Research Article Dynamics of interaction between solitary and rogue of the space-time fractional Broer–Kaup models arising in shallow water of harbor and coastal zone

Selina Akter1,2 · Ratan Kumar Sen1 · Harun‑Or‑ Roshid1

Received: 26 May 2020 / Accepted: 28 October 2020 / Published online: 16 November 2020 © Springer Nature Switzerland AG 2020

Abstract Under inquisition, we consider the space-time fractional Broer–Kaup model that replicate the real coastal profle to describe bidirectional wave transmission of long in shallow water. We integrate of the model via the unifed scheme to derive exact solitary wave solutions. The achieved results present the hyperbolic, trigonometric, and rational func- tions solutions. All of these solutions convey the periodic and solitonic natures. In particular, we present combo waves that imply a curvy periodic waves in which rogue waves occur in both sides of each waves. The dynamics of obtained nonlinear wave solutions are demonstrated in 3-D and 2-D shapes with defnite parametric values. It is shown that the obtained solutions of the model pact a very rich dynamical behavior and can be used to describe seismic type nonlinear wave motion in shallow water waves of the coastal and harbor region.

Keywords The space-time fractional Broer–Kaup equation · The unifed method · Conformable derivative · Traveling wave solution

1 Introduction on numerical modeling of wave propagation [3–5]. Non- linear fractional derivatives are the appropriate tools for Generally, overall the world, configurations in the low modeling highly non-linear problems including seismic lying area of coastal region are often afected by waves in harbor and coastal region. One of this model is wave pressure or cyclonic and such massive the space-time fractional Broer–Kaup model (FBKM) [3]. waves of the and consequently damages. Mostly In this article, we shed light on the space-time fractional broken tsunami wave moves inland in the form of tur- Broer–Kaup model (FBKM) which is signifcant model to bulent hydraulic bores [1]. During tsunami type seismic describe bidirectional wave transmission of long waves in occur, region is severely damaged. type massive shallow water. The usual form of the space-time FBKM [3] waves create dissipated energy that travels through water is as follows medium and when the depth of the ocean decreases, the T uT  u T  v 0, increase to higher. The velocity of tsunami t + x + x = waves depends on ocean intensity instead of the gap from � � � � � � the source of the wave [2]. It is, therefore, an important Tt v + Tx u + Tx (uv)+Tx Tx Tx u = 0, t > 0, 0 <�, � ≤ 1. event whose study will contribute to better understand- (1) ing of coastal hazards from seismic waves. Right now, such massive wave related research for the most part focuses

* Harun‑Or‑ Roshid, [email protected]; [email protected] | 1Department of Mathematics, Pabna University of Science and Technology, 6600, Pabna, Bangladesh. 2Department of Applied Mathematics, Gono Bishwabidyalay, Savar, Dhaka, Bangladesh.

SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8

Here Tt and Tx indicate conformable fractional derivative sides of each waves and express in terms of rational, trigo- [4] reverence to t and x, respectively. Besides, the seismic nometric and hyperbolic functions. (, tsunami and typhoon) waves are refect as The structure of this paper goes along with: In Sect. 2, the solitary waves. are stable and confned non- we have recalled the unifed method. In Sect. 3, we have linear wave progressed balancing between nonlinearity presented the exact solutions to the space-time fractional and in an arrangement. Due to their rising in a Broer–Kaup equations employing the unifed method. In massive diversity in media, extending from fuids [5, 6] and Sect. 4, we have discussed obtaining solutions with the astrophysical plasmas [7, 8] to semiconductors [9, 10] and graphical representation and in the last section, conclu- nonlinear optics [11, 12], solitons have converted a sensi- sions are given. ble purpose for technical exploration. Such hip in natural structures, solitons can similarly be formed in laboratories as dual plasma schemes [13, 14], Joseph-son junctions [15, 2 The unifed method 16] and so on. With this point of view, deriving and its interaction solution of nonlinear wave models attract a Here, we illustrate the unifed method [29, 30] for govern- much interest of young scientist [17–19]. Currently, many ing exact solutions of nonlinear partial diferential equa- scientists are working with considerable eforts on the tions (NLPDEs) in the subsequent way. Adopt a NPDE in fractional diferential models as it gives the real proper- two independent variables x and t is given by ties of natural phenomena [20, 21]. Tozar et. al. [22], Kurt P u, u , u , u , u , u , 0, [23], Kurt et. al. [24], Tasbozan [25] and Tasbozan et. al. [26] x xxt xxx t tt … = (2) investigated new solutions for the time fractional integra- where p is a polynomial in u = u(x, t) and its several deriva- ble dispersive model that arises in ocean engineering, the tives including the highest order derivatives and non- fractional Bogoyavlensky-Konopelchenko model that linear terms. arising in fuid dynamics, the fractional coupled Burgers’ Step-1: To convert the NLPDE into Ordinary diferential equation that arises as a model of poly-dispersive sedi- equation (ODE), we operate the traveling wave variable mentation, the time fractional Benjamin-Ono model that arises internal waves in deep water and the fractional u(x, t) = U( ), = kx − t, (3) Drinfeld-Sokolov-Wilson system in shallow water waves respectively. Iyiola et. al. [27] studied on the analytical where k is constant and ω is the wave velocity. Now switch- solutions of the system of conformable time-fractional ing (3) into (2) we attain an ODE Robertson equations with 1-D difusion. Cenesiz et. al. [28] R u u u uu u , , , , … = 0. (4) established new exact solutions of Burgers’ type equations with conformable derivative. Step-2: Now we integrate (4) as many times as possible. Various approaches are used to study the above frac- Keep the integrating constant to zero. tional models. Recently, Gozukizil et. al. [29] proposed a unifed scheme that receiving much attention of young Assume that the solution of ODE (4) takes the follow- researcher to handle non-linear fractional diferential mod- ing form els. Beside this, some researcher has compared the all tanh N methods [8, 9] with the unifed method [29]. Akcagil and U( ) = l + l S( )i + m S( )−i , Aydemir [30] have exemplifed that the unifed method 0 i i (5) i=1 is not only more general than the family of tanh function   methods but also it gives many more general solutions whose li (i = 0, 1, 2, …, N) and mi(i = 0, 1, 2, …, N) are con- than the new members of the family of (G′/G)-expansion stants to be calculated subsequently on the situation that method. They come to verdict that the unifed method lN and mN cannot be zero at a time. The function S(ξ) satis- gives many more general solutions in an elegant way than fes the Riccati equation, the family of the tanh-function methods. S� S 2 Therefore, the aim of this paper is to solve the space- = ( ( )) + , (6) time fractional Broer–Kaup equation for finding exact whose solutions are specifed as follows: solitary wave solutions via the efective unifed method Case-1: Hyperbolic function (when λ < 0): [29, 30]. This method produces combo waves that imply a curvy periodic waves in which rogue waves occur in both

Vol:.(1234567890) SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8 Research Article

