ON THE COMPLETE INTEGRABILITY OF AN EQUATION HAVING SOLITONS BUT NOT KNOWN TO HAVE A LAX PAIR

It is usually assumed that a system having N-soliton solutions is completely integrable. Here we have analyzed a set of equations occuring in case of capillary gravity waves. Though the system under discussion has N-soliton solutions, it has yet to be shown that the system is completely integrable. No Lax pair is known for the system. Here we show that the system is not completely integrable in the sense of Ablowitz et al.


I. INTRODUCTI
In recent years there have been tremendous studies for the understanding of the complete integrability of non-linear partial differential equations.Usually equations having N-soliton solutions do possess an Inverse Scattering Transform (IST).But for some equation, it is still not possible to get hold of an IST but one can find N-soliton solution by techniques like those of Hirota.One of the most interesting equations is that of capillary gravity waves initially deduced by KAWAHARA et al [i] and analysed for N-soliton solution by Ma  [2].As far as we know no IST has been found for this equation.So here is an example whose solitary wave solutions have been found but whose complete integrability is still unsettled due to the lack of IST.
In the current literature there has come out two different [3,4] approaches to test the complete integrability of non-linear partial differential equation.Both of these are really variant of the celebrated Painlve test for the ordinary differential equation.
In the approach of Weiss et al, [4] it is required to proceed exactly at every stage of proving the compatibility conditions for the assumed series solution of the non- linear field variable (x,t).The whole procedure becomes quite tricky and cumbersome after certain stages of calculation.On the other hand in the methodology of Ablowitz et al [5,6] it is required to proceed with the leading singularities for the purpose of avoiding moving singularities in the solution manifold; it is only required to deter- mine the position of "resonances" and to obtain the expansion coefficients in arbitrary form.If it can be demonstrated that the expansion coefficients and the wave front of the solution manifold is arbitrary then the system is completely integrable.Here we have carried out an analysis of the above mentioned equations (written below in equa- tion (2.1)) from this point of view have concluded that the system is not completely integrable.

BASIC EQUATIONS.
The non-linear equations under consideration read The second of this set is really the complex conjugate of the first one.
Following the procedure of Ablowitz, et al To determine to cominant behavior, we initially assume E-.Pao, G qb o, n-#Sc o So matching the most singular terms in (2.1) for #(x,t) 0 we get s -2, p + q s 2 -4.We proceed with p -2, q -2, s -2.We also get c o 6, aobo - Now to determine the next to leading order terms, we set, Co-2 + crr-2 in the reduced set of equations and obtain ar(r 2)(r 3) arCo + aoC r br(r 2)(r 3) brco + bocr (2.5) Cr(r 2)(r 3)(r 4) -aobr(r 4) arbo(r 4)   This set of homogeneous equations can have a non-vanishing solution only if the deter- Using equations (2.3), we get the resonance positions at r O, -i, 4, 5, 6 As has been elaborately discussed in the paper by Ablowitz et al., the resonance at r -i corresponds to the arbitrariness of wavefront.
3. DETERMINATION OF COEFFICIENTS AT RESONANCE POSITIONS.
We now proceed to determine the coefficients at the resonance positions.With no loss of generality we assume (x,t) x f(t) and all the co-efficients aj, bj and cj are functions of t only.We then have [(hf where c 3 is given by the expression (3.3).
Though these equations give the coefficients a3, b 3 and c 3 explicitly yet the appearance of the arbitrary function h(t) in each of them, introduces some arbitrary- ness in them.
At the resonance j 4, we get the following matrix equation when substituted from equations (3.7) this leads to another equation for the functions f(t) and h(t), and hence coupled with (3.8) determine f and h.So the arbitrariness in all the coefficients and the wave front are lost.
For the resonance at 6, we get I 0-al I a 6 1 1 3 i a s i4 24 / c 6 2c4f a 3 / That is 6a 6 -aoC6 3ia5i 4 6b 6 -boC6 3ib5 + i 4 2boa6 + 2aob6 + 24c6 2c4-&3 Combining these equations we get another differential equation between h and f and this leads to an inconsistency when compaired with the relation (3.8).So that at the resonance positions the compatibility condition is not satisfied.
CONCLUSION: In the above discussions, we have argued that the system described by equation (2.1) is not completely integrable in the sense of Ablowitz et al. [6], and the system is not known to have inverse scattering transform.So one can arise a serious question: If a system has N-soliton solution, does it have a Lax pair always?Our present notion of n.p.d.e's having soliton solution may be very limited and may have to be extended in the future.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

c o 6 ,(
a obo -36 (3.1) .6Let a o h(t) which is an arbitrary function of t Hence b o h(t)" For, j I we now consider the recurrence relation obtained by linearization with respect to the non-leading terms consequence of fixing the function f(t).So we try to keep nonleading terms in equation(3.4)  which is modified to: arbitrary along with 61 12CLC3.The differential equation connecting h and f, which originated from the non-trivial solution of

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