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Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation

Published Online: 14 Oct 2022
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Received: 01 Aug 2020
Accepted: 31 Jan 2021
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

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 k is d > 1/(2k). The equations which describe the free surface elevation and the free surface velocity are nonlinear and dispersive system of equations. Surface wave propagation in deep water stands mainly to the nonlinear modulation of wave vectors. It was derived in [1] by using the the time scales through the perturbation technique, when dealing with the fluid equations. In the last decades, researchers have developed numerous methods such as the Hirota bilinear method [7], the inverse scattering method [8], the Darboux and Bücklund transformations [9], the Lie symmetry analysis [10, 11, 12, 13], the dynamical system methods [14, 15, 16, 17] and the F-expansion method [18,19] to find exact solutions of NLPDEs.

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] δuxxtk2ux+32k(uux)xx=0,,δ=2kg, \delta {u_{xxt}} - {k^2}{u_x} + {3 \over 2}k{\kern 1pt} {(u{\kern 1pt} {u_x})_{xx}} = 0,\;,\delta = 2\sqrt {{k \over g}}, where g is the gravity acceleration and k is the wave number. We mention that 12g/k {1 \over 2}\sqrt {g/k} is the speed of traveling wave. On the other hand there is no balance in (1). The geometric structures of the solutions are visualized. We remark that (1) is integrable and we get δuxtk2u+32k(uux)x=S(t), \delta {u_{xt}} - {k^2}u + {3 \over 2}k{\kern 1pt} {(u{\kern 1pt} {u_x})_x} = S(t), where S(t) is a self-(free)-source to the surface waves in deep water. By using the transformation u(x,t) = vx(x,t), and (2) is rewritten (δvxtk2v+32kvxvxx)x=S(t). {(\delta {v_{xt}} - {k^2}v + {3 \over 2}k{\kern 1pt} {v_x}{\kern 1pt} {v_{xx}})_x} = S(t).

By using the approach presented in [4] for the k^p {\hat k_p} extension operator, (3) is extended to the (1+2) dimensional equation, (δvxtk2v+32kvxvxx)x+μvyy=S(t). {(\delta {v_{xt}} - {k^2}v + {3 \over 2}k{\kern 1pt} {v_x}{\kern 1pt} {v_{xx}})_x} + \mu {v_{yy}} = S(t).

In (4) the last term stands for transverse dispersion. Here we take μ = 1. We mention that (4) is new. It is worth noticing that (3) may be also extended to (3+1) dimensions as (δvxtk2v+32kvxvxx)x+μvyy+σvzz=S(t). {(\delta {v_{xt}} - {k^2}v + {3 \over 2}k{\kern 1pt} {v_x}{\kern 1pt} {v_{xx}})_x} + \mu {v_{yy}} + \sigma {v_{zz}} = S(t).

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 (25) we have the following theorems.

Theorem 1

The necessary conditions that a NLPDE has a periodic (or elliptic) solutions are that all terms are of even or odd derivatives when p = 2 in (7,8), and the equation is invariant under the transformation (x,y,z,t) ← (−x,−y,−z,−t).

Proof

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.

Theorem 2

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 (35) 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 ξ = x ω(t),t := t, and u(x,t) = U(ξ,t). The extended unified method used, suggests solutions in a rational form in an auxiliary function that satisfies an auxiliary equation, U(ξ,t)=a0(t)+a1(t)g(ξ,t)s0(t)+s1(t)g(ξ,t),gξ(ξ,t=γi=02jcigii(ξ,t),gt(ξ,t)=β(t)i=02jcigii(ξ,t). \matrix{{U(\xi,t) = {{{a_0}(t) + {a_1}(t)g(\xi,t)} \over {{s_0}(t) + {s_1}(t)g(\xi,t)}},\,{g_\xi}(\xi,t = \gamma \sqrt {\sum\limits_{i = 0}^{2j} {c_i}g_i^i(\xi,t)},\,} \cr {{g_t}(\xi,t) = \beta (t)\sqrt {\sum\limits_{i = 0}^{2j} {c_i}g_i^i(\xi,t)}} \cr}.

By substituting (6) when p = 2 and j = 1 into (2), we have found that the calculations give rise to inconsistent results. Thus (2) does not have periodic (elliptic), when p = 2 and j = 2, solutions

The equations (35) do not satisfy the necessary conditions.

In what follows we find the self-similar and semi-self-similar of (3) when p = 1.

Self-Similar solutions

First, we find the solutions of (3). To this end, it is integrated to give δvxtk2v+32kvxvxx=S0(t)x+S(t). \delta {v_{xt}} - {k^2}v + {3 \over 2}k{\kern 1pt} {v_x}{\kern 1pt} {v_{xx}} = {S_0}(t){\kern 1pt} x + S(t).

