Printer Friendly

The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three.

1. Introduction

The Johnson equation was introduced in 1980 by Johnson [1] to describe waves surfaces in shallow incompressible fluids [2, 3]. This equation was derived for internal waves in a stratified medium [4]. The Johnson equation is dissipative; it is well known that there is no solution with a linear front localized along straight lines in the (x, y) plane. This Johnson equation is, for example, able to explain the existence of the horseshoe-like solitons and multisoliton solutions quite naturally.

We consider the Johnson equation (J) in the following normalization:

[([u.sub.t] + [6uu.sub.x] + [u.sub.xxx] + u/2t]).sub.x] - 3 [u.sub.yy]/[t.sup.2] = 0 (1)

where as usual subscripts x, y, and t mean partial derivatives.

The first solutions were constructed in 1980 by Johnson [1]. Other types of solutions were found in [5]. A new approach to solve this equation was given in 1986 [6] by giving a link between solutions of the Kadomtsev-Petviashvili (KP) [7] and solutions of the Johnson equation. In 2007, other types of solutions were obtained by using the Darboux transformation [8]. More recently, in 2013, other extensions have been considered as the elliptic case [9].

Here, we consider the famous Kadomtsev-Petviashvili (KPI), which can be written in the following form:

[([4u.sub.t] - [6uu.sub.x] + [u.sub.xxx]).sub.x] - [3u.sub.yy] = 0 (2)

The KPI equation first appeared in 1970 [7] in a paper written by Kadomtsev and Petviashvili. This equation is considered as a model for surface and internal water waves [10] and in nonlinear optics [11].

In the following, we will use the KPI equation to construct solutions to the Johnson equation but in another way different from this used in [6]. Indeed, these last authors consider another representation of KPI equation given by

[([u.sub.t] + [6uu.sub.x] + [u.sub.xxx]).sub.x] - [3u.sub.yy] = 0, (3)

and so the transformations between solutions of (3) and (1) are different from those we use to transform solutions to (2) in solutions to (1).

In fact, to obtain solutions to (1) from solutions to (2), we use the following transformation:

(x; y; t) [??] ([x.sub.1] = -ix - i[y.sup.2]t/12;[y.sub.1] = yt; [t.sub.1] = 4it) (4)

In this paper, we give solutions by means of Fredholm determinants of order 2N depending on 2N - 1 parameters and then by means of Wronskians of order 2N with 2N - 1 parameters. So we construct an infinite hierarchy of solutions to the Johnson equation, depending on 2N - 1 real parameters.

New rational solutions depending a priori on 2N - 2 parameters at order N are constructed, when one parameter tends to 0.

We obtain families depending on 2N - 2 parameters for the Nth order as a ratio of two polynomials of degree 2 N(N + 1) in x, t and of degree 4N(N + 1) in y.

In this paper, we construct only rational solutions of order 3, depending on 4 real parameters; we construct the representations of their modulus in the plane of the coordinates (x, y) according to the four real parameters [a.sub.i] and [b.sub.i] for 1 [less than or equal to] i [less than or equal to] 2 and time t.

2. Solutions to Johnson Equation Expressed by Means of Fredholm Determinants

Some notations are given. We define first real numbers [[lambda].sub.j] such that -1 < [[lambda].sub.v] < 1, v = 1, ..., 2N; they depend on a parameter [epsilon] and can be written as

[[lambda].sub.j] = 1 - 2[[epsilon].sup.2][j.sup.2],

[[lambda]sub.N+j] = -[[lambda].sub.j],

1 [less than or equal to] j [less than or equal to] N. (5)

Then, we define [[kappa].sub.v], [[delta].sub.v], [[gamma].sub.v], and [x.sub.r,v]; they are functions of [[lambda].sub.v], 1 [less than or equal to] v [less than or equal to] 2N, and are defined by the following formulas:

[mathematical expression not reproducible] (6)

[e.sub.v], 1 [less than or equal to] v [less than or equal to] 2N are defined by

[mathematical expression not reproducible] (7)

[[epsilon].sub.v], 1 [less than or equal to] v [less than or equal to] 2N are defined by

[[epsilon].sub.j] = 1,

[[epsilon].sub.N+j] = 0

1 [less than or equal to] j [less than or equal to] N. (8)

As usual I is the unit matrix and [D.sub.r] = [([d.sub.jk]).sub.1[less than or equal to]j, k[less than or equal to]2N] is the matrix defined by the following:

[mathematical expression not reproducible] (9)

Then we get the following theorem.