3 Solutions for the space‑time FBK model −(l2+d2)−l − cosh 2 −( +E) , via the unifed method l sinh 2√ −( +E�) √+d � ⎧ √ − −(l2+d2)�−l√ − cosh� 2 −( +E) ⎪ , Now, we apply the unifed integral scheme to extract exact ⎪ l sinh 2√−( +E)� +√d � solutions of (1). First, we consider the following transfor- S( ) = ⎪ √ − cosh 2 �−√( +E) −sinh� 2 −( +E) −l mation to convert (1) into ordinary diferential equation ⎪ , ⎪ √ l+�cosh �2√−( +E)�−sinh �2√−( +E)� � (ODE): ⎨ l � �E � �E ⎪ − −cosh √2 −( + ) −sinh √2 −( + ) t x , u x t U a b ⎪ √ l+�cosh 2� −√( +E) +�sinh 2� −√( +E) �� ( , ) = ( ); = + , (7) ⎪   ⎪ � √ � � √ � Case-2⎪ : Trigonometric function (when λ > 0): where a, b are constants. ⎩ Plugging (7) into (1), we have the following ODE (l2−d2)−l  cos 2 ( +E) , aU� + bUU� + bV � = 0. (8) l sin 2√( +E�) √+d � ⎧ √ − (l2−d2)�−√l  cos� 2 ( +E) � � � � 3 ��� ⎪ , aV + bU + b UV + VU + b U = 0.. (9) ⎪ l sin 2 √( +E)� +√d � S( ) = ⎪ √  i cos 2� √( +E) +�sin 2 ( +E) −il One time integration of (8) with respect to ξ, yields ⎪ , � � � � � � ⎪ √ l+cos 2√( +E) −i sin 2√ ( +E) C 2 ⎨ 1 U a  −i cos� √2 ( +E�) +sin� 2√ ( +E�) +il V = − − U. (10) ⎪ , b 2 b ⎪ √ �l+cos �2 √( +E) �+i sin �2√( +E)� � ⎪ Switching (8) into (7) yields a unique equation ⎪ � √ � � √ � where l⎪ ≠ 0 and d, E are real arbitrary constant. b 3a a2 ⎩ b3U�� − U3 − U2 + C + b − U − C = 0, Case-3: Rational function solutions (when λ = 0) 2 2 1 b 2 (11)  1 S( ) =− . here C and C are integration constants. + E 1 2 Now balancing the terms U″ and U3 of (11), unknown N Step-03: By homogeneous balance principle, we deter- becomes N = 1. mine the positive integer N of (4). Inserting (5) into (4) So the trial solution (5) takes the form making use of (6), and then extracting all terms of like U( ) = l + l S( ) + m S−1( ). powers of S(ξ) to zero yields an over-determined sys- 0 1 1 (12) tem of algebraic equations. Solving this algebraic equa- Combining (6), (11) and (12) and collecting the coef- tions for li(i = 0, 1, 2, …, N) and mi(i = 0, 1, 2, …, N) k, ω and ficients of S(ξ), gives the following set of algebraic inserting the values into (5) together with the general equations: solutions of (6), provides us the required solutions of (2). 1 3 3 2b3l − bl3 = 0; − bl l2 − al2 = 0; 1 2 1 2 0 1 2 1

a2 b2 bC l 1 − − 1 1 2b3l − b m l2 + 2l2l + l l2 + 2l m − 3al l − = 0; 1 2 1 1 0 1 1 0 1 1 0 1 b  

a2 b2 bC l 1 3 − − 1 0 − b 4m l l + l l2 + 2l m − a l2 + 2l m − − C = 0; 2 1 0 1 0 0 1 1 2 0 1 1 b 2   

a2 b2 bC m 1 − − 1 1 2b3m + b m l2 + 2l m + 2l2m + l m2 − 3am l − = 0; 1 2 1 0 1 1 0 1 1 1 1 0 b   

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8

3 3 1 − bm2l − am2 = 0; 2b3m 2 − bm3 = 0; 2 1 0 2 1 1 2 1 Solving the above algebraic equations, we obtain the following sets of constraints:

Set-1:

4 2 2 2 2 1 4b4 + a2 + 2b2 1 4b + a + 2b a 2a3 2a − 2b a a C =− , C =− − + − , l =− , l = 2b, m = 0. 1 2 b 2 2b b b b 0 b 1 1     

Set-2:

4 2 2 2 2 1 4b4 + a2 + 2b2 1 4b + a + 2b a 2a3 2a − 2b a a C =− , C =− − + − , l =− , l =−2b, m = 0 1 2 b 2 2b b b b 0 b 1 1     

Set-3:

4 2 2 2 2 1 4b4 + a2 + 2b2 1 4b + a + 2b a 2a3 2a − 2b a a C =− , C =− − + − , l =− , l = 0, m = 2b 1 2 b 2 2b b b b 0 b 1 1     

Set-4:

4 2 2 2 2 1 4b4 + a2 + 2b2 1 4b + a + 2b a 2a3 2a − 2b a a C =− , C =− − + − , l =− , l = 0, m =−2b 1 2 b 2 2b b b b 0 b 1 1     

Set-5:

4 2 2 4 2 2 1 −8b4 + a2 + 2b2 1 −8b + a + 2b a 2a3 24b + 2a − 2b a a C =− , C =− − + − + 24b3a , l =− , l = 2b, m = 2b 1 2 b 2 2b b b b 0 b 1 1     

Set-6:

1 16b4 + a2 + 2b2 C =− , 1 2 b 4 2 2 4 2 2 1 16b + a + 2b a 2a3 −24b + 2a − 2b a a C =− − + − − 24b3a , l =− , l = 2b, m =−2b 2 2b b b b 0 b 1 1     

Vol:.(1234567890) SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8 Research Article

Set-7:

1 16b4 + a2 + 2b2 C =− , 1 2 b 1 16b4 + a2 + 2b2 a 2a3 −24b4 + 2a2 − 2b2 a a C =− − + − − 24b3a , l =− , l =−2b, m = 2b 2 2b b b b 0 b 1 1     

Set-8:

1 −8b4 + a2 + 2b2 C =− , 1 2 b 4 2 2 4 2 2 1 −8b + a + 2b a 2a3 24b + 2a − 2b a a C =− − + − + 24b3a , l =− , l =−2b, m =−2b . 2 2b b b b 0 b 1 1      Assigning the Set-1 into the trial solution (12), we reach 2b l2 − d2 − l cos 2 ( + E) the results a u (x, t) =− − � �, 16 b �� � √ � √ � l sin 2 ( + E) + d (a) Hyperbolic function solutions (when λ < 0): � √ � (18)

2b − l2 + d2 − l − cosh 2 − ( + E) a u (x, t) =− + � �, (13) 11 b � � � √ � √ � l sinh 2 − ( + E) + d � √ �

2b − l2 + d2 − l − cosh 2 − ( + E) a u (x, t) =− − � �, (14) 12 b � � � √ � √ � l sinh 2 − ( + E) + d � √ �

b E E l a 2 − cosh 2 − ( + ) − sinh 2 − ( + ) − u (x, t) =− + , (15) 13 b � � � � � � √l + cosh 2 √− ( + E) − sinh 2 √− ( + E) � √ � � √ � (b) Trigonometric function solutions (when λ > 0):

b l E E a 2 − − cosh 2 − ( + ) − sinh 2 − ( + ) u (x, t) =− + , (16) 14 b � � � � �� √l + cosh 2 −√ ( + E) + sinh 2 −√ ( + E) � √ � � √ � b i E E il a 2 cos 2 ( + ) + sin 2 ( + ) − u (x, t) =− + , 17 b � � � � � � √l + cos 2 √ ( + E) − i sin 2√ ( + E) 2b l2 − d2 − l cos 2 ( + E) � √ � � √ � (19) a u (x, t) =− + � �, b i E E il 15 b �� � √ � √ � a 2 − cos 2 ( + ) + sin 2 ( + ) + l sin 2 ( + E) + d u (x, t) =− + , 18 b � � � � � � √ l + cos 2 √ ( + E) + i sin 2√ ( + E) � √ � (17) � √ � � √ � (20)

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8

(c) Rational function solutions (when λ = 0) b l E d a 2 sin 2 ( + ) + a 2b u (x, t) =− + , u (x, t) =− − , 35 b � � √ � � 19 b + E (21) l2 − d2 − l cos 2 ( + E) t x � � (26)� where = a + b . � � √ √   b l E d Assigning the Set-3 into the trial solution (12), we reach a 2 sin 2 ( + ) + the results u36(x, t) =− − , b � � √ � � l2 − d2 − l cos 2 ( + E) (a) Hyperbolic function solutions (when λ < 0): �� � √ � √ (27)� b l E d a 2 sinh 2 − ( + ) + u (x, t) =− + , 31 b � � √ � � − l2 + d2 − l − cosh 2 − ( + E)

� � � √ � √ (22)� b l E d a 2 sinh 2 − ( + ) + u (x, t) =− − , 32 b � � √ � � − l2 + d2 − l − cosh 2 − ( + E)