For bounded solutions at x → ±∞, we set S0(t) = 0. We use the transformation ξ = x ω(t),t = t, and v(x,t) = V(ξ,t).

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; Vξ,t)=i=0nai(t)gi(ξ,t),gξ(ξ,t)p=γi=0jpcigii(ξ,t),gt(ξ,t)p=β(t)i=0jpcigii(ξ,t),p=1,2, \matrix{{V\xi,t) = \sum\limits_{i = 0}^n {a_i}(t){g^i}(\xi,t),{\kern 1pt} {g_\xi}{{(\xi,t)}^p} = \gamma \sum\limits_{i = 0}^{jp} {c_i}g_i^i(\xi,t),\,} \cr {{g_t}{{(\xi,t)}^p} = \beta (t)\sum\limits_{i = 0}^{jp} {c_i}g_i^i(\xi,t),p = 1,2,} \cr} or in the rational form; V(ξ,t)=a0(t)+a1(t)g(ξ,t)s0(t)+s1(t)g(ξ,t),gξ(ξ,t)p=γi=0jpcigii(ξ,t),gt(ξ,t)p=β(t)i=0jpcigii(ξ,t). \matrix{{V(\xi,t) = {{{a_0}(t) + {a_1}(t)g(\xi,t)} \over {{s_0}(t) + {s_1}(t)g(\xi,t)}},\,{g_\xi}{{(\xi,t)}^p} = \gamma \sum\limits_{i = 0}^{jp} {c_i}g_i^i(\xi,t),} \cr {{g_t}{{(\xi,t)}^p} = \beta (t)\sum\limits_{i = 0}^{jp} {c_i}g_i^i(\xi,t).} \cr}

For details see [2,3]. We mention that the compatibly equation gξt= gtx holds. Here, we find a class of two self-similar solutions to (3).

(i) By substituting from (8) into (6) with S0 = 0, and j = 2. The calculations give rise to a set of nonlinear equations where each one’s possesses many solutions. The continuation in the calculations is not straight forward. Further, we can have, for example, two equations in a1(t) and a1'(t) a_1^{'}(t) , so we use the compatibly equation a1'(t)(a1(t))'=0 a_1^{'}(t) - {({a_1}(t))^{'}} = 0 . By symbolic calculations, the coefficients in (8) are given by a1(t)=2c2(2+m2)S(t)s0(t)k2(c1+2c1m),a0(t)=S(t)s0(t)k2,s1(t)=2mc2s0(t)(c1+k0),c0=(c12k02)4c2,k0=(c(2+m+m2)(2+m2),β(t)=12c2k2(2+m^2)3(c12(1+2m)(c12m(8+30m12m2+m3)2c2(168m+18m24m3+5m4)δγ2ω(t)),c2=(c12m(8+30m12m2+m3)(20+8m+18m2+4m3+4m4), \matrix{{{a_1}(t) = {{- 2{c_2}(2 + {m^2})S(t){s_0}(t)} \over {{k^2}({c_1} + 2{c_1}m)}},} \cr {} \cr {{a_0}(t) = - {{S(t){\kern 1pt} {s_0}(t)} \over {{k^2}}},} \cr {} \cr {{s_1}(t) = {{2m{c_2}{s_0}(t)} \over {({c_1} + {k_0})}},\;{c_0} = {{(c_1^2 - k_0^2)} \over {4{c_2}}},\;{k_0} = {{(c(- 2 + m + {m^2})} \over {(2 + {m^2})}},} \cr {\beta (t) = {{12{c_2}{k^2}{{(2 + m{{\kern 1pt}^\^}2)}^3}} \over {(c_1^2(1 + 2m)(c_1^2m(8 + 30m - 12{m^2} + {m^3}) - 2c2(16 - 8m + 18m2 - 4{m^3} + 5{m^4})\delta {\kern 1pt} {\gamma^2}{\kern 1pt} \omega (t))}},} \cr {} \cr {\;{c_2} = {{(c_1^2m(8 + 30m - 12{m^2} + {m^3})} \over {(20 + 8m + 18{m^2} + 4{m^3} + 4{m^4})}},} \cr {} \cr {} \cr} and g(x,t)=(5+2m+2m2)(B0h1(x,t)h2(x,t)(1+2m))c1(h2(x,t)(4+m)+B0mh1(x,t))(28m+m2),h1(x;t)=e2c1m2(24+31m+33m2+12m3)((γxω(t)+0tβ(t1)dt1))(40+176m+80m2+144m3+22m4+28m54m6),h2(x,t)=e4c1(10+39m+m66)((γxω(t)+0tβ(t1)dt1)(40+176m+80m2+144m3+22m4+28m54m6)., \matrix{{\matrix{{\matrix{{} \cr {g(x,t) = {{(5 + 2m + 2{m^2})({B_0}{h_1}(x,t) - {h_2}(x,t){\kern 1pt} (1 + 2m))} \over {{c_1}({h_2}(x,t){\kern 1pt} (- 4 + m) + {B_0}m{\kern 1pt} {h_1}(x,t))(- 2 - 8m + {m^2})}},} \cr {} \cr {} \cr {} \cr}} \cr {{h_1}(x;t) = {e^{{{2{c_1}{m^2}(24 + 31m + 33{m^2} + 12{m^3})(- (\gamma x\omega (t) + _0^t\beta ({t_1})d{t_1}))} \over {(40 + 176m + 80{m^2} + 144{m^3} + 22{m^4} + 28{m^5} - 4{m^6})}}}},} \cr {} \cr {} \cr}} \cr {{h_2}(x,t) = {e^{{{4{c_1}(10 + 39m + {m^6}{\kern 1pt} {\kern 1pt} 6)(- (\gamma {\kern 1pt} x{\kern 1pt} \omega (t) + _0^t\beta ({t_1})d{t_1})} \over {(40 + 176m + 80{m^2} + 144{m^3} + 22{m^4} + 28{m^5} - 4{m^6})}}.}}} \cr},