Theorem 1. The function v defined by

v (x, y, t) = -2 [[absolute value of (n (x, y, t))].sup.2]/d [(x, y, t).sup.2] (10)

with

n (x, y, t) = det (I + [D.sub.3] (x, y, t)), (11)

d (x, y, t) = det (I + [D.sub.1] (x, y, t)), (12)

and [D.sub.r] = [([d.sub.jk]).sub.1[less than or equal to]j,k[less than or equal to]2N] is the matrix

[mathematical expression not reproducible] (13)

is a solution to (1), depending on 2N - 1 parameters [a.sub.k], [b.sub.k], 1 [less than or equal to] k [less than or equal to] N - 1 and [epsilon].

Proof. The solution v to the KPI equation can be written as follows by using [12]:

v (x, y, t) = -2 [[absolute value of (n (x, y, t))].sup.2]/d [(x, y, t).sup.2] (14)

where

n (x, y, t) = det (I + [D.sub.3] (x, y, t)), (15)

d (x, y, t) = det (I + [D.sub.1] (x, y, t)), (16)

and [D.sub.r] = [([d.sub.jk]).sub.1[less than or equal to]j,k[less than or equal to]2N]] is the matrix

[mathematical expression not reproducible] (17)

where [[kappa].sub.v], [[delta].sub.v], [x.sub.r,v], [[gamma].sub.v], [[tau].sub.v], [e.sub.v], and [[epsilon].sub.v] are defined in (6), (5), (7), and (8).

The connection between the solutions to the Johnson equation and these to the KPI equation was already explained in [6] but with another expression of the KPI equation (3).

Here, the knowledge of a solution u to the KPI equation (2) gives a solution to the Johnson equation (1). Let us consider u(x, y, t) a solution of the KPI equation (2), then the function

[??] ([x.sub.1],[y.sub.1], [t.sub.1]) (18)

for

[x.sub.1] = -ix - i[y.sup.2]t/12,

[y.sub.1] = yt,

[t.sub.1] = 4it (19)

is a solution to the KPI equation (2). Using this crucial transformation, the solution to the Johnson equation takes the form

v(x,y,t) = [[absolute value of (det(I + [D.sub.3] (x, y, t)))].sup.2]/ det [(I + [D.sub.1] (x, y, t)).sup.2] (20)

with the matrix [D.sub.r] defined in (17).

So we get the solutions to (14) by means of Fredholm determinants.

3. Solutions to the Johnson Equation by Means of Wronskians

We use the following notations:

[[phi].sub.r,v] = sin [[PHI].sub.r,v], 1 [less than or equal to] v [less than or equal to] N,

[[phi].sub.r,v] = cos [[PHI].sub.r,v], N + 1 [less than or equal to] v [less than or equal to] 2N, r = 1,3, (21)

with

[mathematical expression not reproducible] (22)

[W.sub.r](w) is the Wronskian of the functions [[phi].sub.r,1], ..., [[phi].sub.r,2N] defined by

[mathematical expression not reproducible] (23)

We consider the matrix [mathematical expression not reproducible] defined in (17).

Then we have the following result.

Theorem 2.

det (I + [D.sub.r]) = [k.sub.r] (0) x [W.sub.r] ([[phi].sub.r,1], ..., [[phi].sub.r,2N]) (0), (24)

where

[mathematical expression not reproducible] (25)

Proof. First, we remove the factor [mathematical expression not reproducible] in each row v in the Wronskian [W.sub.r](w) for 1 [less than or equal to] v [less than or equal to] 2N.

Then

[mathematical expression not reproducible] (26)

with

[mathematical expression not reproducible] (27)

The determinant [[??}.sub.r] can be written as

[[??}.sub.r] = det ([[alpha].sub.jk][e.sub.j] + [[beta].sub.jk]), (28)

where [mathematical expression not reproducible], and [[beta].sub.jk] = [([i[gamma].sub.j]).sup.k-1], 1 [less than or equal to] j [less than or equal to] N, 1 [less than or equal to] k [less than or equal to] 2N;

[mathematical expression not reproducible], and [[beta].sub.jk] = [([i[gamma].sub.j]).sup.k-1], N [less than or equal to] 1 [less than or equal to] j [less than or equal to] 2N, 1 [less than or equal to] k [less than or equal to] 2N.