� � � √ � √ (23)�

b E E l a 2 − cosh 2 − ( + ) − sinh 2 − ( + ) − u (x, t) =− + , (24) 33 b � � � � � � √l + cosh 2 √− ( + E) − sinh 2 √− ( + E) � √ � � √ � (b) Trigonometric function solutions (when λ > 0):

b l E E a 2 + cosh 2 − ( + ) + sinh 2 − ( + ) u (x, t) =− + , (25) 34 b � � � � �� − l − cosh √2 − ( + E) − sinh √2 − ( + E) √ � � √ � � √ ��

Fig. 1 The profle of solution u11 of the space-time FBK equation sketched for the parameters l = a = E = 1, b = 0.5, λ = − 2, α = β = 3/4: (a) 3D plot; (b) 2D plots

Vol:.(1234567890) SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8 Research Article

Fig. 2 The profle of rogue into a periodic waves of the solution u17 for the parameters l = 2, a = b = λ = 1, α = β = 3/4, E = 0: (a) 3D (upper), contour (lower) plots of real part; (b) 3D (upper), contour (lower) plots of imaginary part; (c) 2D plots real part; (d) 2D plots of imaginary part

b l E i E a 2 + cos 2 ( + ) − sin 2 ( + ) u (x, t) =− + , 37 b � � � � �� i cos 2 √ ( + E) + sin 2 √( + E) − il √ � � √ � � √ � (28)�

(c) Rational function solutions (when λ = 0)

b l E i E a 2 + cos 2 ( + ) + sin 2 ( + ) u (x, t) =− + , (29) 38 b � � � � �� −i cos 2 √ ( + E) + sin 2 √( + E) + il √ � � √ � � √ � �

a u (x, t) =− − 2b( + E), 39 b (30)

t x where = a + b .   Assigning the Set-5 into the trial solution (12), we reach the results

(a) Hyperbolic function solutions (when λ < 0):

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8

Fig. 3 The profle of rogue into a periodic waves of the solution u18 for the parameters l = 2, a = b = λ = 1, α = β = 3/4, E = 0: (a) 3D (upper), contour (lower) plots of real part; (b) 3D (upper), contour (lower) plots of imaginary part; (c) 2D plots real part; (d) 2D plots of imaginary part

Fig. 4 The profle of solution u19 of the space-time FBK equation sketched for the parameters E = 3, a = 1, b = 0.5, λ = 0, α = β = 1/2: (a) 3D plot; (b) 2D plots

Vol:.(1234567890) SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8 Research Article

2 2 2b − l + d − l − cosh 2 − ( + E) 2b l sinh 2 − ( + E) + d a (31) u51(x, t) =− + �� � + , b � � √ � √ � � � √ � � l sinh 2 − ( + E) + d − l2 + d2 − l − cosh 2 − ( + E)

� √ � � � � √ � √ �

2 2 2b − l + d − l − cosh 2 − ( + E) 2b l sinh 2 − ( + E) + d a (32) u52(x, t) =− − �� � − , b � � √ � √ � � � √ � � l sinh 2 − ( + E) + d − l2 + d2 − l − cosh 2 − ( + E)

� √ � � � � √ � √ �

b E E l b E E l a 2 − cosh 2 − ( + ) − sinh 2 − ( + ) − 2 − cosh 2 − ( + ) − sinh 2 − ( + ) − u (x, t) =− + + , (33) 53 b � � � � � � � � � � � � √l + cosh 2 √− ( + E) − sinh 2 √− ( + E) √l + cosh 2 √− ( + E) − sinh 2 √− ( + E) � √ � � √ � � √ � � √ � (b) Trigonometric function solutions (when λ > 0):

b l E E b l E E a 2 − − cosh 2 − ( + ) − sinh 2 − ( + ) 2 + cosh 2 − ( + ) + sinh 2 − ( + ) u (x, t) =− + + , (34) 54 b � � � � �� � � � � �� √l + cosh 2 −√ ( + E) + sinh 2 −√ ( + E) − l − cosh 2√ − ( + E) − sinh 2√ − ( + E) � √ � � √ � √ � � √ � � √ ��

2 2 2b l − d − l cos 2 ( + E) 2b l sin 2 ( + E) + d a (35) u55(x, t) =− + �� � + , b � � √ � √ � � � √ � � l sin 2 ( + E) + d l2 − d2 − l cos 2 ( + E)