By bearing in mind that v(x,t) = V(ξ,t), thus V (ξ,t) is found first and then the results are put into u = vx. Finally, we have v(x,t)=P/Q,P=2B0h(x,t)c1(4+m)m(2+m)2(1+2m)γS(t)ω(t),Q=k2(2+m2)(3Boh2(x,t)m+h1(x,t)(4+7m2m2))2,h(x,t)=ep0q0,p0=c1(20+78m+24m2+31m3+33m4+12m5+2m6)(0tβ(t1)dt1+xω(t)),q0=(20+88m+40m2+72m3+11m4+14m52m6), \matrix{{v(x,t) = - P/Q,\;P = 2{B_0}h(x,t){c_1}(- 4 + m)m{{(2 + m)}^2}(1 + 2m)\gamma S(t){\kern 1pt} \omega (t),\;} \cr {} \cr {Q = {k^2}(2 + {m^2})(3Bo{h_2}(x,t)m + {h_1}(x,t)(4 + 7m - 2{m^2}{{))}^2},} \cr {} \cr {h(x,t) = {e^{{{{p_0}} \over {{q_0}}}}},} \cr {{p_0} = {c_1}(20 + 78m + 24{m^2} + 31{m^3} + 33{m^4} + 12{m^5} + 2{m^6})(_0^t\beta ({t_1})d{t_1} + x{\kern 1pt} \omega (t)),} \cr {{q_0} = (20 + 88m + 40{m^2} + 72{m^3} + 11{m^4} + 14{m^5} - 2{m^6}),} \cr} where h1(x,t) and h2(x,t) are given in (11). We mention also that ω(t)is free and m,γ,k and B0 are free parameters. Now the self source is free. Here it is shown two cases when S(t) is oscillatory or double humps. In the first case we take ω(t)=1e(0.04t)(3.5+2.5Cos(2t)+1.3Sin(3t))),S(t)=e0.03t(2.5sin(0.7t)+1.3sin(0.3t)). \matrix{{\omega (t) = {1 \over {{e^{(0.04t)(3.5 + 2.5Cos(2t) + 1.3Sin(3t))}})}},} \cr {} \cr {S(t) = - {e^{- 0.03t}}(2.5sin(0.7t) + 1.3sin(0.3t)).} \cr} and for the double humps source ω(t)=0.5e0.04t,S(t)=(at2+b)eμt2 \omega (t) = 0.5{e^{- 0.04t}},S(t) = (a{\kern 1pt} {t^2} + b){e^{- \mu {t^2}}}

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

Fig. 1

In fig. 1a m = −5,c1 = 1.5, B0 = 5, k = 1.7, γ = 2, δ=2k/g \delta = 2\sqrt {k/g} g = 9.8. In fig. 1b m = −5, c1 = 1.5, Bo = 5, k = 1.7, γ = 2, δ=2k/g \delta = 2\sqrt {k/g} , g = 9.8, a = 1.5, b = 0.2, μ = 1/8.