We have to calculate [[??].sub.r]. So, we use the following lemma.

Lemma 3. Let A = [([a.sub.ij]).sub.i,j[member of][1, ..., N]], let B = [([b.sub.ij]).sub.i,j[member of][1, ..., N]], and let [([H.sub.ij]).sub.i,j[member of][1, ..., N]] be the matrix formed by replacing in A the jth row of A by the ith row of B. Then

det ([a.sub.ij][x.sub.i] + [b.sub.ij]) = det ([a.sub.ij]) x det ([[delta].sub.ij][x.sub.i] + det ([H.sub.iij])/det ([a.sub.ij])) (29)

Proof. Let [mathematical expression not reproducible] be the transposed matrix in cofactors of A. Then A x [??] = det A x I.

So det([??]) = [(det(A)).sup.N-1].

Then the general term of the product [mathematical expression not reproducible] can be expressed by

[mathematical expression not reproducible] (30)

We obtain

[mathematical expression not reproducible] (31)

So det([a.sub.ij][x.sub.i] + [b.sub.ij]) = det(A) x det([[delta].sub.ij][x.sub.i] + det([H.sub.ij])/ det(A)).

We use the notations U = [([[alpha].sub.ij]).sub.i,j[member of][1, ..., 2N]] and V = [([[beta].sub.ij]).sub.i,j[member of][1, ..., 2N]].

Using the preceding lemma, we get

[mathematical expression not reproducible] (32)

where [([H.sub.ij]).sub.i,j[member of][1, ..., 2N]] is the matrix obtained by replacing in U the jth row of U by the ith row of V defined previously.

U is the classical Vandermonde determinant that is equal to

[mathematical expression not reproducible] (33)

We have to compute det([H.sub.ij]) to evaluate the determinant [[??].sub.r]. To do that, we study two cases.

(1) For 1 [less than or equal to] j [less than or equal to] N, the matrix [H.sub.ij] is a Vandermonde matrix, where the jth row of U in U is replaced by the ith row of V. Then we have

det ([H.sub.ij]) = [(-1).sup.N(2N+1)+N-1] [(i).sup.N(2N-1)] x M, (34)

with M = M([m.sub.1], ..., [m.sub.2N]) being the determinant defined by [m.sub.k] =[y.sub.k] for k [not equal to] j and [m.sub.j] = -[[gamma].sub.i]. Thus we get

[mathematical expression not reproducible] (35)

To compute [[??].sub.r], we have to simplify the quotient [q.sub.ij] = det([H.sub.ij])/det(U):

[mathematical expression not reproducible] (36)

[q.sub.ij] is equal to [r.sub.ij] defined by -[[PI].sub.l[not equal to]j] ([[gamma].sub.l] + [[gamma].sub.i])/ [[PI].sub.l[not equal to]i] ([[gamma].sub.l] + [[gamma].sub.i]), because det([[delta].sub.ij][x.sub.i] + det([qsub.ij])/ det(A)) = det([[delta].sub.ij][x.sub.i] + det([r.sub.ij])/ det(A)).

Thus [r.sub.ij] can be written as

[mathematical expression not reproducible] (37)

with the notations given in (17).

(2) We can do the same estimations for N + 1 [less than or equal to] j [less than or equal to] 2N are made; det [H.sub.ij] is first as follows:

det ([H.sub.ij]) = [(-1).sup.N(2N+1)+N-1] [(i).sup.N(2N-1)] x M, (38)

with M = M([m.sub.1], ..., [m.sub.2N]) being the determinant defined by [m.sub.k] = [[gamma].sub.k] for k [not equal to] j and [m.sub.j] = -[[gamma].sub.i]. Then we get

[mathematical expression not reproducible] (39)

Then [q.sub.ij] := det([H.sub.ij])/ det(U) can be expressed as

[mathematical expression not reproducible] (40)

[q.sub.ij] is replaced by [r.sub.ij] defined by [[PI].sub.l[not equal to]j] ([[gamma].sub.l] + [[gamma].sub.i])/ [[PI].sub.i[not equal to]j] ([[gamma].sub.l] - [[gamma].sub.i]), for the same reason as previously exposed.

Then [r.sub.ij] can be written as

[mathematical expression not reproducible] (41)

with notations given in (17).

