Benjamin-Ono equation: Rogue waves, generalized breathers, soliton bending, fission, and fusion
BBenjamin-Ono equation: Rogue waves, generalized breathers,soliton bending, fission, and fusion
Sudhir Singh a , K. Sakkaravarthi b , K. Murugesan a , R. Sakthivel c a Department of Mathematics, National Institute of Technology, Tiruchirappalli – 620015, Tamil Nadu, India b Department of Physics, National Institute of Technology, Tiruchirappalli – 620015, Tamil Nadu, India c Department of Applied Mathematics, Bharathiar University, Coimbatore – 641046, Tamil Nadu, India
Abstract
In this work, we construct various interesting localized wave structures of Benjamin-Onoequation describing the dynamics of deep water waves. Particularly, we extract the rogue waveand generalized breather solutions with the aid of bilinear form and by applying two appropriatetest functions. Our analyses reveal the control mechanism of rogue wave with arbitraryparameters to obtain both bright and dark type first and second order rogue waves. Additionally, ageneralization of homoclinic breather method also known as the three-wave method is used forextracting the generalized bright-dark breathers and solitons. Interestingly, we have observed themanipulation of breathers as well as soliton bending, fission and fusion. Our results are discussedcategorically with the aid of clear graphical demonstrations.
Keywords:
Benjamin-Ono equation; Rogue Waves; Breather; Soliton.
1. Introduction
The study of water waves gain tremendous interest over more than two centuries; theoretical aswell as experimental investigations are performed to understand the dynamics of these waves [1].These wave equations are modeled using prototype ordinary / partial / delay / fractional di ff erentialequations [2, 3]. The nonlinear dynamics of waves associated with such model equations exploreseveral exciting phenomena including wave-mixing / breaking and interaction of several localizedstructures like solitons, breathers, lump and rogue waves, which have several applications in fluiddynamics, plasma, Bose-Einstein condensate, fibre optics and even in finance [3, 4]. These wavemodels also studied in the sense of wick type fractional stochastic systems, where the analysis ofthese models are more general [5]. One of the most agreeable localized structure is soliton. It canbe viewed as a classical solution structure in the integrable models. Also, it is well known that thestability or identity-preserving nature of these solitons even after the collisions enable them to havephenomenal applications in diverse areas of science and technology. Especially, these nonlinearwave properties help in tackling the behaviour of real wave structures, including DNA, plasma, Email addresses: [email protected] (Sudhir Singh), [email protected] (K. Sakkaravarthi), [email protected] (K. Murugesan), [email protected] (R. Sakthivel)
Preprint submitted to Elsevier a r X i v : . [ n li n . S I] A p r ave transmissions in optical fibre and many more. Mathematically, these solitons are nothingbut the solution prototype of integrable nonlinear partial di ff erential equations. Di ff erent types ofanalytical solutions for such models can also be obtained to unearth the dynamics of various otherlocalized structures.One natural question arises here, why other localized solution structures apart from solitons,such as rogue waves and breathers, multi-shock waves and lumps, are required? It can beexplained as, because of several highly unstable practical phenomena such nonlinear wavestructures are necessary to explain them completely. Rogue waves are the comparatively newkind of localized structures and they are also known as “monster waves, killer waves, extremewaves” and “freak waves” [6]. Their behaviour is mysterious and it can also be explained withchaotic phenomenon. A famous saying for rogue waves are ‘coming from nowhere and disappearwith no trace’ [7], because of the reason that the rogue waves are temporally and spatiallylocalized disturbance and amplitude is increasing on the background by a few order of magnitude[7, 8], which can also be