� √ � �� � √ � √ �

2 2 2b l − d − l cos 2 ( + E) 2b l sin 2 ( + E) + d a (36) u56(x, t) =− − �� � − , b � � √ � √ � � � √ � � l sin 2 ( + E) + d l2 − d2 − l cos 2 ( + E)

� √ � �� � √ � √ �

b i E E il b l E i E a 2 cos 2 ( + ) + sin 2 ( + ) − 2 + cos 2 ( + ) − sin 2 ( + ) u (x, t) =− + + , (37) 57 b � � � � � � � � � � �� √l + cos 2 √ ( + E) − i sin 2√ ( + E) i cos 2 √( + E) + sin 2 (√ + E) − il � √ � � √ � √ � � √ � � √ � � (c) Rational function solutions (when λ = 0)

b i E E il b l E i E a 2 − cos 2 ( + ) + sin 2 ( + ) + 2 + cos 2 ( + ) + sin 2 ( + ) u (x, t) =− + + , (38) 58 b � � � � � � � � � � �� √ l + cos 2 √ ( + E) + i sin 2√ ( + E) −i cos 2 √ ( + E) + sin 2 √( + E) + il � √ � � √ � √ � � √ � � √ � �

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8

a 2b including combination of sine-cosine, combination of u (x, t) =− − − 2b( + E), 59 b + E (39) sine hyperbolic-cosine hyperbolic and rational function

t x solutions for the space-time FBK are derived by applying where = a + b .   unifed method. But, by means of unifed approach, we The other sets of solutions can obtain in the similar to determined solutions which are diferent from Yaslan [3] the above three sets and we avoid these for simplicity. solutions. We observed that their solutions (40), (41) and

(44) are similar to our solutions u11, u15 and u19 respectively Remark: The solutions obtained in this paper have been (See the Appendix). Other solutions are completely new verifed with Maple-18 by placing them back to the con- with more complex phenomena and were not obtained sider models (1). by Yaslan [3]. It is worth declaring that exp (−φ (ξ))- expansion method is special case of the unifed method. Any researcher can compare their experimental/numerical 4 Clarifcation and graphical results with our exact solutions to require actual investiga- representations of the solutions tion like Ref. [1, 2].

In this segment, we discuss the physical exposition of securing solutions to the space-time FBKM by manipulat- 6 Conclusion ing new vigorous method namely the unifed method. We describe the behavior of all possible solutions with 3-D In this paper, we have fruitfully determined many new and 2-D plots. The solutions u11, u12, u13, u14, u31, u32, u33, exact traveling wave solutions including combo periodic- u34, u51, u52, u53, u54 express hyperbolic function signifes solitary wave of the space-time FBKM by using the unifed solitonic nature. All the solutions have similar confgura- method. The dynamics of the obtained wave solutions are tions and depicted only Fig. 1 of u11. demonstrated in 2-D, 3-D and contour shapes. The Fig. 1 From Fig. 1 we can see that the wave amplitudes displayed that the wave amplitudes abruptly changed and abruptly change and go to infinity after reaching the gone to infnity after reaching the beach of coaster area as beach of coaster area as x, t → 0. The solutions u15, u16, x, t → 0, but the Fig. 4 showed that the wave amplitudes u17, u18, u35, u36, u37, u38, u55, u56, u57, u58 express trigono- abruptly changed to reduces after reaching shallow water metric function that have periodic behavior is depicted as x, t → 0. Moreover, we have presented curvy periodic by Fig. 2 of u17 only. From Fig. 2 we can see that the wave waves in which rogue waves occur in both sides of each amplitudes remain unchanged in the deep ocean area (i.e. waves (see Figs. 2 and 3). The researchers can compare when x, t are larger) but when waves reaches to shallow their experimental/numerical results with our exact results water in harbor region even in beach (i.e. when x, t → 0 are like Ref. [1, 2], and civil engineer’s may convey to predict going to smaller values), then wave amplitudes abruptly disaster situations of harbor regions by taking preventive changes and some rogue type waves get into the periodic initiatives with help of our results. The future prospect of waves i.e. peaks and valleys arises into the each waves. the fractional models holds strong. A lot of research lies We also provide the Fig. 3 of solution u18 which has the ahead of the models such as study of conservation law, similar behavior of Fig. 2. The solutions u19, u39, u59 express complexiton solutions, multi-soliton solutions and lie sym- rational polynomial function with similar characteristics metry analysis etc. and we only plotted the fgure of u19 in Fig. 4. From Fig. 4 we can see that the wave amplitudes abruptly change to reduces after reaching the beach of coaster area as x, t → 0. Compliance with ethical standards