[e.sub.i] is replaced by [mathematical expression not reproducible]. Then det [[??].sub.r] can be rewritten as

[mathematical expression not reproducible] (42)

We compute the two members of the last relation (42) in y = 0. Using (33), we get

[mathematical expression not reproducible] (43)

Thus, the Wronskian [W.sub.r] given by (26) can be rewritten as

[mathematical expression not reproducible] (44)

Then

det (I + [D.sub.r]) = [k.sub.r] (0) [W.sub.r] ([[phi]sub.1], ..., [[phi].sub.2N]) (0). (45)

This finishes the proof of Theorem 2.

Then the solution v to the Johnson equation can be rewritten as

v (x, y, t) = -2 [[absolute value of (det (I + [D.sub.3] (x, y))].sup.2]/[(det (I + [D.sub.1] (x, y, t))).sup.2]. (46)

With (24), the following link between Fredholm determinants and Wronskians is obtained:

det (I + [D.sub.3]) = [k.sub.3] (0) x [W.sub.3] ([[phi].sub.r,1], ..., [[phi].sub.r,2N]) (0) (47)

and

det (I+ [D.sub.1]) = [k.sub.1] (0) x [W.sub.1] ([[phi].sub.r,1], ..., [[phi].sub.r,2N]) (0). (48)

As [[PHI].sub.3,j](0) contains N terms [x.sub.3,j] 1 [less than or equal to] j [less than or equal to] N and N terms - [x.sub.3,j] 1 [less than or equal to] j [less than or equal to] N, we have the relation [k.sub.3](0) = [k.sub.1](0), and we get the following theorem.

Theorem 4. The function v defined by

v (x, y, t) = -2 [[absolute value of ([W.sub.3] ([[phi].sub.3,1], ..., [[phi].sub.3,2N]) (0))].sup.2]/ [([W.sub.1] ([[phi].sub.1,1], ..., [[phi].sub.1,2N]) (0)).sup.2] (49)

is a solution of the Johnson equation which depends on 2N - 1 real parameters [a.sub.k], [b.sub.k], and [epsilon], with [[phi].sup.r.sub.v] defined in (21).

[mathematical expression not reproducible] (50)

where [[kappa].sub.v], [[delta].sub.v], [x.sub.r,v], [[gamma].sub.v], and [e.sub.v] are defined in (6), (5), and (7).

4. Study of the Limit Case When [epsilon] Tends to 0

4.1. Rational Solutions of Order N Depending on 2N - 2 Parameters. An infinite hierarchy of rational solutions to the Johnson equation depending on 2N - 2 parameters is obtained. For this, we take the limit when the parameter e tends to 0.

We get the following statement.

Theorem 5. The function v

[mathematical expression not reproducible] (51)

is a rational solution to the Johnson equation. It is a quotient of two polynomials n(x, y, t) and d(x, y, t) depending on 2N - 2 real parameters [[??].sub.j] and [[??].sub.j], 1 [less than or equal to] j [less than or equal to] N - 1, of degrees 2N(N+ 1) in x, t and 4N(N +1) in y.

4.2. Families of Rational Solutions of Order 3 Depending on 4 Parameters. Here we construct families of rational solutions to the Johnson equation of order 3 explicitly; they depend on 4 parameters.

We only give the expression without parameters and we give it in the appendix because of the length of the solutions.

We construct the patterns of the modulus of the solutions in the plane (x, y) of coordinates in functions of parameters [a.sub.i], b, 1 [less than or equal to] i [less than or equal to] 2, and time t.

The role of the parameters [a.sub.i] and [b.sub.i] for the same integer i is the same one; one will be interested primarily only in parameters [a.sub.i].

The study of these configurations makes it possible to give the following conclusions. The variation of the configuration of the module of the solutions is very fast according to time t. When time t grows from 0 to 0,01, one passes from a rectilinear structure with a height of 98 to a horseshoe structure with a maximum height equal to 4. The role played by the parameters [a.sub.i] and [b.sub.i] is the same for same index i. When variables x, y, and time tend towards infinity, the modulus of the solutions tends towards 2 in accordance with the structure of the polynomials which will be studied in a forthcoming article.

5. Conclusion

We have constructed solutions to the Johnson equation, starting from the solutions of the KPI equation, what makes it possible to obtain rational solutions. These solutions are expressed by means of quotients of two polynomials of degree 2N(N + 1) in x, t and 4N(N +1) in y depending on 2N - 2 parameters.

Here we have given a new method to construct solutions to the Johnson equation related to previous results [12-14].

