H.I.Adel-Gwd , M.Tntwy , , M.S.Mni Rjn
a Department of Mathematics, Faculty of Science, Cairo University,Giza, Egypt
b Department of Basic Sciences, Faculty of Engineering, October 6 University, Giza, Egypt
c Department of Physics, University College of Engineering, Anna University, Ramanathapuram, India
ABSTRACT The oscillatory motion on the ocean surface is a combination of a variety of different types of waves.The regularized waves are among them.Here, it is shown that they arise as solutions of the (1+2)-dimensional Benjamin-Bona-Mahony equation (BBME).Numerous works on (1+1)-dimensional BBME were carried in the literature.In this paper, we consider the (1+2)-dimensional non-autonomous BBME, with timedependent coefficients.The model equation is completely new.Our objective is to find the exact solutions and investigate the relevant phenomena.To solve this issue, the extended unified method is used to find the exact solutions in the form of semi-self similar and self similar solutions.To solve this issue, similarity transformations are introduced.Here, the generalized unified methods (GUM) are also used in the symbolic computations.The numerical results of these solutions are evaluated and are shown graphically.Different wave patterns of regularized waves in shallow water, near ocean shores, are observed.Oscillatory waves and vector of lumps with troughs are shown.The time-dependent coefficients are used, here,to control the different wave patterns that take the forms of the multi-U shaped wave with basins with a trough.Further pattern formation occurs, which is in the form of two layers of lumps with troughs.Wave tunneling is also observed.These waves patterns are novel.The stability of the steady state solutions is analyzed.It is found that the stability depends significantly on the dispersion coefficient.
Keywords:(1+2)-dimensional Benjamin-Bona-Mahony equation Extended unified method Wave patterns Similariton solutions
Regularized waves formation may occur near ocean shores.The nomenclature regularized stems from the fact that the BBME is considered as a regularized version of KdVE.The regularized waves in shallow water waves and in certain theoretical investigations may be considered as a superior model for long waves.Also, the BBME is used to the study of drift waves in plasma or the Rossby waves in rotating fluids, the analysis of the surface waves of long wavelength in liquids, hydromagnetic waves in cold plasma and to describe the propagation of acoustic-gravity waves in compressible fluids.In recent years, attentions were focused on analytical studies of various nonlinear evolution equations (NLEEs) with time dependent coefficients.They are of great interest in many fundamental areas of physics, biology, cosmology and fluid dynamics.Many complex phenomena are revealed via the solution of these non-autonomous equations.Relevant works in these areas were performed in [1-7] .In fact, the latest results found in this area are on soliton propagation, that arises by arguing to the effects of the presence of dispersivity and nonlinearity [8-12] .The controlling parameters play a remarkable role in self-similar long waves patterns [13-15] .Self-similar pico second and femto second rogue waves in the nonlinear Schrödinger equation (NLSE) has been observed in [16] .Further, the group velocity, the dispersion and nonlinear effects, with the presence of an external potential had been investigated in [17-25] .Indeed, the later effects mentioned induce a nonlinear change in the refractive index.Nonlinear tunneling[26-28] and non-autonomous solitons with background were discussed in [29] .
In [30] , the first integral method for analytic treatment of the modified BBME where exact new solutions are formally derived.In [31] , the Lie-group formulation was applied to inspect the symmetries of the BBME with variable coefficients.The exp-function method was applied to find some new soliton solutions of the modified BBM equations [32] .In [33] , the bifurcation method of dynamical systems was used to investigate the nonlinear wave solutions of the modified BBME.Numerical computations for the BBME, were, also, performed [34-36] .In [37] , the ´G /G -expansion method was proposed, and used to obtain the traveling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations and the Hirota-Satsuma equations.In [38] , a generalized ´G /G -expansion method was proposed to search for exact solutions of BBME, the (2+1) dimensional generalized Zakharov-Kuznetsov equation, the variant Bousinessq equations and three well-known nonlinear lattice equations[39] .Triangular periodic wave solutions, hyperbolic function solutions, and Jacobi elliptic function solutions were found.Moreover,it can also be used for many other nonlinear evolution equations in mathematical physics.Very recent works are also of interest in this area [40-48] .
