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).