Conflict of interest The authors declare that they have no any con- 5 Comparison and discussion fict of interest.

To best of our knowledge, only Yaslan [3] investigated the space-time model by using the exp (−φ (ξ))-expansion Appendix method and obtained fve solutions only (40), (41), (42), (43), (44) (See the Appendix). They got only fve solutions Yaslan [3] derived the exact solutions the space-time FBK including a tan, a tan hyperbolic, a combination of sine model by using exp(−φ(ξ)) expansion method. He derived the following fve solutions: hyperbolic-cosine hyperbolic and two rational function 2 solutions. Moreover, in this paper seventy-two solutions For λ − 4μ > 0, μ ≠ 0,

Vol:.(1234567890) SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8 Research Article

4b u1(x, t) = a0 + , 2 t x (40) − − 2 − 4 tanh  −4 b −a + b + b + C 2 0 √   √ � � � � �� For λ2 − 4μ < 0, μ ≠ 0,

4b u2(x, t) = a0 + , 2 t x (41) − − 4 − 2 tan 4 − b −a + b + b + C 2 0 √   √ � � � � �� For λ2 − 4μ > 0, μ = 0, λ ≠ 0,

2b u3(x, t) = a0 + , t x t x (42) cosh b −a + b + b + C + sinh b −a + b + b + C − 1 0   0           2 For λ − 4μ = 0, μ ≠ 0, λ ≠ 0, 9. Wazwaz AM (2004) The tanh method for travelling wave solu- tions of nonlinear equations. Appl Math Comput 154:713–723 2 t x 2b b −a0 + b + b + C 10. Liu D (2005) Jacobi elliptic function solutions for variant Bouss-   inesq equations. Chaos Soliton Fract 24:1373–1385 u4 = a0 − , (43) t x  2 b −a + b + b + C + 4 11. Akgül A, Kjijcman A, Inc M (2013) Improved (G′/G) Expansion 0   method for the space and time fractional foam drainage and    KdV equations. Abstr Appl Anal 2013:414353 For λ2 − 4μ = 0, μ = 0, λ = 0, 12. Roshid HO, Alam MN, Hoque MF, Akbar MA (2013) A new extended (G′/G)-expansion method to find exact traveling 2b u = a + . wave solutions of nonlinear evolution equations. Math Stat 5 0 t x b −a + b + b + C (44) 1(3):162–166 0   13. Zayed EME, Abdelaziz MAM (2012) The two-variable ((G′/G),  (1/G))-expansion method for solving the nonlinear KdV-mKdV equation. Math Probl Eng 2012:1–14 14. Bekir A (2008) Application of the (G′/G)-expansion method for References nonlinear evolution equations. Phys Lett A 372:3400–3406 15. L-Xiao L, E-Qiang L, M-Liang W (2010) The ((G′/G), (1/G))− expansion method and its application to travelling wave solu- 1. Nistor I, Palermo D, Cornett A, Al-Faesly T (2010) Experimental tions of the Zakharov equations. Appl Math-A J Chinese Univ and numerical modeling of tsunami loading on structures. In: 25(4):454–462 Proceedings of the coastal engineering conference, 1–14. https​ 16. Kudryashov NA (2009) On new travelling wave solutions of the ://doi.org/10.9753/icce.v32.currents.2​ . KdV and KdV-Burgers equations. Commun Nonlinear Sci Numer 2. Zaitsev AI, Kovalev DP, Kurkin AA, Levin BW, Pelinovsky EN, Simul 14:1891–1900 Chernov AG, Yalciner A (2008) The Nevelsk tsunami on August 17. Hossen MB, Roshid HO, Ali MZ (2017) Modifed double sub- 2, 2007: Instrumental data and numerical modeling. Dokl Earth equation method for fnding complexiton solutions to the (2+1) Sci 421(1):867–870. https://doi.org/10.1134/S1028​ 334X0​ 80503​ ​ dimensional nonlinear evolution equations. Int