The novelty of this work, the unified method [49] presented by the author, was used.After this nomenclature, this method cover most of all known methods such as, the tanh, modified,and extended versions, the F-expansion, the exponential, the ´G /G-expansion methods [50-52] .On the other hand, the extended unified method was proposed [53] , also, is efficient to replace the analysis of inspecting the symmetries of PDEs that arise by using Lie groups.
The (1+1)-dimensional BBME reads,
Eq.(1) describes long waves in a nonlinear dispersive system.The(1+2) dimensional BBM equation with time-dependent coefficients[31] reads,
Here, we consider Eq.(2) with time dependent coefficients,
where α(t) , β(t) and γ(t) are arbitrary functions.In (3), tand x, y represent temporal and spatial variables.We solve Eq.(3) to get single and multiple semi-similar and self-similar waves solutions.To this end, we use suitable similarity variables.
The outlines of this paper are in what follows: In Section 2 , the outlines of the unified method is presented.In Section 3 semi-selfsimilar solutions are obtained, while the self-similar solutions are found in Section 4 .Multiple solutions are found in Section 5 .Stability of the steady state solution is analyzed in Section 6 , while Section 7 is devoted to conclusions.
By this method, solutions are obtained as polynomial or rational forms in an auxiliary functions that satisfies an auxiliary equations[49] .
2.1.Polynomial solutions
We consider the (NLEE) with time dependent coefficients,
(i ) Semi-self-similar solutions
To find semi-self-similar solutions, we use the transformations ξ= κ1x + κ2y , τ= tand u (x, y, t) = U(ξ, τ) , whereare the characteristic wave lengths and ξand τare independent variables.
By substituting from into Eq.(4) , it reduces to nonlinear differential equation,
The solution of Eq.(5) is expressed in form of polynomial by,
where n and k are determined by using the balance condition, that results from balancing the highest order derivative term and the nonlinear term in Eq.(3) .This condition reads n = 2(k -1) .While the limiting value of k is determined from the consistency condition, namely the condition that relates the difference between:
(i) The number of equations that results from substituting from Eq.(6) into Eq.(5) and by setting the coefficients of φi(ξ, τ) =0 , i = 0 , 1 , ..., say r(k ) .Here r(k ) = 6(k -1) .
(ii) The number of parameters in Eq.(6) which is n + k + 2 and the highest order derivative, say m in Eq.(5) .When Eq.(3) is integrable this condition reads r(k ) -(n + k + 2) ≤m, where m is the highest order derivative (here m = 4 ).The consistency condition reads 1 ≤k ≤3 .
2.2.Rational solutions
Here, the solution is in a rational form, in an auxiliary function that satisfies an a auxiliary equation, namely,
We will briefly present the main steps of the (EUM) method applied to evolution equations with variable coefficients.
2.3.Steps of computations:
(a) We substitute from Eq.(6) (or (7)) into Eq.(3) , we get a set of equations in the parameters in ai(τ) , qiand cj.
(b) These equations are solved to find the premeditation parameters.
(c) The auxiliary equations are then solved to get φ (ξ) .
(d) The exact solution given by Eq.(6) (or Eq.(7) ) is calculated
(d) Verification is done via Eq.(3) .
For the balance of rational function solutions of Eq.(3) to exist,we use the transformation u = w x , into Eq.(3) .By integrating on x and by setting the constant of integration zero, we have,
For semi-self-similar solutions we use the transformations ξ=κ1x + κ2y , τ= tand w (x, y, t) = W (ξ, τ) .Thus Eq.(7) is,
Hereafter, we set γ1(τ) =+γ(τ) .
3.1 Polynomial solutions
In this case the balance condition for Eq.(10) is n = (k -1) and the consistency condition gives, 1 ≤k ≤7 / 2 .
We consider k = 2 , n = 1 , then we write,
From Eq.(10) into Eq.(9) and by using the symbolic computational, the solution is,
3.2 Rational solutions
In this case we write rational solutions which take the form,
From Eq.(12) into Eq.(9) and by using the symbolic computational, the solution is,
For self-similar solutions, we use the similarity transformations w (x, y, t) = W (ξ, τ) , ξ= ω1(τ) x + ω2(τ) y , τ= tinto Eq.(8) and self-similar solutions (similarity solutions) are found via polynomial and rational functions.The same step of computations is used.The equation Eq.(8) becomes,
where γ2(τ) =τ) +(τ) γ(τ) .