Figure 1a shows that the crest of the wave is lumps coupled to soliton moving along the characteristic curve (t) = const.. Which is entailed by troughs with cavities. Figure 1b shows coupled hump-soliton with a trough which is interior of the hump. Figure 1c shows the steepness function which is defined here by stp=Maxxu(x,t)2|u(x,t)|dx stp = {{\mathop {Maxu}\limits_{x \in {\rm{\mathbb R}}} {{(x,t)}^2}} \over {\int_{ - \infty }^\infty |u(x,t)|dx}}

(ii) To find a second solution we use the transformation u(x,t) = 1/w(x,t), into the equation (2), and we get 2k2w(x,t)3+2S(t)w(x,t)4+9kwx(x,t)22δw(x,t)2wxt(x,t)+w[x,t](4δwt(x,t)wx(x,t)3kwxx(x,t)=0. \matrix{{- 2{k^2}w{{(x,t)}^3} + 2S(t)w{{(x,t)}^4} + 9k{w_x}{{(x,t)}^2} - 2\delta w{{(x,t)}^2}{w_{xt}}(x,t)} \cr {} \cr {+ w[x,t](4{\kern 1pt} \delta {\kern 1pt} {w_t}(x,t){w_x}(x,t) - 3k{w_{xx}}(x,t) = 0.} \cr}

We use the transformation ξ = x ω(t), t = t, and w(x,t) = W (ξ,t)., and we have 2k2W(ξ,t)3+2S(t)W(ξ,t)4+9kω(t)2Wξ(x,t)22δω(t)W(ξ,t)2Wξt(ξ,t)+W(ξ,t)(4δω(t)Wt(ξ,t)Wξ(ξ,t)3kω(t)2Wξξ(ξ,t)=0. \matrix{{- 2{k^2}W{{(\xi,t)}^3} + 2S(t)W{{(\xi,t)}^4} + 9k\omega {{(t)}^2}{W_\xi}{{(x,t)}^2} - 2\delta \omega (t)W{{(\xi,t)}^2}{W_{\xi t}}(\xi,t)} \cr {+ W(\xi,t)(4{\kern 1pt} \delta {\kern 1pt} \omega (t){W_t}(\xi,t){W_\xi}(\xi,t) - 3k\omega {{(t)}^2}{W_{\xi \xi}}(\xi,t) = 0.} \cr}

We mention that (8) holds but W(ξ,t) replaces V(ξ,t), Wξ,t)=i=01ai(t)gi(ξ,t)i=01si(t)gi(ξ,t),gξ(ξ,t)=γi=0jcigi(ξ,t),gt(ξ,t)=β(t)i=0jcigi(ξ,t) W\xi,t) = {{\sum\limits_{i = 0}^1 {\kern 1pt} {a_i}(t){\kern 1pt} {g^i}(\xi,t)} \over {\sum\limits_{i = 0}^1 {\kern 1pt} {s_i}(t){\kern 1pt} {g^i}(\xi,t)}},\,{g_\xi}(\xi,t) = \gamma \sum\limits_{i = 0}^j {\kern 1pt} {c_i}{\kern 1pt} {g^i}(\xi,t),\,{g_t}(\xi,t) = \beta (t)\sum\limits_{i = 0}^j {\kern 1pt} {c_i}{\kern 1pt} {g^i}(\xi,t)

