NEARLY CONCONCENTRIC KORTEWEG-DE VRIES EQUATION AND PERIODIC TRAVELING WAVE SOLUTION

The generalized nearly concentric Korteweg-de Vries equation [u+u/(2/)+Cu-] + 00/rfl 0 is considered. The author converts the equation into the power Kadomtsev-Petviashvili ,equation Jut + unu= +==x] + uw 0. Solitary wave solutions and cnoidal wave solutions are obtained The cnoidal wave solutions are shown to be representable as infinite sums of solitons by using Fourier series expansions and Poisson’s summation formula


INTRODUCTION
The concentric Koneweg-de Vries equation (also called cylindrical Korteweg-de Vries equation), u n + /(2r/) + uu + u 0, (1.1) was first derived by Maxon and Viecelli in 1974 from the study of propagation of radically ingoing acoustic waves in cylindrical geometry [1].In the equation, u=u(,),7=s/2v,t, and -l/9(r/AD + v,t), where is the expansion parameter, AD the Debye length, o, the ion plasma frequency, r the radial distance, and the time.
As the one-dimensional Korteweg-de Vries equation (KdV equation for short) can be extended to the two-dimensional KdV equation, so the concentric KdV equation can be generalized to some higher dimensional equations.Considering the nearly straight wave propagation which varies in a very small angular region, Johnson derived the following nearly concentric KdV equation which is one of the generalized equations from Eq. (1.1) [u n + u/(2) + uu.+ tq(] + /n o, where u u(r/, , 0) and 0 is the angular variable which varies in a small region [2,3].
In this paper, the author considers the power nearly concentric KdV equation of the form [, + /() + , + ] + 0/ 0, (1.3) where n is a positive integer.Eq. (1.3) is converted to the power Kadomtsev-Petviashvili equation (KP equation for short).Chen and Weds method [4] is applied to the power KP equation to obtain ordinary differential equations The solitary wave solutions and cnoidal wave solutions can be expressed as sums of infinite number of solitons by using Fourier series expansions and Poisson's summation formula.The author has also established a criterion for the existence of a single soliton solution, it is C > 0, where C (ore b2)/a (see Section 3).

FORMULATION OF THE PROBLEM
We start from the power nearly concentric KdV equation [un + u/(2r/) + u"u.+ u] + /r/2 0, ( where n is a positive integer.Eq. (2.1) reduces to the usual nearly concentric KdV equation when n 1 Considering that Eq. (2.1) is more analogous to the two-dimensional case and motivated by the results obtained by Chert and Wen [5] and Johnson [2,3,6], the author imroduces the transformations r/= t, x + y/(4t), and 0 y/t.One can argue that since tan 0 y/z and 0 is the variable in a very small angular sector, 0 can be used to approximate y/x.Thus when x and t are large and of the same order, it seems to be reasonable to assume 0 y/t.Therefore, u(, r, 0) u(x, y, t), and -, () (), Hence Eq. (2.1) can be converted to the power KP equation where n is a positive integer [7].
We now look for the real-valued traveling wave solution of the form U(z)= u(z,y,t) with z az + by w-t, where a, b, and w are real constants.Without loss of generality we assume a > 0.
Substitution into Eq.(2.2) yields the fourth order nonlinear ordinary differential equation of U Integrating Eq. (2.3) twice with respect to z yields the second order equation: a (aw-b2)U + U n+ + a4U" Az + Ba (2 4) n+l where A and B are integration constants.

SOLITARY WAVE SOLUTIONS
For the existence of solitary wave solution, we impose the boundary conditions U, U', U", U"' 0 when z 4-oo.These conditions imply A B 0 in Eq. (2.4).Using the fact U" d(U') /(2dU), we can obtain from Eq. (2.4) where C (aw b2)/a2.It can be verified that when C _< 0, solutions to Eq. (3.1) exist only when n is odd and these solutions are unbounded.Therefore, they are not of much physical interest.Eq. (3.1) has nontrivial solitary wave solution when C > 0, and the solitary wave solution is U(z) where z0 is an integration constant Since sech2X 1/cosh2X 4/(e x + e-X)2, the solitary wa,e described by the solution decays exponentially for z 4-oo Furthermore, we note that C > 0 gives a condition under which a nontrivial solitary wave solution exists.This condition indicates a relationship that must be satisfied by the three constants a, b, and w, namely, aw > b2.On the other hand, if aw <_ b2, either no real solution exists or the solutions are unbounded.
For the existence of periodic traveling wave solution, the cubic function F(U) in the right-hand side of Eq. ( 41) plays an important role.It is shown in Reference [4] that a cnoidal wave solution exists only if F(U) has three distinct real simple zeros U1, U2, and U3 such that U3 < U2 < U1 and U2 <_ U(z) <_ UI In this case, we can write Eq. (4.1) as the following by separating variables where z is a value at which U(zl) Ut.The period T in z is given by T 2V/a v/(U, U)(U U2)(U U3) (4 3) Eq (4.2) can also be expressed in terms of elliptic and trigonometric functions as 1 (Z Z)-2 _1(sine, k 2 / v/: u " Vu, v F(, ), where sin -1 v/(UI U)/(U1 U2), k (UI U2)/(U U3), and F(, k) sn-(sin,k) is the normal elliptic integral of the first kind with modulus k [8].Denote F(, k) by v, we then have 1 ,- x/v u (, ,), 2x/a sn(v,k) sine x/(U U)/(Ui U) and hence the cnoidal wave solution is obtained where cn(v, k) cos d dn(v, i k2sin2. It should be noted that for the estence of cnoid wave solution there is no refiion on C wch cm be positive, zero or negative as long as C ] (U + U2 + U).
Using the Fourier series emsion of dn2(v, k) [9] md the Poisson's suation foula [10], we have where K f/ dO/v ksin0 is the complete elliptic integral of the first kind with modulus k" K'= f/dO/v/1 k'sinO is the complete elliptic integral of the first kind with modulus k' v/1 k " and E f/ v/1 ksin0 dO is the complete elliptic integral of the second kind with modulus k.Therefore, the cnoidal wave solution U () in Eq. (4.5) can be expressed as U(z) P + Q E sech2R(z-zl + roT), where In Eq. (4.7), U is clearly a periodic function of z with period T, and each term in the infinite series is a soliton.This gives a representation of a periodic function by an infinite number of solitons.It should be mentioned that cn(v,k) cosy + k2[v-(sinv)(cosv)](sinv)/4 when k << 1 [8].Thus under the limiting case of k -, 0+, e. U1 --, U2, we have u() u2 + (u u)cos2v 1 U_ + (U1 U)(1 + cos2v), when U U.
Under the limiting case k 1-, i.e.U2 Us, since cn(v, k) sech v, we shall obtain U() .u + (ua U) sehv A + B seth C(z-z), when U2 Us, U, v/V, 5. CNOIDAL WAVE SOLUTION WHEN n 2 When n 2 we assume A 0 and obtain Eq.where D is an imegration constant.Suppose that we choose the constants B, C, and D so that the function F(U) in the right-hand side of Eq. ( 5.1) has four distinct simple real zeros Ut > U2 > Us > U4 with U4 U1, Us U2, and U < U < U. From Eq. (5.1) we can derive the cnoidal wave solution