4.1 Polynomial solutions
When k = 2 , n = 1 , and by using Eq.(10) , into Eq.(14) calculations give rise to the solution,
4.2 Rational solutions
From Eq.(12) into Eq.(14) we find that the solution is,
The numerical results of Eq.(16) fort u (x, y, t) =w (x, y, t) , are shown in Fig.1 (a-c), when y = 2 .
We observe that waves are transmitted periodically as depicted in Fig.1 (a).When we properly vary the time functions α(t) and ω(t) , the frequency varies remarkably.As an example, the functions α(t) and ω(t) are taken periodic functions, which imply periodic waves behavior.Especially, when ω(t)= 5 cos (7 t) , the frequency of wave is increased as shown in Fig.1 (c).When we closely observe the Fig.1 (a)-(c), we may conclude that oscillating period is mainly affected by the control parameters.
Here we analyze the interactions between the combination of waves.For multi-wave solutions to exist via rational function, it is required that a balance between the higher order and the convective term holds.So that we write the solution,
(i ) Multi semi-self-similar solutions
Here we take the multi-auxiliary equations,
By substituting from Eqs.(17) to (18), into Eq.(8) we find,
(ii ) Multi self-similar solutions
In this case, we write the solution in the form,
together with the multi-auxiliary equations,
By substituting from Eqs.(20) and (21), into Eq.(14) , we get,
Fig.1.(a)-(c) The 3D plots are displayed for different values of the parameters.When 1(a): A = 4 , a 0 = 3 , a 1 = -1 .2 , c 0 = 0 .8 , c 1 = 5 .2 , c 2 = 1 .8 , λ= 2 α(t) = 2(4 cos (2 .7 t) +sin (3 .5 t)) β(t) , ω 1 (t) = 5 cos (5 t) , ω 2 (t) = sech (2 t) (1 .5 cos (5 t) + 3 sin (7 t)) .1(b): The same caption as in 1(a) but α(t) = 5 sech (2 t ) β(t ) .1(c): The same caption as in 1(b), but ω 1 (t) = 5 cos (7 t) .
Fig.2.(a)-(c).The 3D plot of u (x, y, t) is displayed against x and t.In Fig.2 (a): A = 4 , a 0 = 0 .5 , a 1 = 3 .2 , a 2 = 6 , b 0 = 2 ., b 1 = 2 .8 b 2 = 0 .4 , c 0 = 0 .2 , c 1 = 1 , c 2 = 1 .2 , λ1 =2 , λ2: = 1 .5 , κ1 = 3 , d 0 = 0 .5 , κ2 = 0 .2 , y = 2 , d 1 = 0 .9 .α(t) = (2 sin (5 t) + 5 sin (7 t)) β(t) , ω 1 (t) = 5 cos (4 t) , ω 2 (t) = 7 sech (2 t) cos (5 t) .In Fig.2 (b) the same caption as in Fig.2 (a), but α(t) = 2 sech (5 t ) 2 β(t ) .In Fig.2 (c) the same caption as in 2(a), but ω 1 (t) = 5 sech (2 t) sin (3 t) .
The numerical results of Eq.(22) , by bearing in mind that u =wxand w (x, t) = W (ξ, τ) , are shown in Fig.2 (a-c)
Fig.2 (a) shows two layers of lumps while Fig.2 (b) shows lumps with small cavities, by varying α(t) .Fig.2 (c) shows a single lump, by varying ω1(t) .
Fig.3 (a) and (3(b) shows multiple U-shaped waves with basins.Fig.3 (c) shows small U-shaped waves.
The results of Eq.(17) , by bearing in mind that u = wxand w (x, t) = W (ξ, τ) , are shown in Fig.4 (a-c).
Fig.4 (a) clearly depicts the soliton fission in which solitons are initially merged and when tincreases, solitons split.Moreover, one of them oscillates.The solitons amplitudes are invariant during the propagation.Since μ2(t) contains only the periodic function, soliton varying periodically.In Fig.4 (b) when we include exponential function in the term μ2(t) with periodic function, soliton dynamics change significantly in portraits.Moreover, soliton collision occur at a specific distance.Also, it indicates that soliton collision can be controlled through the nonuniform effects.Fig.4 (c) obviously illustrates combination of periodic functions the collision of solitons is elastic.In addition, attraction and repulsion between the solitons are periodically varying in time.By closely observing Fig.4 (b)and 4 (c), we may conclude that soliton collision takes place at very short distance due to the effect of combination with the term μ2(t) .