considered as a possible explanation for its chaotic behaviour. Further,these rogue waves appear due to several reasons including a universal route of modulationinstability [8]. These waves appear as substantial, large localized structures compared to otherlocalized waves. Their height is approximately two to three times height of the surroundingwaves. In ocean their occurrence damages ships and oil drilling platforms. Although, theappearance of rogue waves are not limited to ocean but in finance [9], plasma [10], superfluid[11], Bose-Einstein condensate [12] and well known optical rogue waves [13]. A rigoroustreatment for rogue waves including modelling and experimental observations are being donecontinuously with di ff erent models [14, 15, 8]. Various rational solution for di ff erent evolutionequations were constructed using di ff erent analytical methods including the famous Inversespectral transform, Darboux transformation, Hirota method, dressing method and several ansatzapproaches. It is clear that the exact solutions of the nonlinear evolution equations helpsignificantly to understand the dynamics of the waves and these solutions with di ff erent physicalstructures have phenomenal applications in a broader range of science and engineering [16, 17].Being motivated by the increasing interest on the rogue waves, we devote our investigation inunderstanding them in the following familiar Benjamin-Ono (BO) equation [18]: u tt + β ( u x ) + β uu xx + γ u xxxx = , (1)where β and γ are arbitrary nonlinearity and dispersion coe ffi cients, respectively. TheBenjamin-Ono equation describes one-dimensional internal waves in deep water [19, 20] and it ismathematically an important nonlinear partial integro-di ff erential equation possessingintegrability and solvable by inverse scattering transform as well as B¨acklund transformation andsingularity structure analysis admitting N -soliton solutions [21, 22, 23, 24] and have infiniteconserved quantities apart from special rational and periodic solutions [25, 26]. Its varioussolutions including nonlocal symmetries [27] and rogue wave solutions [28] are available in theliterature. Our aim of the present work is to explore various physically interesting solutions suchas rogue waves and breathers with a newly developed method and to study their controlmechanism along with bending, fusion and fission of nonlinear waves especially solitons.Now by using a bilinearizing transformation we write the BO equation into a compact bilinearform and then solve it by introducing a polynomial test function of required order. For this purpose,2y making use of the transformation u ( x , t ) = u + γβ ( ln f ) xx , (2)where u is ar arbitrary background while f ( x , t ) is a arbitrary function to be determined, the BOequation (1) is transformed into the following Bilinear form:( D t + u β D x + γ D x ) f · f = . (3)Here D represents the standard Hirota di ff erential operator [28, 29, 30] and it can be defined as D ax D by f · g = (cid:32) ∂∂ x − ∂∂ x (cid:48) (cid:33) a (cid:32) ∂∂ y − ∂∂ y (cid:48) (cid:33) b f ( x , y , t ) · g ( x (cid:48) , y (cid:48) , t (cid:48) ) | ( x , y , t ) = ( x (cid:48) , y (cid:48) , t (cid:48) ) . In recent years, localized structures such as lump, breather and rogue wave arising in variousnonlinear systems / models are highly appreciated and they become interesting both inmathematical and physical perspective due to their occurrence in diversified areas like plasmas,optics, Bose-Einstein condensate and financial systems [6, 7, 8].The present work is organized as follows. In Sec. 2, the first and second order rogue wavesolutions are constructed using the Hirota bilinear form and polynomial test functions [34]. InSec. 3, a generalized breather solutions of di ff erent wave structures are obtained by using thethree-wave method [35] along with their dynamics. Conclusions are provided in the final section.
2. Rogue Waves
In this section, we will construct first and second order rogue wave solutions by using thebilinear form (3) and appropriate polynomial test functions and investigate their evolution.
