Waves can be created by the movement of fluids with different densities which are classified as interface and surface waves. They are characterized by high and low parts that correspond to crests and troughs respectively.

Here we recall the definition of wave steepness as the ratio of the height to the wave length. In an analog to this definition, the steepness function is suitably defined hereafter (cf (14)). The surface waves arise in deep as well as in shallow water. In deep water they are characterized by the fact that the depth is greater than one-half the wave length, that is the relation between the depth and the wave number

The Lie symmetry group applied to the surface wave equation was used to find exact similarity and traveling wave solution in [20]. Solutions of the periodic waves via nonlinear Schrodinger NLS equation was carried in [21]. The surface waves were also studied by using the NLS equation in [23,24]. The formation of three-dimensional surface waves in deep water was considered in [22]. The method used here is the extended unified method proposed by the author in [2, 3, 4, 5, 6].

In this paper, the solutions of the integrable evolution equation for surface waves in deep water [1]
_{x}

By using the approach presented in [4] for the

In (4) the last term stands for transverse dispersion. Here we take

Here self- (and semi-self-) similar solutions of (3) and (4) are obtained.

In what concerning the existence of periodic and elliptic solution of the equations (2–5) we have the following theorems.

The necessary conditions that a NLPDE has a periodic (or elliptic) solutions are that all terms are of even or odd derivatives when

In the case of even order derivatives, then after calculation each term is free from the square root (see (7,8)), so that the equations result are also free. In the case of odd order derivatives all terms factored by the square root and also the equations result are, thus, free.

A sufficient condition for the existence of periodic (elliptic) solutions is that the solutions of the equation obtained are consistent.

After theorem 1, we find that the equation (2) satisfies the necessary conditions while (3–5) do not satisfy them. Thus (2) may have periodic (elliptic) solutions. This requires to verify the sufficient condition. To do this we search for self- similar by assuming that

By substituting (6) when

The equations (3–5) do not satisfy the necessary conditions.

In what follows we find the self-similar and semi-self-similar of (3) when

First, we find the solutions of (3). To this end, it is integrated to give

For bounded solutions at _{0}(

The extended unified method used, suggests solutions in polynomial or rational forms in an auxiliary function that satisfies an auxiliary equation. In the polynomial form;

For details see [2,3]. We mention that the compatibly equation _{ξt}_{tx}

(i) By substituting from (8) into (6) with _{0} = 0, and _{1}(

By bearing in mind that _{x}_{1}(_{2}(_{0} are free parameters. Now the self source is free. Here it is shown two cases when

The solution given in (12) is displayed against x and t in figure 1a

In fig. 1a _{1} = 1.5, _{0} = 5, _{1} = 1.5,

Figure 1a shows that the crest of the wave is lumps coupled to soliton moving along the characteristic curve

(ii) To find a second solution we use the transformation

We use the transformation

We mention that (8) holds but

By substituting from (18) into (17), symbolic calculations give

From the third equation in (19), we find that ^{2}, where

(i) We use the transformations _{0}(_{x},v

We remark that in the present case _{0}(

Finally the solution is

We remark that the solution in (23) describes periodic waves when 16 <

The results in (23) are displayed against _{0}(

In fig. 2. The 3D plot has been don when _{1} = 1.5,

(ii) We use the transformations _{0}(

When substituting from (23) and (24) into (17) and then we use _{0}(^{−0.2t}(1 + 7

Figs 3a and 3b. In Fig. 4a _{0} = −15, _{1} = 0.8, _{2} = −0.7,

Here, we find the self-Similar solution of the extended equation (4). We also use the transformations _{1}(

The coefficients _{1}(_{0}(

We mention that _{1}(_{x}

In (2) for traveling waves, we put _{0} and

It integrates and gives

The Phase portrait is shown in figures 5a, 5b, and 5c.

Figures 4a, 4b, and 5c. The phase portrait is displayed.. In Fig. 5a _{1} = 2,_{0} = 2,

In Fig c _{1} = 2, _{0} = −2,

We remark that the trajectories are open, so the traveling waves are unstable.

The equation for surface waves in deep water has been extended to (2+1) dimensions. The equation obtained is new. Self-similar and semi-self-solutions of (1+1) equation and its extended form have been found. A variety of different geometric structures are shown, Waves with crest formed of coupled lumps and soliton wave moving along the characteristic curve and entailed by troughs are found. Also, multi-periodic, coming and incoming waves, Periodic waves with troughs and lump wave coupled to soliton are shown.. These results are of interest in ocean engineering and sciences. The steepness function is defined and it exhibits mainly the same behavior in all cases studied in this work. The bifurcation is also analyzed and it is found that traveling wave solutions are unstable.

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