We have given two types of representations of the solutions to the Johnson equation. An expression by means of Fredholm determinants of order 2N depending on 2N - 1 real parameters is given. Another expression by means of Wronskians of order 2N depending on 2N-1 real parameters is also constructed. Also rational solutions to the Johnson equation depending on 2N - 2 real parameters are obtained when one of parameters (e) tends to zero.

The patterns of the modulus of the solutions in the plane (x, y) and their evolution according to time and parameters have been studied in Figures 1, 2, 3, 4, and 5.

In another study, we will give a more general representation of rational solutions to the Johnson equation. It can be written without limit at order N depending on 2N - 2 real parameters. We will prove that these solutions can be written as a quotient of polynomials of degree 2N(N + 1) in x, t and 4N(N+ 1) in y.

Appendix

The solutions to the Johnson equation can be written as

[mathematical expression not reproducible] (A.1)

with

[mathematical expression not reproducible] (A.2)

[a.sub.12] = 8916100448256,

[a.sub.11] = 8916100448256[ty.sup.2] + 2567836929097728t,

[a.sub.10] = 4086546038784[t.sup.2][y.sup.4] + 2139864107581440[t.sup.2][y.sup.2] + 338954474640900096[t.sup.2] + 40122452017152,

[a.sub.9] = 1135151677440[t.sup.3][y.sup.6] + 802449040343040[t.sup.3][y.sup.4] + 231105323618795520[t.sup.3][y.sup.2] + 27116357971272007680[t.sup.3] + 33435376680960[ty.sup.2] + 13909116699279360t,

[a.sub.8] = 212840939520[t.sup.4][y.sup.8] + 178322008965120[t.sup.4][y.sup.6] + 69545583496396800[t.sup.4][y.sup.4] + 14790740711602913280[t.sup.4][y.sup.2] + 12538266255360[t.sup.2][y.sup.4] + 1464283330448688414720[t.sup.4] + 6419592322744320[t.sup.2][y.sup.2] + 1964395250759761920[t.sup.2] - 125382662553600,

[a.sub.7] = 28378791936[t.sup.5][y.sup.10] + 26005292974080[t.sup.5][y.sup.8] + 12125896609628160[t.sup.3][y.sup.6] + 3492258223572910080[t.sup.5][y.sup.4] + 2786281390080[t.sup.3][y.sup.6] + 621211109887322357760[t.sup.5][y.sup.2] + 802449040343040[t.sup.3][y.sup.4] + 56228479889229635125248[t.sup.5] + 539245755110522880[t.sup.3][y.sup.2] + I55302777471830589440[t.sup.3] - 83588441702400[ty.sup.2] - 24073471210291200t,

[a.sub.6] = 2759049216[t.sup.6][y.sup.12] + 2600529297408[t.sup.6][y.sup.10] + 1352275234652160[t.sup.6][y.sup.8] + 467916951524474880[t.sup.6][y.sup.6] + 406332702720[t.sup.4][y.sup.8] + 112163117062988759040[t.sup.6][y.sup.4] - 104021171896320[t.sup.4][y.sup.6] + 17890879964754883903488[t.sup.6][y.sup.2] - 18188844914442240[t.sup.4][y.sup.4] + 1574397436898429783506944[t.sup.6] + 25883796245305098240[t.sup.4][y.sup.2] - 24379962163200[t.sup.2][y.sup.4] + 7765138873591529472000[t.sup.4] - 18456327927889920[t.sup.2][y.sup.2] - 1714031150172733440[t.sup.2] + 376147987660800,

[a.sub.5] = 197074944[t.sup.7][y.sup.1]4 + 180592312320[t.sup.7][y.sup.12] + 100553799499776[t.sup.7][y.sup.10] + 39230841972326400[t.sup.7][y.sup.8] + 40633270272[t.sup.5][y.sup.10] + 11298482488030003200[t.sup.7][y.sup.6] - 45509262704640[t.sup.5][y.sup.8] + 2402016291564313116672[t.sup.7][y.sup.4] - 240734712l0291200[t.sup.5][y.sup.6] + 357817599295097678069760[t.sup.7][y.sup.2] - 3030047576335319040[t.sup.5][y.sup.4] - 4063327027200[t.sup.3][y.sup.6] + 32387604416196269832142848[t.sup.7] + 776513887359152947200[t.sup.5][y.sup.2] - 5717449412424160[t.sup.3][y.sup.4] + 259417759488945816600576[t.sup.5] - 722204136308736000[t.sup.3][y.sup.2] - 52692013785085378560[t.sup.3] + 188073993830400[ty.sup.2] + 78238781433446400t,