To extract the rogue wave solution of order-one we choose the form of f ( x , t ) as [34]: f ( x , t ) = k + ( α x + β t ) + ( α x + β t ) . (4)Substituting the above f form (4) into bilinear equation (3) and collecting the coe ffi cients ofdi ff erent powers of { x i y j , i , j = , , } , we get the following set of equations:Coe ffi cient of x : ( − u α β − u α α β − u α β − α + β + α β − α α β β + α β − α β ) = , (5a)Coe ffi cient of t : ( − u α ββ + u α ββ − β − u α α ββ β + u α ββ − u α ββ − β β − β ) = , (5b)Coe ffi cient of tx : ( − u α ββ − u α α ββ − α β − u α α ββ − u α ββ − α β β − α β β − α β ) = , (5c)Constants : (4 k u α β + k u α β + k β + k β + α γ + α α γ + α γ ) = . (5d)3olving the above system of equations (5a)-(5d), we obtain the following relations among theparameters resulting to the first order rogue wave solution as u β > , β = ± (cid:112) u βα , β = ∓ (cid:112) u βα , k = − α + α ) γ u β . (6)From Eqs. (6) and (4), we get the explicit form of f as f ( x , t ) = (cid:16) α x + (cid:112) u β α t (cid:17) + (cid:16) α x − (cid:112) u β α t (cid:17) − α + α ) γ u β . (7)Thus we can obtain the first order rouge wave solution from (2) and (7) as u ( x , t ) = u + u γ ( − u x β + t u β − γ )(2 u x β + t u β − γ ) , (8)under the constraint condition u β >
0. The above solutions carries three arbitrary parameters u , β and γ . It is of natural interest to understand the importance and roles of these arbitraryparameters in defining the dynamics of the rogue wave (8). We can obtain two types of roguewaves namely bright and dark by tuning these parameters as shown in Fig. 1 by retaining thenecessary condition u β >
0. Mainly, we obtain bright single peak doubly localized excitationwith the choice u = − . β = − .
01 and γ = .
05. On the other hand, a dark type first orderrogue wave structure is depicted for the choice u = . β = . γ = − . Figure 1: Bright (left panel) and dark (right panel) type first order rogue waves through solution (8). The bottom panelshows the corresponding contour plots.
For a much clear inference of the arbitrary parameters, we have demonstrated their rolegraphically for bright rogue wave with three di ff erent set of values in Fig. 2. Our analysis showsthat the increase in the magnitude of β decreases the width of the rogue waves, while the γ parameter is altering its width in direct proportion with an appreciable change in their tail andwithour a ff ecting the amplitude. However, the parameter u is much simpler which increases theamplitude of the rogue wave along with a shifts from its constant background. Importantly, thereoccurs appreciable change in the significant wave height (amplitude) of the amplitude. Similar4 ff ects can also be observed in the case of dark rogue wave, where the depth / darkness,background, width and tail of the dark rogue waves are controlled by tuning β , γ and u parameters. Figure 2: Impact of β , γ and u parameters in the first order bright rogue wave (8) for fixed values of other parameters. Next natural step is to construct and look for the dynamical behaviour of higher order roguewaves in the present BO system. Here we construct a simplest higher order rogue wave, that isthe rogue wave of order two. For this purpose, we have adopted the following polynomial testfunction: f ( x , t ) = β + (cid:16) β x + β t (cid:17) + β x + β x t + β t + β x + β t , (9)where β j ( j = , , . . . ,
