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

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] (1) δuxxt−k2ux+32k (u ux)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 (2) δuxt−k2u+32k (u ux)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) = v_x(x,t), and (2) is rewritten (3) (δvxt−k2v+32k vx vxx)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, (4) (δvxt−k2v+32k vx vxx)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 (5) (δvxt−k2v+32k vx vxx)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 (2–5) we have the following theorems.

First, we find the solutions of (3). To this end, it is integrated to give (7) δvxt−k2v+32k vx vxx=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; (8) Vξ,t)=∑i=0nai(t)gi(ξ,t), gξ(ξ,t)p=γ∑i=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; (9) V(ξ,t)=a0(t)+a1(t)g(ξ,t)s0(t)+s1(t)g(ξ,t), gξ(ξ,t)p=γ∑i=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 g_ξt= g_tx 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 (10) 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=(c12−k02)4c2, k0=(c(−2+m+m2)(2+m2),β(t)=12c2k2(2+m ^2)3(c12(1+2m)(c12m(8+30m−12m2+m3)−2c2(16−8m+18m2−4m3+5m4)δ γ2 ω(t)), c2=(c12m(8+30m−12m2+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 (11) g(x,t)=(5+2m+2m2)(B0h1(x,t)−h2(x,t) (1+2m))c1(h2(x,t) (−4+m)+B0m h1(x,t))(−2−8m+m2),h1(x;t)=e2c1m2(24+31m+33m2+12m3)(−(γxω(t)+0tβ(t1)dt1))(40+176m+80m2+144m3+22m4+28m5−4m6),h2(x,t)=e4c1(10+39m+m6 6)(−(γ x ω(t)+0tβ(t1)dt1)(40+176m+80m2+144m3+22m4+28m5−4m6)., \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 = v_x. Finally, we have (12) 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+7m−2m2))2,h(x,t)=ep0q0,p0=c1(20+78m+24m2+31m3+33m4+12m5+2m6)(0tβ(t1)dt1+x ω(t)),q0=(20+88m+40m2+72m3+11m4+14m5−2m6), \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 (13) ω(t)=1e(0.04t)(3.5+2.5Cos(2t)+1.3Sin(3t))),S(t)=−e−0.03t(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 (14) ω(t)=0.5e−0.04t,S(t)=(a t2+b)e−μt2 \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 xω(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 (15) stp=Maxx∈ℝu(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 (16) −2k2w(x,t)3+2S(t)w(x,t)4+9k​wx(x,t)2−2δw(x,t)2wxt(x,t)+w[x,t](4 δ wt(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 (17) −2k2W(ξ,t)3+2S(t)W(ξ,t)4+9k​ω(t)2Wξ(x,t)2−2δω(t)W(ξ,t)2Wξt(ξ,t)+W(ξ,t)(4 δ ω(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), (18) Wξ,t)=∑i=01 ai(t) gi(ξ,t)∑i=01 si(t) gi(ξ,t), gξ(ξ,t)=γ∑i=0j ci gi(ξ,t), gt(ξ,t)=β(t)∑i=0j ci gi(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 (4k a0(t)+9(c12+k02γ2s0(t)ω(t)2)12 k (c12−k02), c0=(c12−k02)4c2,a0(t)=(9c1(c12+k02)γ2s0(t) ω(t)2)4k (c1+2k0),s1(t)=(c2s0(t)(2k3(c12+c1k0−2k02)+9c1k0(c12+k02γ2S(t) ω(t)2))k3 (c13+2c12k0−c1k02−2k03),S(t)=−2k2(3c15+c14k0−7c13k02+9c12k03 3+2c1k04−8k05)27c12k0(c12+k02)2ω(t)2,β(t)=(k2(9c16−3c15k0+11c14k02−5c13k03−6c12k04−14c1k05+8k06)−18c12k0(c12+k02)2δ γ ω'(t)(9c12k02(c12+k02)2δ γ ω(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 (19) g(x,t)=−P1Q1, P1=c1(−1+A0e(ko(γ x ω(t)+0tβ(t1)dt1))+k0(1+A0e(k0(γ x ω(t)+0tβ(t1)dt1)),Q1=c2(−1+A0e(ko(γ x ω(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 (20) u(x,t)=p, q,.p=4k(c1−k0)(c1+2k0)(3c1(c13+c12ko+c1k02+k03)+4c2ko(c12+c1k0−2k02)g(x,t)), q=27c12(c1+k0)(c1 2+ko 2) 2γ2(c1−k0+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).