J Appl Comput 46 Math 3(1):679–697 3. Yaslan HC (2018) New analytic solutions of the space-time frac- 18. Roshid HO, Ma WX (2018) Dynamics of mixed lump-solitary tional Broer–Kaup and approximate long water wave equations. waves of an extended (2+1)-dimensional shallow water wave J Ocean Eng Sci 3:295–302 model. Phys Lett A 382(45):3262–3268 4. Hammad MA, Khalil R (2014) Abel’s formula and wronskian for 19. Roshid HO, Khan MH, Wazwaz AM (2020) Lump, multi-lump, conformable fractional diferential equations. Int J Difer Equ cross kinky-lump and manifold periodic-soliton solutions for Appl 13(3):177–183 the (2+1)-D Calogero_Bogoyavlenskii_Schif equation. Heliyon 5. Hirota R (1971) Exact solutions of the KdV equation for multiple 6:e03701 collisions of solutions. Phys Rev Lett 27:1192–1194 20. Guner O, Bekir A (2018) Solving nonlinear space-time fractional 6. Miura MR (1978) Backlund transformation. Springer, Berlin, diferential equations via ansatz method. Comput Methods Dif Germany Equ 6(1):1–11 http://cmde.tabrizu.ac.ir​ 7. Roshid HO (2017) Novel solitary wave solution in shallow water 21. Güner Ö, Bekir A (2015) Exact solutions of some fractional difer- and ion acoustic plasma waves in-terms of two nonlinear mod- ential equations arising in mathematical biology. Int J Biomath els via MSE method. J Ocean Eng Sci 2(3):196–202 8(1):1550003 8. Wazwaz AM (2005) The tanh method: exact solutions of the Sine 22. Tozar A, Kurt A, Tasbozan O (2020) New wave solutions of time Gordon and the Sinh-Gordon equations. Appl Math Comput fractional integrable dispersive wave equation arising in ocean 49:565–574 engineering models. Kuwait J Sci 47(2):22–33

Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2000 | https://doi.org/10.1007/s42452-020-03779-8

23. Kurt A (2020) New analytical and numerical results for fractional 27. Iyiola OS, Tasbozan O, Kurt A, Cenesiz Y (2017) On the analytical Bogoyavlensky-Konopelchenko equation arising in fuid dynam- solutions of the system of conformable time-fractional Robert- ics. Appl Math J Chin Univ 35:101–112 son equations with 1-D difusion. Chaos Soliton Fract 94(1):1–7 24. Kurt A, Senol M, Tasbozan O, Chand M (2019) Two reliable meth- 28. Cenesiz Y, Baleanu D, Kurt A, Tasbozan O (2017) New exact solu- ods for the solution of fractional coupled Burgers’ equation aris- tions of Burgers’ type equations with conformable derivative. ing as a model of poly-dispersive sedimentation. Appl Math Non Waves Random Complex Media 27(1):103–116 Sci 4(2):523–534 29. Gozukizil OF, Akcagil S, Aydemir T (2016) Unifcation of all hyper- 25. Tasbozan O (2019) New analytical solutions for time fractional bolic tangent function methods. Open Phys 14:524–541 Benjamin-Ono equation arising internal waves in deep water. 30. Akcagil S, Aydemir T (2018) A new application of the unifed China Ocean Eng 33(5):593–600 method. NTMSCI 6(1):185–199 26. Tasbozan O, Solen M, Kurt A, Ozkan O (2018) New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water Publisher’s Note Springer Nature remains neutral with regard to waves. Ocean Eng 161(1):62–68 jurisdictional claims in published maps and institutional afliations

Vol:.(1234567890)