8) are parameters of second order rogue wave solution, will be determinedlater. Substituting (9) into the bilinear form (3) and equating to zero the coe ffi cients of { x m y n , m , n = , , , , , , , , } , we obtained the following algebraic nonlinear system ofequations:Coe ffi cient of x : − u ββ + β β = , (10a)Coe ffi cient of t x : − u ββ β + β β = , (10b)Coe ffi cient of t x : − u ββ β + β + β = , (10c)Coe ffi cient of t x : 24 u ββ β − β β = , (10d)Coe ffi cient of t x : 36 u ββ β − β β = , (10e)Coe ffi cient of t : 12 u ββ β − β = , (10f)Coe ffi cient of x : − u ββ β + β β β + β β + β γ = , (10g)Coe ffi cient of t x : − u ββ β β + β β β + u ββ β − β β β + β β + β β γ = , (10h)Coe ffi cient of t x : − u ββ β β + β β − u ββ β β − β β β + u ββ β − β β β − β β γ = , (10i)5oe ffi cient of t x : 24 u ββ β − u ββ β β + β β + u ββ β β − β β β + β γ = , (10j)Coe ffi cient of t : 4 u ββ β + u ββ β β − β β + β β γ = , (10k)Coe ffi cient of x : − u ββ + β β + u ββ β + β β β + β β + β β γ = , (10l)Coe ffi cient of t x : − u ββ β − β + β β − u ββ β β + β β β + u ββ β − β β β + β β β γ − β β γ = , (10m)Coe ffi cient of t x : − u β beta + u ββ β − β β − u ββ β β + β β + u ββ β β − β β β − β β β γ − β β β γ + β β γ = , (10n)Coe ffi cient of t : 4 u ββ β − β + u ββ β + u ββ β β + β β + β β γ + β β β γ + β β β γ = , (10o)Coe ffi cient of x : 60 u ββ β + β β β − u ββ β + β β + β β + β γ − β β γ = , (10p)Coe ffi cient of x : 24 u ββ β + β β − u ββ + β β + β β γ − β β γ = , (10q)Coe ffi cient of t x : 72 u ββ β β + β β β − u ββ β + β β + u ββ β − β β − β β γ − β β β γ + β β γ = , (10r)Coe ffi cient of t : 12 u ββ β β + β β + u ββ β + u ββ β − β β + β γ + β β γ + β β β γ + β β β γ = , (10s)Coe ffi cient of t : 4 u ββ β + β β + u ββ β − β + β β β γ + β β γ + β β γ = , (10t)Constants : 4 u ββ β + β β + β β γ + β γ = . (10u)By solving the above system of equations, we get the following set of relations among theparameters: β = − β γ u β ; β = β u β ; β = − β γ u β ; β = − β γ u β ; β = − β γ u β ; β = − β γ u β ; β = β γ u β . (11)From Eqs. (2), (9) and (11), we obtained the two-rogue wave solution of Benjamin-Ono equation(1) as follows: u ( x , t ) = u + γβ (cid:32) F F (cid:33) , (12a)where F = u β (4608 t u x β + t u β + u x βγ + γ + u β γ ( − x + t γ ) + t u x β ( − x + t γ ) − t u β ( − x + t γ ) − u x β γ (7 x + t γ ) + u β γ (3 x + t x γ − t γ ) − u x β ( x + t x γ − t γ ) + t u β ( − x + t x γ + t γ )) , (12b) F = t u x β + t u β − u x βγ − γ + u x β ( x − t γ ) − u β γ ( x − t γ ) + t u β (3 x − t γ ) . (12c)6 igure 3: Bright (left panel) and dark (right panel) type second order rogue waves through solution (12c). The bottompanel shows the corresponding contour plot. The parameter choice for bright rogue wave is u = − . β = − . β = . γ = .
5, while that of dark rogue wave is u = . β = . β = . γ = − . Our categorical analysis on the above second order rogue wave solution reveal that there existtwo types of rogue waves, namely bright and dark rogue waves, as appeared in the case of firstorder rogue wave solutions. For completeness, we have shown such a bright and dark type secondorder rogue wave structures in Fig. 3 and the choices of parameters are given in the caption.Further, the impact of arbitrary parameters β , γ , and u give additional freedom in manipulatingthe amplitude or depth, width, background and tail of the second order rogue waves too whichis shown in Fig. 4 for the second order bright rogue wave. Similarly, such e ff ects can also beobserved in the dark rogue wave which is not given here by considering the length of the article. Figure 4: Manipulation of the second order bright rogue wave (8) by controlling the parameters β , γ and u .
