equations}, IMA Journal of Applied Mathematics, Vol. 32 (1984), pp 125-145.


Ablowitz, J. M. and Villarroel, J., Solutions of the Time Dependent Schrödinger and the Kadomtsev-Petviashvili Equations, Phys. Rev. Let., Vol. 78, No 4 (1997).


Bratsos A. G., Twizell E.H., An Explicit Finite-difference scheme for the solution of the Kadomtsev-Petviashvili equation, Intern. J. Computer Math., 68 (1998), pp 157-187.


ii.             Davey-Stewartson


J. Satsuma, M. J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys., 20, No 7 (1979), pp 1496-1503.


D. Anker, N. C. Freeman, On the soliton solutions of the Davey-Stewartson equation for long waves, Proc. R. Soc. Lond., A. 360 (1978), pp 529-540.


A. Davey, K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. Lond., A. 338 (1974), pp 101-110.


iii.            Boussinesq


J. Argyris and M. Haase, An Engineer's Guide to Soliton Phenomena: Application of the Finite Element Method, Comput. Methods Appl. Mech. Engrg. 61 (1987), pp 71-122.


A.G. Bratsos, The solution of the Boussinesq equation using the method of lines, Comput. Methods Appl. Mech. Engrg. 157 (1998), pp 33-44.


R. Hirota, Exact N-soliton solutions of the wave of long waves in shallow-water and in nonlinear lattices, J. Math. Phys. 14  (1973), pp 810-814.


V.S. Manoranjan, A.R. Mitchell and J.Li. Morris, Numerical solutions of the Good Boussinesq equation, SIAM J. Sci. Stat. Comput. 5 (1984), pp 946-957.


iv             κυβική εξίσωση Schrödinger


E.H. Twizell, A.G. Bratsos and J.C. Newby, A finite-difference method for solving the cubic Schrödinger equation, Mathematics and Computers in Simulation, 43 (1997), pp 67-75.


J.A.C Weideman and B.M. Herbst, Split-step methods for the solution of the non-linear Schrödinger equation, SIAM J. Numer.  Anal. 23(3) (1986), pp 485-507.


G.B. Whitham, Linear and nonlinear Waves, New York: Wiley-Interscience (1974).


v              coupled Schrödinger


M. S. Ismail, Finite difference method with cubic spline for solving nonlinear Schrödinger equation, Int. J. Computer Math., 62 (1996), pp 101-112.


D. F. Griffiths , A. R. Mitchell, J. Li Morris, A numerical study  of the nonlinear Schrödinger equation, Comput. Meths. Appl. Mech. Engrg. 45 (1984), pp. 177-215.


M. Wadati, T. Izuka, M. Hisakado, A coupled Nonlinear Schrödinger equation and optical solitons, J. of Phys. Soc. of Japan., 61, No 7 (1992).



b.            Equations describing real physical phenomena


Bayram, A. & Larson, M. Wave transformation in the nearshore zone: comparison between a Boussinesq model and field data, Coastal Eng. Vol. 39 (2000), pp. 149-171

Beji, S. & Nadaoka, K. A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth, Ocean Eng., Vol. 23 (1996), pp. 691-704.

Larsen, J & Dancy, M., Open Boundaries in Short Wave Simulations - A New Approach. Coastal Engineering, Vol 7 (1983), pp 285-297.

Madsen, P A, Murray, R & Sørensen, O R, A New Form of the Boussinesq Equations with Improved Linear Dispersion Characteristics (Part 1). Coastal Engineering, Vol 15, No 4 (1991), pp 371-388.

Madsen, P A & Sørensen, O R., A New Form of the Boussinesq Equations with Improved Linear Dispersion Characteristics, Part 2: A Slowly-Varying Bathymetry. Coastal Engineering, Vol 18, No 1 (1992), pp 183-204.

Nadaoka, K. & Raveenthiran, K., A phase-averaged Boussinesq model with effective description of carrier wave group and associated long wave evolution. Ocean Eng., Vol. 29 (2002), pp. 21-37.

Oliveira, F.S.B.F., Improvement on open boundaries on a time dependent numerical model of wave propagation in the nearshore region. Ocean Eng., Vol. 28 (2000), pp. 95-115.

Zou, Z.L., High order Boussinesq equations, Ocean Eng., Vol. 26 (1999), pp. 767-792.