[a.sub.4] = 10264320[t.sup.8][y.sup.16] + 8599633920[t.sup.8][y.sup.14] + 5056584744960[t.sup.8][y.sup.12] + 2139864107581440[t.sup.8][y.sup.10] + 2821754880[t.sup.6][y.sup.12] + 697595699071549440[t.sup.8][y.sup.8] - 6501323243520[t.sup.6][y.sup.10] + 177488888539234959360[t.sup.8][y.sup.6] - 3845068318310400[t.sup.6][y.sup.8] + 34787822153690052034560[t.sup.8][y.sup.4] - 1044253684499742720[t.sup.6][y.sup.6] - 423263232000[t.sup.4][y.sup.8] + 4907212790332768156385280[t.sup.8][y.sup.2] - 109389853179563212800[t.sup.6][y.sup.4] - 947335672627200[t.sup.4][y.sup.6] + 485814066242944047482142720[t.sup.8] + 14909066637295736586240[t.sup.6][y.sup.2] - 102312252643737600[t.sup.4][y.sup.4] + 5903990388369111688151040[t.sup.6] + 13866319417127731200[t.sup.4][y.sup.2] + 39182082048000[t.sup.2][y.sup.4] - 249593749508299161600[t.sup.4] + 34104084214579200[t.sup.2][y.sup.2] + 9605315012906188800[t.sup.2] + 493694233804800,

[a.sub.3] = 380160[t.sup.9][y.sup.18] + 268738560[t.sup.9][y.sup.16] + 171992678400[t.sup.9][y.sup.14] + 75951966781440[t.sup.9][y.sup.12] + 134369280[t.sup.7][y.sup.14] + 27104945362698240[t.sup.9][y.sup.10] - 511678218240[t.sup.7][y.sup.12] + 7806224264457093120[t.sup.9][y.sup.8] - 288534917283840[t.sup.7][y.sup.10] + 1814330860623290695680[t.sup.9][y.sup.6] - 100216969038397440[t.sup.7][y.sup.8] - 28217548800[t.sup.5][y.sup.10] + 340778665995331121971200[t.sup.9][y.sup.4] - 20508457607060520960[t.sup.7][y.sup.6] - 91715095756800[t.sup.5][y.sup.8] + 44164915112994913407467520[t.sup.9][y.sup.2] - 1804470366815555420160[t.sup.7][y.sup.4] - 3789342690508800[t.sup.5][y.sup.6] + 5182016706591403173142855680[t.sup.9] + 178908799647548839034880[t.sup.7][y.sup.2] + 8152882249885286400[t.sup.5][y.sup.4] + 4353564672000[t.sup.3][y.sup.6] + 90783436621156210893127680[t.sup.7] + 1358899302878517657600[t.sup.5][y.sup.2] + 5934779360870400[t.sup.3][y.sup.4] + 29285666608973768294400[t.sup.5] + 1420334801407180800[t.sup.3][y.sup.2] + 526149786772124467200[t.sup.3] + 164564744601600[ty.sup.2] + 119615060076134400t,

[a.sub.2] = 9504[t.sup.10][y.sup.20] + 4976640[t.sup.10][y.sup.18] + 3941498880[t.sup.10][y.sup.16] + 1651129712640[t.sup.10][y.sup.14] + 4199040[t.sup.8][y.sup.16] + 653847366205440[t.sup.10][y.sup.12] - 23648993280[t.sup.8][y.sup.14] + 205426954327818240[t.sup.10][y.sup.10] - 10990332149760[t.sup.8][y.sup.12] + 54232715942544015360[t.sup.10][y.sup.8] - 4695812902748160[t.sup.8][y.sup.10] - 1175731200[t.sup.6][y.sup.12] + 11359288866511037399040[t.sup.10][y.sup.6] - 1230421861859328000[t.sup.8][y.sup.8] - 5224277606400[t.sup.6][y.sup.10] + 2249139195569185405009920[t.sup.10][y.sup.4] - 212000616866308423680[t.sup.8][y.sup.6] + 376147987660800[t.sup.6][y.sup.8] + 235546213935972871506493440[t.sup.10][y.sup.2] - 13134177751903386992640[t.sup.8][y.sup.4] + 1033554363961835520[t.sup.6][y.sup.6] + 272097792000[t.sup.4][y.sup.8] + 37310520287458102846628560896[t.sup.10] + 1226803197583192039096320[t.sup.8][y.sup.2] + 246897520732746547200[t.sup.6][y.sup.4] + 515462057164800[t.sup.4][y.sup.6] + 905380759816395724853084160[t.sup.8] + 27954499944929506099200[t.sup.6][y.sup.2] + 141431643360460800[t.sup.4][y.sup.4] + 862595998300681902489600[t.sup.6] + 44295187026935808000[t.sup.4][y.sup.2] + 20570593075200[t.sup.2][y.sup.4] + 13630591987036559769600[t.sup.4] - 10155995666841600[t.sup.2][y.sup.2] + 8350485326069760000[t.sup.2] - 246847116902400,