(i) We use the transformations ξ = β x + ω₀(t), t = t, and, u = v_x,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). (21) c0=(c12−k02)4c2,s1(t)=(mc22s0(t))(c13−c1k02),k0=c14−1/6c12c2m−(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))β δ γ,g(x,t)=−(1/(4c1(32+7m)))m(−688+37m+5ptanh(K(x,t)),p=52384−40m+m2,K(x,t)=P2Q2, P2=25c1384−40m+m2h β δ γ2(β x +ω0(t))+8m(−688+37m)(−688+37m)4k2t),h=(c12m(−2342912+364544m−20176m2+493m3), .Q2=2(−688+37m)c12m h β δ γ \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 (22) u(x,t)=pq, p=(4k(688−37m)2(−16+m)(−208+17m)Sech(K(x,t)2),q=(15c12(11264−832m+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 (23) a1(t):=1/12(3c1c2(4ka0(t)+9(c12+k02)γ2s0(t)β212/k(c12−k0),a0(t)=9c1(c12+k02)γ2s0(t)β24k(c1+2k0),S(t)=−((4k 3(c1+2ko))9c1(c12+k02)β2γ2,β(t)=k2(3c13−c1k02−8k03)3c1k02(c12+k02)δβγ, s1(t)=P3Q3, P3=(c2(2k3(c1 2+c1ko−2k02)+9c1k0(c12+k02)β2γ2S(t))s0(t)),Q3=k3(c13+2c12k0−c1k02−2k03), \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 (24) g(ξ,t)=−((c1(−1+Aoek0(γ ξ +0tβ(t1)dt1)+(1+Aoek0(z γ+0tβ(t1)dt1)2c2(−1+A0ek0(z γ+0tβ(t1)dt1)),ξ=β x +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

Here, we find the self-Similar solution of the extended equation (4). We also use the transformations ξ = x ω(t) + yω₁(t), t = t, and v(x,t) = V(ξ,t). into (4), and we have (25) ω(t)(δω(t)Vξt−k2V+32k ω(t)3(Vξ Vξξ)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 (26) s1(t)=2nc2s0(t)c1, k0=−nn−1,c1=7n22(n−1),c2=−15n344(−1+n)2,S(t)=−77R, R=15(−1+n)3(−n4(1764735−10336305n+10972570n2+38638950n3−104619116n4+64308384n5+91430374n6−107883846n7−54172755n8+8053425n9))3/2γ ω(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 (27) g(x,y,t)=(−22−154]+(−22+154K(x,y,t))/(−1+n)))(−1+n)15n2(1−K(x,y,t)),K(x,y,t)=A1en(−γ(x ω(t)+yω1(t))+∫0tβ(t1)dt1)n−1 \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 = v_x, we get (28) u(x,y,t)=P3Q3, P3=−(123271611K(x,y,t)(−1+n)14(−(k2n64(−49+98n+17n2)2(7−14n+29n2) (−n4(1764735−10336305n+10972570n2+38638950n3−104619116n4+64308384n5+91430374n6−107883846n7−54172755n 8+8053425n 9))γ9s0(t)24ω(t)15/2)1/2−536895n7)2)1/3, Q3=R3R4, R3=(−1+n)43(252105−972405n+162925n2+3761975n3−7072513n4, R4=(15kn22(343−1372n+518n2+1708n3+255n4)+3103261n5+4112619n6−536895n7)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).

In (2) for traveling waves, we put z = β x + γ t, S(t) = A₀ and u(x,t) = U(z), it becomes (29) δ γ βU″−k2U+32k β2(U U')'=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 (30) U'=±6A0kU2β2+4k3U3β2+8A0Uδ γ+4k2U2δ γ+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.

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.