3. Generalized Breathers
As given in the introduction, the second part of this work is to investigate the dynamics ofgeneralized breathers of Benjamin-Ono equation (1), which we carryout in this section. For thispurpose, first we obtain the breather solutions of BO model (1) by adopting a generalized threewave test function approach suggested in Ref. [31] which is also referred as three-wave method.7ere the following form of homoclinic breather (two-wave) ansatz [32] is generalized to obtainedthe homoclinic breathers as well as traveling wave solutions: f ( x , t ) = m e ( p x + k t ) + m cos( q x + a t ) + m e − ( p x + k t ) (13)were m , m , p , k , q , a are arbitrary constants. The above one is a combination of two waves, aperiodic wave cos( q x + a t ) and a soliatry wave cosh( p x + k t ), which gives rise to homoclinicbreathers. For more generalized breather wave solutions, we consider a combination of threewaves as initial test function as given below [31]. f ( x , t ) = b e ( px + kt ) + b cos( qx − at ) + b e − ( px + kt ) + b cosh( mx + ct ) (14)where b , b , b , b , p , k , q , a , m and c are unknown parameters to be determined for generalizedbreathers. Here the choice b = ff erent nonlinear wave structures ranging from bright / dark / gray solitons,breathers, etc. by following the above three-wave interaction method. It is also clear that thenumber of arbitrary parameters in the three-wave method are higher than two-wave homoclinicbreathers.Further, one can also generalize the three-wave method as suggested in Ref. [33] for a KdVtype system f ( x , t ) = b cosh( Ξ ) + b cosh( Ξ ) + b cosh( Ξ ) , (15)where Ξ i = k i ( n i x + r i y + s i t + α i ) , i = , , , and n i , r i , s i , α i are the arbitrary parameters. Thisnewly introduced test function is a combination of homoclinic breather test function and threewave solution test function. Also, in the similar way a generalized N-wave method can also beused to find various solutions of nonlinear wave equations, which is beyond the scope of the currentwork and can be investigated separately to look for other types of possible nonlinear wave entities. Breather Solution using Three-Wave method
Considering the three wave test function (14) and collecting the coe ffi cients of e i ( px + kt ) , cos( qx − at ), sin( qx − at ), cosh( mx + ct ), sinh( mx + ct ) , ( i = − , , (8 b b k + b b p u β + b b p γ ) + ( − a b − b q u β + b q γ ) + (2 b c + b m u β + b m γ ) = , (16a)( − a b b + b b k + b b p u β − b b q u β + b b p γ − b b p q γ + b b q γ ) = , (16b)( − a b b + b b k + b b p u β − b b q u β + b b p γ − b b p q γ + b b q γ ) = , (16c)(2 b b c + b b k + b b m u β + b b p u β ) + (2 b b m γ + b b m b γ + b b b γ ) = , (16d)(2 b b c + b b k + b b m u β + b b u β + b b m γ + b b m p γ + b b p γ ) = , (16e)( − a b b + b b c + b b m u β − b b q u β + b b m γ − b b m q γ + b b q γ ) = , (16f)(4 ab b k − b b pqu β − b b p q γ + b b pq γ ) = , (16g)( − ab b k + b b pqu β + b b p q γ − b b pq γ ) = , (16h) − b b ck − b b mpu β − b b m p γ − b b mp γ ) = , (16i)(4 b b ck + b b mpu β + b b m p γ + b b mp γ ) = , (16j)(4 ab b c − b b mqu β − b b m q γ + b b mq γ ) = . (16k) After solving the above nonlinear algebraic system of equations, di ff erent classes of solutionscan be obtained for the arbitrary parameters which we discuss one by one in the following part. Case 1:
When b = b = b = b (cid:44)
0, Eqs. (16) give a = (cid:112) q γ − q u β ) with b as anarbitrary free parameter. This results into the following form of f : f = b cos (cid:16) qx − (cid:112) q γ − q u β ) t (cid:17) . (17)Thus we get a singular solution of the Benjamin Ono equation (3) as u ( x , t ) = u − γβ (cid:16) q sec ( qx − t (cid:112) q γ − q u β ) (cid:17) . (18)The above solution always result into unbounded singluar form without much advantage orapplications. So, here we do not discuss any further details of this solution (18). Case 2:
When b = b = b , b (cid:44)
0, the explicit form of f is obtained from Eqs. (16) as f = b cos (cid:34)(cid:32) √ u β + m γ √ γ (cid:33) (cid:16) √ x − m √ γ t (cid:17)(cid:35) + (cid:113) − b ( u β + m γ (cid:112) u β + m γ × cosh (cid:34) mx − (cid:114) γ (cid:16) u β + m γ (cid:17) t (cid:35) , (19a)where b = (cid:115) − b ( u β + m γ ) u β + m γ , q = (cid:115) u β + m γ )2 γ , c = − (cid:114) γ ( u β + m γ ), and a = (cid:112) m ( u β + m γ ) while the other parameters ( u , β, γ, b and m ) are arbitrary. From the above f and Eqn. (2), we can obtain the exact solution as below. u ( x , t ) = u + γβ ( ln f ) xx . (19b)This clearly shows that the contribution from both ‘cos’ and ‘cosh’ parts giving rise to the breathingnature of soliton which may be of either bright or anti-dark solitons for appropriate choice. Furtherthe breathing (period of) oscillations can be controlled in addition to the manipulation of breathervelocity (direction of propagation), amplitude / depth and width by tuning the available arbitraryparameters. To be specific, u parameter controls the background energy and amplitude of thebreather while γ influences the amplitude and velocity. However, β parameter varies the widthof the soliton breather in addition to its velocity changes. For illustrative purpose, we have givenbright soliton breather on a constant non-zero energy / amplitude (anti-dark soliton breathers) in9 igure 5: Breathing soliton with periodical oscillation in amplitude and space for (a) m = .
75, (b) m = . m = − .
75. Other arbitrary parameters are fixed as u = γ = . β = .
50, and b = . Fig. 5 traveling with positive, zero and negative velocity having di ff erent amplitudes. Case 3:
For the choice b = b = b , b (cid:44)
0, Eqs. (16) reduces to an explicit form of f as f = b e ( px − √ − p u β − p γ ) t ) + b e − ( px − √ − p u β − p γ t ) , (20)along with a condition k = − (cid:112) − p ( u β + p γ ). Further, by setting b = b >
0, the function f ( x , t ) becomes f ( x , t ) = b cosh (cid:16) px − (cid:112) − p ( u β + p γ t (cid:17) , (21a)Hence the solution of BO equation (1) is obtained as: u ( x , t ) = u + γβ p sech (cid:16) px − t (cid:112) − p ( u β + p γ ) (cid:17) → Bright S oliton . (21b)On the other hand, when b = d , b = − d with d > f ( x , t ) reduces to f ( x , t ) = d sinh (cid:16) px − (cid:112) − p ( u β + p γ t (cid:17) , (22a)Now, the corresponding solution of BO equation (1) can be derived as below. u ( x , t ) = u − γβ p cosech (cid:16) px − t (cid:112) − p ( u β + p γ ) (cid:17) → S ingular S olution . (22b)From the above solutions Eqs. (21b) and (22b) one can understand that they correspond tobright / dark soliton and unbounded singular structures, respectively. Though the singular solutionsare of no further interest, solitons found multifaceted applications due to their stable propagationwith variety of localized profiles of salient features. In the present case, the solution (21b) furtherdivided into two categories namely bright and dark soliton based on the choice of arbitraryparameter β and γ apart form u parameter which controls the background and amplitude / depthof this bright / dark soliton. Particularly when βγ >
0, we obtain a standard bright soliton for10 igure 6: Bright soliton, anti-dark soliton (bright soliton on a constant background u (cid:44)
0) and W-shaped (doublewell) dark soliton of BO equation for the choice u = γ = − . u = γ = .
5, and u = γ = . β = − . b = .