[a.sub.1] = 144[t.sup.11][y.sup.22] + 41472[t.sup.11][y.sup.20] + 59719680[t.sup.11][y.sup.18] + 17199267840[t.sup.11][y.sup.16] + 77760[t.sup.9][y.sup.18] + 9906778275840[t.sup.11][y.sup.14] - 604661760[t.sup.9][y.sup.16] + 2853152143441920[t.sup.11][y.sup.12] - 180592312320[t.sup.9][y.sup.14] + 821707817311272960[t.sup.11][y.sup.10] - 121358033879040[t.sup.9][y.sup.12] - 27993600[t.sup.7][y.sup.14] + 236651851385646612480[t.sup.11][y.sup.8] - 31028029559930880[t.sup.9][y.sup.10] - 162855567360[t.sup.7][y.sup.12] + 34077866599533112197120[t.sup.11][y.sup.6] - 6881802969981911040[t.sup.9][y.sup.8] + 37614798766080[t.sup.7][y.sup.10] + 9814425580665536312770560[t.sup.11][y.sup.4] - 1242422219774644715520[t.sup.9][y.sup.6] + 47210751873515520[t.sup.7][y.sup.8] + 9069926400[t.sup.5][y.sup.10] + 565310913446334891615584256[t.sup.11][y.sup.2] - 17038933299766556098560[t.sup.9][y.sup.4] + 28592864205503201280[t.sup.7][y.sup.6] + 22348298649600[t.sup.5][y.sup.8] + 162809543072544448785288265728[t.sup.11] + 3680409592749576117288960[t.sup.9][y.sup.2] + 425233795458583756800[t.sup.7][y.sup.4] + 16216157690265600[t.sup.5][y.sup.6] + 5299789813559389608896102400[t.sup.9] + 188493199628667526840320[t.sup.7][y.sup.2] + 3450530873475072000[t.sup.5][y.sup.4] + 1142810726400[t.sup.3][y.sup.6] + 10121126380061334322544640[t.sup.7] + 654028065841191321600[t.sup.5][y.sup.2] - 4184646362726400[t.sup.3][y.sup.4] + 164399083009466381107200[t.sup.5] - 675561785838796800[t.sup.3][y.sup.2] + 332358343529280307200[t.sup.3] - 41141186150400[ty.sup.2] - 59243308056576000t,

[a.sub.0] = [t.sup.12][y.sup.24] + 497664[t.sup.12][y.sup.20] + 648[t.sup.10][y.sup.20] + 103195607040[t.sup.12][y.sup.16] - 6635520[t.sup.10][y.sup.18] - 447897600[t.sup.10][y.sup.16] + 11412608573767680[t.sup.12][y.sup.12] - 1651129712640[t.sup.10][y.sup.14] - 291600[t.sup.8][y.sup.16] - 113927950172160[t.sup.10][y.sup.12] - 2149908480[t.sup.8][y.sup.14] + 709955554156939837440[t.sup.12][y.sup.8] - 136951302885212160[t.sup.10][y.sup.10] + 935210188800[t.sup.8][y.sup.12] - 6162808629834547200[t.sup.10][y.sup.8] + 594406696550400[t.sup.8][y.sup.10] + 125971200[t.sup.6][y.sup.12] + 23554621393597287150649344[t.sup.12][y.sup.4] - 3786429622170345799680[t.sup.10][y.sup.6] + 304663152316907520[t.sup.8][y.sup.8] + 386983526400[t.sup.6][y.sup.10] + 153350399697899004887040[t.sup.10][y.sup.4] + 274450410981965168640[t.sup.8][y.sup.6] + 727916013158400[t.sup.6][y.sup.8] + 325619086145088897570576531456[t.sup.12] - 26223983281671965245440[t.sup.8][y.sup.4] - 47790298402652160[t.sup.6][y.sup.6] + 23808556800[t.sup.4][y.sup.8] + 13850117379435204844581814272[t.sup.10] + 175158576526078771200[t.sup.6][y.sup.4] - 208971104256000[t.sup.4][y.sup.6] + 45085017511182307436789760[t.sup.8] + 68145477097881600[t.sup.4][y.sup.4] + 732141665224344207360000[t.sup.6] - 1714216089600[t.sup.2][y.sup.4] + 5839742646192439296000[t.sup.4] - 629671731344179200[t.sup.2] - 30855889612800 (A.3)

