Abstract

A direct
rational exponential scheme is proposed to investigate multi-soliton solutions
of nonlinear evolution equations. To do that the truncated Painleve expansion,
bilinear differential equations of the (1+1)-dimensional Benjamin-Ono equation
are presented.  As a result, the multi-soliton
wave interaction solutions of the equation are explicitly given and fission
phenomenon is found, which are difficult to find by other traditional methods. The
different solutions are established with different dispersive relations between
the wave number and wave speed; a possible condition for fission is proposed. Furthermore,
three-dimensional plots of some obtained wave solutions are given out to show
the properties of the explicit analytic interaction solutions. Two classes of
rational solutions to the Benjamin-Ono equation are constructed via Maple
search which are generated from polynomial solutions to the corresponding
generalized bilinear equation. A conjecture is made that the two presented
classes of rational solutions contain all rational solutions to the considered
equation. From a class of polynomial generating functions, a Maple search tells
us two classes of rational solutions to the considered KdV-like equation, along
with some special interesting solutions.

Keywords: Multi-soliton;
rational solution; bilinear form; Benjamin-Ono equation.

Mathematics Subject Classification: 35K99, 35P05, 35P99.

Introduction

Soliton
equations connect affluent histories of exactly solvable systems constructed in
mathematics, fluid physics, microphysics, cosmology, field theory, etc. To
explain some physical phenomenon further, it becomes more and more important to
seek exact solutions and interactions among solutions of nonlinear wave
solutions. Owing to the wide applications of soliton theory in mathematics, hydrodynamics,
plasma physics, nuclear physics, biology, astrophysics and geophysics, the
study on integrable models has attracted much attention of many researchers. To
find some exact explicit soliton solutions for nonlinear integrable partial differential
equations (NLPDEs) models is one of the most key and significant tasks. There
are many methods exist in the literature such as -expansion method 1, exp-function
2, -expansion method 2, 3
by which single soliton solutions can be found. But the most remarkable
property of exactly integrable equations is the presence of exact solitonic
solutions, and the existence of one-soliton solution is not itself a specific
property of integrable partial differential equations, many non-integrable equations
also possesses simple localized solutions that may be called one-solitonic. Nevertheless,
there are integrable equations only, which posses exact multi-soliton solutions
which describe entirely elastic interactions between individual solitons 4. A variety of efficient methods like the
inverse scattering transformation (IST) 5, 6,
Backlund transformation 7, 8, Hirota’s
direct method 9, mapping and deformation
approach 10 etc. have been well developed
to find exact solutions for integrable models in which interactions between
soliton solutions gained for integrable models are to be completely elastic. Yet
some soliton models gives completely non-elastic soliton interactions when
specific conditions between the wave vectors and velocities are satisfied; at a
specific time, one soliton may fission to two or more solitons (soliton fission
phenomena) or two or more solitons will fusion to one soliton (soliton fusion
phenomena). Actually, for numerous genuine physical models such as in organic
membrane and macromolecule material 11, in
even-clump DNA 12 and in many physical
fields like plasma physics, nuclear physics and so on 13,
people have observed the same phenomena. Wazwaz 14-16
investigated multiple soliton solutions such type of non-elastic phenomena.
Burgers equation and Sharma-Tasso-Olver equation are such types of model in which Wang et al. 17 found non-elastic
soliton fission and fusion phenomena with only two dispersion relations. Neyrame
18 established some periodic and soliton
solutions of Benjamin-Ono equation via basic -expansion method.

In this paper,
we would like to investigate non-elastic fission phenomenon of the Benjamin-Ono
equation. We also would like to discuss their polynomial solutions which
generate rational solutions to scalar nonlinear differential equations by
focusing on the Benjamin-Ono equation. Finally, we propose a specific condition
on parameters for which the fission phenomena will occurs.

2. Multi-Soliton
solution of Benjamin-Ono equation and its fission      

In this section, we bring to bear a direct rational
exponential approach to the Benjamin-Ono equation,

                                                                                                                (1)

where H is the Hilbert transform, we consider it as free from
 and  that is constant. The
BO equation describes internal waves. It is a completely integrable equation
that gives N-soliton solutions.

For the (1+1)-dimensional BO equation (1), there exists a
truncated Painleve expansion

                                                                                                                             (2)

with being functions of  and , the function is the equation of singularity manifold. Inserting Eq. (2)
into Eq. (1) and balancing all the coefficients of different powers of , we get

            and                                                                         (3)

Setting the value of in Eq. (2) and truncating we reach to

or .                                                                                     (4)

Directly substituting Eq. (4) into the Eq. (1), we arrive at the
bilinear form

                                                                                          (5)

For single soliton solution we first consider 

                                                                                  (6)

and the corresponding potential field reads .

Inserting Eq. (6) with Eq. (4) into bilinear form Eq. (5), we
can find the value of  as and thus the solution is

                                                                             (7)

and corresponding potential
function is read as

                            (8)

(a)
 

(b)

 

Fig. 1(a) 3D surface of the single solitary wave solution Eq. (7) of the Benjamin-Ono
equation, (b) potential field Eq. (8) with .

To achieve two soliton solution of Eq. (1), we
just suppose

,                                      (9)

where  and the corresponding
potential field reads

Setting Eq. (9) with Eq. (4) into the bilinear form Eq. (5), we
gain some polynomials which are functions of the variables and  Equating all the
coefficients of different power of exponential to zero, we can obtained a
system of algebraic equations in terms of and Solving the system of algebraic equations with the help of symbolic
computation system Maple-13, we attain some solution sets of the unknown
parameters which are given below.