0, and p = − . u = (cid:112) − p ( u β + p γ and amplitude 6 γβ p , while for the choice u (cid:44) u + γβ p and traveling with same velocity. In contrary, for the choice βγ < u (cid:44)
0, Eq.(21b) yields dark soliton solution with di ff erent shapes ranging from a single well or double well(W-shaped) profiles. Also, by tuning these β and γ parameters one can manipulate their widthand amplitude along with the direction of propagation appropriately. For completeness and betterunderstanding, we have shown such bright, anti-dark and dark solitons in Fig. 6. Case 4:
When b = b , b , b (cid:44)
0, we can obtain the explicit form of f from the set ofequations (16) as below. f ( x , t ) = b cos( qx − at ) + b e − ( px + kt ) + b cosh( mx + ct ) , (23a)where the parameters m , p , k , a and q take the following form: m = √ γ (cid:114) − u β − (cid:113) u β − c γ, p = (cid:115) γ (cid:114) − u β + (cid:113) u β − c γ, (23b) k = − c γ (cid:113) u β − c γ ) (cid:114) − (cid:18) u β + (cid:113) u β − c γ (cid:19) (cid:114) − (cid:18) u β − (cid:113) u β − c γ (cid:19) , (23c) a = c γ (cid:18) u β − (cid:113) u β − c γ (cid:19) (cid:115) − (cid:18) u β + (cid:113) u β − c γ (cid:19) , (23d) q = √ γ (cid:114) u β + (cid:113) u β − c γ. (23e)In the above solution (23), u , β , γ , c , b , b and b are arbitrary parameters through which wecan manipulate the resultant nonlinear wave pattern of soliton breather. Compared to the previous11 igure 7: Soliton breathers appearing on a constant background through (23a). The parameter choices are (left panel) u = β = − . γ = − . b = . b = . b = .
5, and c = − .
0; (middle panel) u = β = − . γ = − . b = − . b = − . b = .
5, and c = .
0; and (right panel) u = β = . γ = − . b = . b = . b = .
5, and c = . / increase in the intensity due to increased / decreased widthof the soliton given by Eqn. (23a) for c = − . c = . u = β = − . γ = . b = . b = .
5, and b = . solutions, the above solution reveal interesting patterns of soliton breather explaining variousphenomena. It starts from the breathing of solitons with periodic oscillations in its amplitude andalso solitons with single or double hump / well structure undergoing fusion and fission processesin addition to their bending characteristics. Such type of soliton breather is shown in Fig. 7, whilethe bending, fission and fusion nature of solitons are demonstrated in Figs. 8–10 with appropriatechoice of arbitrary parameters for illustrative purpose. Further, dynamics on these solitonicbending, fission and fusion properties require a separate dedicated investigation which shallunravel di ff erent features. Here also one can control the amplitude, period ofoscillations / breathing, width, and velocity of solitons / breathers by tuning the arbitraryparameters. 12 igure 9: Fission of a W-shaped dark soliton into two single well solitons (for u = β = − . γ = . b = . b = . b = .
5, and c = .
0) and a single well soliton and double well gray solitons (for u = β = − . γ = . b = . b = . b = .
5, and c = − .
0) through Eqn. (23a).Figure 10: Fusion of two single well solitons (for u = β = − . γ = . b = . b = . b = .
5, and c = . u = β = − . γ = . b = . b = . b = .
5, and c = .
0) to form amplified W-shaped dark soliton through Eqn. (23a).
4. Conclusions
We have considered the familiar Benjamin-Ono equation and constructed various localizedwave solutions starting from the rogue waves to breathers and solitons by employing a polynomialtest functions and a recently proposed three-wave method with the aid of bilinear form. Throughthe obtained solutions we are able to control and manipulate the constructed localized waves withthe available arbitrary parameters to realize their multifaceted nature such as tailoring of theiramplitude, width, velocity and tail of both bright and dark type first as well as second order roguewaves, developing bright and dark solitons, exploring their fusion, fission and bending propertieswith clear demonstrations using graphical examples. The reported results will be helpful for acomplete understanding on the dynamics of the considered system and further the analysis can beextended to other related nonlinear models. 13 cknowledgments
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