By substituting from (18) into (17), symbolic calculations give a1(t)=(3c1c2(4ka0(t)+9(c12+k02γ2s0(t)ω(t)2)12k(c12k02),c0=(c12k02)4c2,a0(t)=(9c1(c12+k02)γ2s0(t)ω(t)2)4k(c1+2k0),s1(t)=(c2s0(t)(2k3(c12+c1k02k02)+9c1k0(c12+k02γ2S(t)ω(t)2))k3(c13+2c12k0c1k022k03),S(t)=2k2(3c15+c14k07c13k02+9c12k033+2c1k048k05)27c12k0(c12+k02)2ω(t)2,β(t)=(k2(9c163c15k0+11c14k025c13k036c12k0414c1k05+8k06)18c12k0(c12+k02)2δγω'(t)(9c12k02(c12+k02)2δγω(t)), \matrix{{{a_1}(t) = {{(3c1c2{\kern 1pt} (4k{\kern 1pt} {a_0}(t) + 9(c_1^2 + k_0^2{\gamma^2}{s_0}(t)\omega {{(t)}^2})} \over {12{\kern 1pt} k{\kern 1pt} (c_1^2 - k_0^2)}},\;{c_0} = {{(c_1^2 - k_0^2)} \over {4{c_2}}},} \cr {} \cr {{a_0}(t) = {{(9{c_1}(c_1^2 + k_0^2){\gamma^2}{s_0}(t){\kern 1pt} \omega {{(t)}^2})} \over {4k{\kern 1pt} ({c_1} + 2{k_0})}},} \cr {} \cr {{s_1}(t) = {{({c_2}{s_0}(t)(2{k^3}(c_1^2 + {c_1}{k_0} - 2k_0^2) + 9{c_1}{k_0}(c_1^2 + k_0^2{\gamma^2}S(t){\kern 1pt} \omega {{(t)}^2}))} \over {{k^3}{\kern 1pt} (c_1^3 + 2c_1^2{k_0} - {c_1}k_0^2 - 2k_0^3)}},} \cr {} \cr {S(t) = {{- 2{k^2}(3c_1^5 + c_1^4{k_0} - 7c_1^3k_0^2 + 9c_1^2k_0^3{\kern 1pt} {\kern 1pt} 3 + 2{c_1}k_0^4 - 8k_0^5)} \over {27c_1^2{k_0}{{(c_1^2 + k_0^2)}^2}\omega {{(t)}^2}}},} \cr {} \cr {\beta (t) = {{({k^2}(9c_1^6 - 3c_1^5{k_0} + 11c_1^4k_0^2 - 5c_1^3k_0^3 - 6c_1^2k_0^4 - 14{c_1}k_0^5 + 8k_0^6) - 18c_1^2{k_0}{{(c_1^2 + k_0^2)}^2}\delta {\kern 1pt} \gamma {\kern 1pt} {\omega^{'}}(t)} \over {(9c_1^2k_0^2{{(c_1^2 + k_0^2)}^2}\delta {\kern 1pt} \gamma {\kern 1pt} \omega (t))}},} \cr} and g(x,t)=P1Q1,P1=c1(1+A0e(ko(γxω(t)+0tβ(t1)dt1))+k0(1+A0e(k0(γxω(t)+0tβ(t1)dt1)),Q1=c2(1+A0e(ko(γxω(t)+0tβ(t1)dt1)).. \matrix{{g(x,t) = - {{{P_1}} \over {{Q_1}}},\;{P_1} = {c_1}(- 1 + {A_0}{e^{(ko(\gamma {\kern 1pt} x{\kern 1pt} \omega (t) + _0^t\beta ({t_1})d{t_1})}}) + {k_0}(1 + {A_0}{e^{({k_0}(\gamma {\kern 1pt} x{\kern 1pt} \omega (t) + _0^t\beta ({t_1})d{t_1})}}),} \cr {} \cr {{Q_1} = {c_2}(- 1 + {A_0}{e^{(ko(\gamma {\kern 1pt} x{\kern 1pt} \omega (t) + _0^t\beta ({t_1})d{t_1})}}).} \cr}.

From the third equation in (19), we find that S(t) behaves as 1/ω(t)2, where ω(t) is the only free function in this case. The solution u = 1/w gives rise to u(x,t)=p,q,.p=4k(c1k0)(c1+2k0)(3c1(c13+c12ko+c1k02+k03)+4c2ko(c12+c1k02k02)g(x,t)),q=27c12(c1+k0)(c12+ko2)2γ2(c1k0+2c2g(x,t))ω(t)2 \matrix{{u(x,t) = {{p,{\kern 1pt}} \over q},.p = 4k({c_1} - {k_0})({c_1} + 2{k_0})(3{c_1}(c_1^3 + c_1^2ko + {c_1}k_0^2 + k_0^3) + 4{c_2}ko} \cr {(c_1^2 + {c_1}{k_0} - 2k_0^2)g(x,t)),{\kern 1pt} q = 27c_1^2({c_1} + {k_0})(c1{\kern 1pt} {\kern 1pt} 2 + ko{\kern 1pt} {\kern 1pt} 2){\kern 1pt} {\kern 1pt} 2{\gamma^2}({c_1} - {k_0} + 2{c_2}g(x,t))\omega {{(t)}^2}} \cr} where g(x,t) is given by the last equation in (19).

Semi-self-similar solutions.