Set-1:   then

                               (10)

where are arbitrary constants.

The corresponding potential field reads

                                   (11)

 

(a)
 

(b)

 

Fig. 2 (a) 3D surface of two solitary
wave fission solution Eq. (10) of the Benjamin-Ono equation, (b) potential
field Eq. (11) with .

From the Fig. 2, which shows the fission
phenomenon between two solitary waves with the parameters choosing as , we can evidently see that, after interaction one single
solitary waves fission to two solitary wave and decrease wave height with
smaller potential energy. i.e., at a specific time  From careful analysis
of Eq. (10), Eq. (11), it is conclude that for any values of the parameters , only fission occurs. Neither elastic scattering nor fusion
does exist.  

Set-2:   then

               (12)

where , and are arbitrary constants and the corresponding potential field
reads . This solution also has the same phenomena like solution Eq.
(12).

Set-3:   then

                 (13)

where are arbitrary constants.

The corresponding potential field reads .

(a)

(b)

 

Fig. 3 (a) 3D surface of two solitary
wave solution Eq. (13) of the Benjamin-Ono equation, (b) corresponding
potential field v with .

From the Fig.
3 Solution Eq. (13) is completely elastic, before and after collision of
the two salitary waves their shape and size remain same. So, elastic
scattering, no fussion and no fission phenomena does exist for any values of
the parameters. 

To achieve three soliton solution of Eq. (1),
we just suppose

                (14)

where and the corresponding potential field reads .

Setting Eq. (14) with Eq. (4) into the bilinear form Eq. (5),
we gain some polynomials which are functions of the variables and  Equating all the
coefficients of different power of exponential to zero, we can obtained a
system of algebraic equations in terms of  and . Solving the system of algebraic equations with the aid of
symbolic computation system Maple-13, we attain the following constraining solution
sets of the unknown parameters which are given below.

Set-1:   then

     (15)

where are arbitrary constants.

The corresponding potential field reads

(a)

(b)

Fig. 4 (a) Profile of three solitary
wave fusion solution Eq. (15) of the Benjamin-Ono equation, (b) corresponding
potential field  with .

Set-2:

then

 (16)

where ,

and the corresponding potential
field reads .

(a)

(b)

Fig. 5 (a) Profile of three solitary
wave fusion solution Eq. (16) of the Benjamin-Ono equation, (b) corresponding
potential field  with .

Set-3:

then

,  (17)

where ,

and the corresponding potential
field reads .

Set-4:  

then

,   (18)

where ,

and the corresponding potential
field reads .

Set-5:  

then

,               (19)

where and

 and the corresponding
potential field reads .

Set-6:

, then

                  (20)

where are arbitrary constants and the corresponding potential field
reads .

3. Rational
Polynomial function solutions of the Benjamin-Ono equation  

By a Maple computation on

                                                               (21)

we notice that such type of polynomial
solution for the bilinear equation like Eq. (5) does not allow the degree of t
greater than 1.Which is analytically proved by Ma in the article Ref 19. He made a conjecture that this is true,
namely, any polynomial solution  to the bilinear
equation (5) must have the degree of t not greater than 1. In this regard, we
want to consider the trial solution for the rational polynomial is of degree as
follows:

            ,                                                                                                     
(22)

with of degree one and of degree at least three.

According to the bilinear
form Eq. (5), we have to write   

Thus using Eq. (22) into the bilinear
equation like Eq. (5) and solving, we achieve a class of rational polynomial
solutions of the Eq. (1):

            .                                                       (23)

On the other hand, if consider with of degree two and of degree four, then setting withinto the bilinear form Eq. (5) and solving coefficients of
different powers of for , we get

Which are the same results of Eq. (23). Now, we can conclude
that in the rational polynomial solutions does not support as a polynomial of degree more than 3.

Secondly, we considerof degree two and of degree two, then setting with , into the bilinear form Eq. (5) and solving coefficients of
different powers of for , we get   Thus

            .                                                                                 (24)

(a)

     
        (b)

 

Fig. 6 (a) Profile of rational
polynomial solution Eq. (23) of the Benjamin-Ono equation, (b) density plot of
the solution with .

(a)

(b)

Fig. 7 (a) Profile of rational
polynomial solution Eq. (24) of the Benjamin-Ono equation, (b) density plot of
the solution with .

Remark:
All of the solutions available
in this paper have been checked with the help of Maple-13 and we observe that
they satisfy the original Benjamin-Ono equation.

4. Concluding remarks

We successfully used the proposed method in the
Benjamin-Ono equation and found some multi soliton solutions. From the analysis
we observed that when dispersive relation exists between the wave number and
the wave speed with any one or then fission exist in the solution. The obtained solutions
may be significant and important for analyzing the nonlinear phenomena arising
in applied physical sciences.

We considered a generalized bilinear form of the nonlinear
Benjamin-Ono (BO) differential equation, and constructed two classes of rational
solutions to the resulting equation. We remark that it is worth checking if
there exists a kind of Wronskian solutions to the BO equation. We also
conjecture that the two classes of rational solutions in Eq. (23) and Eq. (24)
would contain all rational solutions to the BO equation, generated from
polynomial solutions to the generalized bilinear equation (5) under the link
Eq. (4). 

x

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