We study the stability of the steady state solutions (SSS) of Eq.(3) .We assume that α(t) → α0, β(t) → β0and γ(t) → γ0as t →∞ .Thus for high values of t, Eq.(3) is written,
The SSS of Eq.(3) is found by setting u t = 0.The only bounded solution of Eq.(23) is us(x, y ) = U0.We consider the expansion,
By introducing Eq.(24) into (23), we get,
The eigenvalue problem in Eq.(25) is taken subjected to the boundary condition | U(±∞ , ±∞ ) | = 0 or | U(±∞ , ±∞ ) |≤U 1 .In the first case, the eigen function U(x, y ) takes the form,
By considering the first quarter and inserting Eq.(26) into (25) we find,
Fig.3.(a)-(c) The 3D plots for u (x, y, t) given by Eq.(20) are displayed against x and t.In Fig.3(a) : A = 4 , a = 1 .2 , a 0 = 4 , a 2 = 2 , b 0 = 2 , b 1 = 3 .8 , b 2 = 3 .8 , c 0 = 0 .2 , c 1 =1 , λ1 = 2 , λ2 = 3 , y = 2 (a) α(t) = 2 sin (t) + 20 , ω 2 (t) = 5 , μ1 (t) = 3 , α(t ) = β(t )(2 sech (5 t ) + 7 cos (4 t) , ) , ω 1 (t) = 5 Cos (5 t ) , ω 2 (t ) = 1 + e-0 .5 t cos (7 t) .In Fig.3 (b): The same caption as in Fig.3 (a), but α(t) = β(t )(5 sech (5 t ) 2 cos (4 t )) .In Fig.3 (c) the same caption as in Fig.3 (a), but ω 1 (t) = 5 cos (5 t ) sech (0 .9 t ) 2 .
Fig.4.(a-c) Solutions of Eq.(1) when a 0 = 0 .9 , a 1 = 4 , a 2 = 6 , b 0 = 0 .5 , b 1 = 1 .2 , b 2 = 0 .4 , c 0 = 2 , c 1 = 0 .2 , d 0 = 0 .5 , d 1 = 0 .5 λ1 = 2 , λ2 = 4 , κ1 = 0 .5 , κ2 = 0 .2 , y = 0 , In (a)μ2 (t) = 8 sin (t) + 2 , in (b) μ2 (t) = e -0 .5(t-2) + e -0 .5(t+2) + 8 sin (2 t) + 2 and in (c) μ2 (t) = 8 sin (5 t) + 6 cos (5 t) + 2 .
Fig.5.(i) and (ii) when k 1 = 3 .5 , k 2 = 2 .5 .In (i) β0 = 1 .3 , γ0 = 1 .8 .(ii) β0 = 0 .3 , γ0 = 1 .8 .
The result in Eq.(27) is displayed in Fig.5 (i) and (ii).
After Fig.4 ) (i) and (ii), we find that the SSS is asymptotically stable when α0＜ 0 and unstable when α0＞ 0 .On the other hand the eigenvalue decreases with raising the dispersion coefficient.
In this work, the extended ((1+2) dimensional) nonautonomous Benjamin-Bona-Mahony equation is considered.The exact solutions are obtained together with investigating the relevant physical phenomena.The extended unified method is used as we are concerned with studying a non-autonomous system.Attention is focused on semi-self and self similar (similariton)solutions.To this issue, similarity transformations are introduced.The generalized unified method is used to find multiple soliton solutions.Different wave patterns formation, which are classified:(a) multiple-U shaped waves with basins(b) two lumps layers with troughs.Further patterns are found, which take the form of multiple wave tunneling and the widths of tunnels increase with time.The stability of the steady state solution is analyzed.It is found the solution is asymptotically stable when the dispersion coefficient is negative, otherwise it is unstable.It is worthy to mention that the model equation, considered here, is new and wave patterns shown in this paper are novel.
Declaration of Competing Interest
The authors declare that there is no conflict of interest