(i) We use the transformations ξ = β x + ω0(t), t = t, and, u = vx,v(x,t) = V(ξ,t). The equation (8) is also used and the results in (i) section 2 hold but symbolically ”β ” replaces ω(t). c0=(c12k02)4c2,s1(t)=(mc22s0(t))(c13c1k02),k0=c141/6c12c2m(c22m2)6c1,c2=(2c12(32+7m))/(m(688+37m)),β=(360c12k2(2c12+9c2m))((2c16(64+m)+30c14c2(24+m)m+r),r=3c12c22m2(138+17m)+c23m3(178+23m))βδγ,g(x,t)=(1/(4c1(32+7m)))m(688+37m+5ptanh(K(x,t)),p=5238440m+m2,K(x,t)=P2Q2,P2=25c138440m+m2hβδγ2(βx+ω0(t))+8m(688+37m)(688+37m)4k2t),h=(c12m(2342912+364544m20176m2+493m3),.Q2=2(688+37m)c12mhβδγ \matrix{{{c_0} = {{(c_1^2 - k_0^2)} \over {4{c_2}}},{s_1}(t) = {{(mc_2^2{s_0}(t))} \over {(c_1^3 - {c_1}k_0^2)}},{k_0} = {{\sqrt {c_1^4 - 1/6c_1^2{c_2}m - (c_2^2{m^2})}} \over {\sqrt 6 {c_1}}},} \cr {{c_2} = (2c_1^2(32 + 7m))/(m(- 688 + 37m)),} \cr {} \cr {\beta = {{(360c_1^2{k^2}(2c_1^2 + 9{c_2}m))} \over {((2c_1^6(- 64 + m) + 30c_1^4{c_2}(- 24 + m)m + r)}},} \cr {r = 3c_1^2c_2^2{m^2}(- 138 + 17m) + c_2^3{m^3}(178 + 23m))\beta {\kern 1pt} \delta {\kern 1pt} \gamma,} \cr {g(x,t) = - (1/(4{c_1}(32 + 7m)))m(- 688 + 37m + 5ptanh(K(x,t)),} \cr {p = 5\sqrt 2 \sqrt {384 - 40m + {m^2}},} \cr {K(x,t) = {{{P_2}} \over {Q2}},\;{P_2} = 25c1\sqrt {384 - 40m + {m^2}} h{\kern 1pt} \beta {\kern 1pt} \delta {\kern 1pt} {\gamma^2}(\beta {\kern 1pt} x{\kern 1pt} + {\omega _0}(t))} \cr {+ 8m(- 688 + 37m)(- 688 + 37m)4{k^2}t),} \cr {h = (c_1^2m(- 2342912 + 364544m - 20176{m^2} + 493{m^3}),\;.} \cr {{Q_2} = \sqrt 2 (- 688 + 37m)c_1^2m{\kern 1pt} h{\kern 1pt} \beta {\kern 1pt} \delta {\kern 1pt} \gamma} \cr}

We remark that in the present case ω0(t) is free function.

Finally the solution is u(x,t)=pq,p=(4k(68837m)2(16+m)(208+17m)Sech(K(x,t)2),q=(15c12(11264832m+29m2)β2γ2(96+4m+ptanh(K(x,t))2). \matrix{{u(x,t) = {p \over q},{\kern 1pt} p = (4k{{(688 - 37m)}^2}(- 16 + m)(- 208 + 17m)Sech(K{{(x,t)}^2}),} \cr {q = (15c_1^2(11264 - 832m + 29{m^2}){\beta^2}{\gamma^2}(- 96 + 4m + ptanh{{(K(x,t))}^2}).} \cr}

We remark that the solution in (23) describes periodic waves when 16 < m < 24 (see the fourth equation in (22)). Otherwise it describes soliton waves.

The results in (23) are displayed against x and t in figure 3 when ω0(t) = 0.4cos(0.7t) + 0.3sin(0.5t).

In fig. 2. The 3D plot has been don when m = 19, c1 = 1.5, k = 1.7, γ = 2, δ=2k/g \delta = 2\sqrt {k/g} , g = 9.8, β = 0.5. It It shows incoming and out coming multi-periodic waves.

Fig. 2

Fig. 3

(ii) We use the transformations ξ = β x + ω0(t),t = t, and, u = 1/w, w(x,t) = W (ξ,t). By substituting from (18) into (17), calculations give a1(t):=1/12(3c1c2(4ka0(t)+9(c12+k02)γ2s0(t)β212/k(c12k0),a0(t)=9c1(c12+k02)γ2s0(t)β24k(c1+2k0),S(t)=((4k3(c1+2ko))9c1(c12+k02)β2γ2,β(t)=k2(3c13c1k028k03)3c1k02(c12+k02)δβγ,s1(t)=P3Q3,P3=(c2(2k3(c12+c1ko2k02)+9c1k0(c12+k02)β2γ2S(t))s0(t)),Q3=k3(c13+2c12k0c1k022k03), \matrix{{{a_1}(t): = 1/12{{(3{c_1}{c_2}(4k{a_0}(t) + 9(c_1^2 + k_0^2){\gamma^2}{s_0}(t){\beta^2}} \over {12/k(c_1^2 - {k_0})}},} \cr {{a_0}(t) = {{9{c_1}(c_1^2 + k_0^2){\gamma^2}{s_0}(t){\beta^2}} \over {4k({c_1} + 2{k_0})}},S(t) = - {{((4k{\kern 1pt} {\kern 1pt} 3(c1 + 2ko))} \over {9{c_1}(c_1^2 + k_0^2){\beta^2}{\gamma^2}}},} \cr {} \cr {\beta (t) = {{{k^2}(3c_1^3 - {c_1}k_0^2 - 8k_0^3)} \over {3{c_1}k_0^2(c_1^2 + k_0^2)\delta \beta \gamma}},\;{s_1}(t) = {{{P_3}} \over {{Q_3}}},\,} \cr {} \cr {{P_3} = ({c_2}(2{k^3}(c1{\kern 1pt} {\kern 1pt} 2 + c1ko - 2k_0^2) + 9{c_1}{k_0}(c_1^2 + k_0^2){\beta^2}{\gamma^2}S(t)){s_0}(t)),} \cr {} \cr {{Q_3} = {k^3}(c_1^3 + 2c_1^2{k_0} - {c_1}k_0^2 - 2k_0^3),} \cr {} \cr {} \cr {} \cr} and g(ξ,t)=((c1(1+Aoek0(γξ+0tβ(t1)dt1)+(1+Aoek0(zγ+0tβ(t1)dt1)2c2(1+A0ek0(zγ+0tβ(t1)dt1)),ξ=βx+ω0(t). \matrix{{g(\xi,t) = {{- ((c1(- 1 + Ao{e^{{k_0}(\gamma {\kern 1pt} \xi {\kern 1pt} + _0^t\beta ({t_1})d{t_1})}} + (1 + Ao{e^{{k_0}(z{\kern 1pt} \gamma + _0^t\beta ({t_1})d{t_1})}}} \over {2{c_2}(- 1 + {A_0}{e^{{k_0}(z{\kern 1pt} \gamma + _0^t\beta ({t_1})d{t_1})}})}},} \cr {\xi = \beta {\kern 1pt} x{\kern 1pt} + {\omega _0}(t).} \cr}

When substituting from (23) and (24) into (17) and then we use u = 1/w, we get the require solution. It is too lengthy to be produced here. The results for the solution are displayed here when ω0(t) = e−0.2t(1 + 7cos(6t) + 4sin(3t)). The 3D plot is shown in Fig. 4a and the steepness function is shown in Fig. 4b.

Figs 3a and 3b. In Fig. 4a A0 = −15, k = 0.7, δ=20.7/9.8 \delta = 2\sqrt {0.7/9.8} , g = 9.8, c1 = 0.8, c2 = −0.7, k = 0.6, γ = −0.7. In Fig. 4b, the same caption as in 4a.

Fig. 4

Self-Similar solutions of (4).

Here, we find the self-Similar solution of the extended equation (4). We also use the transformations ξ = x ω(t) + 1(t), t = t, and v(x,t) = V(ξ,t). into (4), and we have ω(t)(δω(t)Vξtk2V+32kω(t)3(VξVξξ)ξ+μω1(t)2Vξξ=S(t). \omega (t)(\delta \omega (t){V_{\xi t}} - {k^2}V + {3 \over 2}k{\kern 1pt} \omega {(t)^3}{({V_\xi}{\kern 1pt} {V_{\xi \xi}})_\xi} + \mu {\omega _1}{(t)^2}{V_{\xi \xi}} = S(t). and by substituting from (8) into (25). Calculations give rise to s1(t)=2nc2s0(t)c1,k0=nn1,c1=7n22(n1),c2=15n344(1+n)2,S(t)=77R,R=15(1+n)3(n4(176473510336305n+10972570n2+38638950n3104619116n4+64308384n5+91430374n6107883846n754172755n8+8053425n9))3/2γω(t), \matrix{{{s_1}(t) = {{2n{c_2}{s_0}(t)} \over {{c_1}}},\;{k_0} = - {n \over {n - 1}},{c_1} = {{\sqrt 7 n} \over {\sqrt {22} (n - 1)}},{c_2} = - {{15{n^3}} \over {44(- 1 + n{)^2}}},} \cr {} \cr {S(t) = - {{\sqrt {77}} \over R},\;R = \sqrt {15} {{(- 1 + n)}^3}(- {n^4}(1764735 - 10336305n + 10972570{n^2} + 38638950{n^3}} \cr {} \cr {- 104619116{n^4} + 64308384{n^5} + 91430374{n^6} - 107883846{n^7} - 54172755{n^8}} \cr {+ 8053425{n^9}{{))}^{3/2}}\gamma {\kern 1pt} \sqrt {\omega (t)},} \cr}

The coefficients a1(t),a0(t) and β (t) are very lengthy to be produced here. The auxiliary function is g(x,y,t)=(22154]+(22+154K(x,y,t))/(1+n)))(1+n)15n2(1K(x,y,t)),K(x,y,t)=A1en(γ(xω(t)+yω1(t))+0tβ(t1)dt1)n1 \matrix{{g(x,y,t) = {{(- 22 - \sqrt {154]} + (- 22 + \sqrt {154} K(x,y,t))/(- 1 + n)))(- 1 + n)} \over {15{n^2}(1 - K(x,y,t))}},} \cr {} \cr {K(x,y,t) = {A_1}{e^{{{n(- \gamma (x{\kern 1pt} \omega (t) + y{\omega _1}(t)) + \int_0^t \beta ({t_1})d{t_1})} \over {n - 1}}}}} \cr}

We mention that ω(t) and ω1(t) are free functions. By substituting from (27) and (28) into (9) and by using u = vx, we get u(x,y,t)=P3Q3,P3=(123271611K(x,y,t)(1+n)14((k2n64(49+98n+17n2)2(714n+29n2)(n4(176473510336305n+10972570n2+38638950n3104619116n4+64308384n5+91430374n6107883846n754172755n8+8053425n9))γ9s0(t)24ω(t)15/2)1/2536895n7)2)1/3,Q3=R3R4,R3=(1+n)43(252105972405n+162925n2+3761975n37072513n4,R4=(15kn22(3431372n+518n2+1708n3+255n4)+3103261n5+4112619n6536895n7)2))1/3(7+(7+154)n+K(x,y,t)(7+(7+Sqrt[154])n))2γ4s0(t)8ω(t)3), \matrix{{u(x,y,t) = {{{P_3}} \over {{Q_3}}},\;{P_3} = - {{(12327}^{16}}\sqrt {11} K(x,y,t)(- 1 + n{)^{14}}(- ({k^2}{n^{64}}{{(- 49 + 98n + 17{n^2})}^2}} \cr {(7 - 14n + 29{n^2})} \cr {{\kern 1pt} (- {n^4}(1764735 - 10336305n + 10972570{n^2} + 38638950{n^3}} \cr {} \cr {- 104619116{n^4} + 64308384{n^5} + 91430374{n^6} - 107883846{n^7} - 54172755n{\kern 1pt} {\kern 1pt} 8} \cr {+ 8053425n{\kern 1pt} {\kern 1pt} 9)){\gamma^9}{s_0}{{(t)}^{24}}\omega {{(t)}^{15/2}}{)^{1/2}}} \cr {- 536895{n^7}{)^2}{)^{1/3}},\,{Q_3} = {R_3}{R_4},{\kern 1pt} {R_3} = (- 1 + n{)^{43}}(252105 - 972405n + 162925{n^2} +} \cr {3761975{n^3} - 7072513{n^4},{\kern 1pt} {R_4} = (\sqrt {15} k{n^{22}}(343 - 1372n + 518{n^2} + 1708{n^3} + 255{n^4})} \cr {+ 3103261{n^5} + 4112619{n^6} - 536895{n^7}{)^2}{{))}^{1/3}}(- 7 + (7 + \sqrt {154})n} \cr {+ K(x,y,t)(7 + (- 7 + Sqrt[154])n)){\kern 1pt} {\kern 1pt} 2{\gamma^4}{s_0}{{(t)}^8}\omega {{(t)}^3}),} \cr {} \cr {} \cr} where K(x,y,t) is defined in (28).

Bifurcation

In (2) for traveling waves, we put z = β x + γ t, S(t) = A0 and u(x,t) = U(z), it becomes δγβUk2U+32kβ2(UU')'=A0. \delta {\kern 1pt} \gamma {\kern 1pt} \beta {U^{''}} - {k^2}U + {3 \over 2}k{\kern 1pt} {\beta^2}{(U{\kern 1pt} {U^{'}})^{'}} = {A_0}.

It integrates and gives U'=±6A0kU2β2+4k3U3β2+8A0Uδγ+4k2U2δγ+A1β(3kUβ2+2δγ) {U^{'}} = \pm {{\sqrt {6{A_0}k{U^2}{\beta^2} + 4{k^3}{U^3}{\beta^2} + 8{A_0}U\delta {\kern 1pt} \gamma + 4{k^2}{U^2}\delta {\kern 1pt} \gamma + {A_1}\beta}} \over {(3kU{\beta^2} + 2\delta \gamma)}}

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

Fig. 5

Figures 4a, 4b, and 5c. The phase portrait is displayed.. In Fig. 5a A1 = 2,β = 1.5,γ = 2.3, δ=2k/g \delta = 2\sqrt {k/g} , A0 = 2,g = 9.8.

In Fig c A1 = 2, β = 1.5, γ = −5.3, δ=2k/g \delta = 2\sqrt {k/g} , A0 = −2, g := 9.8.

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

Conclusions

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.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

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