[b.sub.12] = 0,

[b.sub.11] = 0,

[b.sub.10] = 213986410758144ty,

[b.sub.9] = 178322008965120[t.sup.2][y.sup.3] + 51356738581954560[t.sup.2]y,

[b.sub.8] = 66870753361920[t.sup.3][y.sup.5] + 34237825721303040[t.sup.3][y.sup.3] + 5546527766851092480[t.sup.3]y + 802449040343040ty,

[b.sub.7] = 14860167413760[t.sup.4][y.sup.7] + 9986032502046720[t.sup.4][y.sup.5] + 2875977360589455360[t.sup.4][y.sup.3] + 354977777078469918720[t.sup.4]y + 534966026895360[t.sup.2][y.sup.3] + 154070215745863680[t.sup.2]y,

[b.sub.6] = 2167107747840[t.sup.5][y.sup.9] + 1664338750341120[t.sup.5][y.sup.7] + 633399775844106240[t.sup.5][y.sup.5] + 138046913308293857280[t.sup.5][y.sup.3] + 156031757844480[t.sup.3][y.sup.5] + 14909066637295736586240[t.sup.5]y + 59916195012280320[t.sup.3][y.sup.3] + 12941898122652549120[t.sup.3]y - 2006122600857600ty,

[b.sub.5] = 216710774784[t.sup.6][y.sup.11] + 173368619827200[t.sup.6][y.sup.9] + 77035107872931840[t.sup.6][y.sup.7] + 22186111067404369920[t.sup.6][y.sup.5] + 26005292974080[t.sup.4][y.sup.7] + 4141407399248815718400[t.sup.6][y.sup.3] + 7489524376535040[t.sup.4][y.sup.5] + 429381119154117213683712[t.sup.6]y + 2978690837753364480[t.sup.4][y.sup.3] + 621211109887322357760[t.sup.4]y - 1003061300428800[t.sup.2][y.sup.3] - 442951870269358080[t.sup.2]y,

[b.sub.4] = 15049359360[t.sup.7][y.sup.13] + 11557907988480[t.sup.7][y.sup.11] + 5646863617228800[t.sup.7][y.sup.9] + 1917318240392970240[t.sup.7][y.sup.7] + 2708884684800[t.sup.5][y.sup.9] + 468373455867425587200[t.sup.7][y.sup.5] + 79515022065577261793280[t.sup.7][y.sup.3] + 72755379657768960[t.sup.5][y.sup.5] + 8587622383082344273674240[t.sup.7]y + 98604938077352755200[t.sup.5][y.sup.3] - 208971104256000[t.sup.3][y.sup.5] + 18636333296619670732800[t.sup.5]y - 160489808068608000[t.sup.3][y.sup.3] - 17332899271409664000[t.sup.3]y + 4513775851929600ty

[b.sub.3] = 716636160[t.sup.8][y.sup.15] + 481579499520[t.sup.8][y.sup.13] + 257576235171840[t.sup.8][y.sup.11] + 97007172877025280[t.sup.8][y.sup.9] + 180592312320[t.sup.6][y.sup.11] + 27938065788583280640[t.sup.8][y.sup.7] - 86684309913600[t.sup.6][y.sup.9] + 6152948136026811924480[t.sup.8][y.sup.5] - 32811249649582080[t.sup.6][y.sup.7] + 954180264786927141519360[t.sup.8][y.sup.3] + 410853908655636480[t.sup.6][y.sup.5] - 23219011584000[t.sup.4][y.sup.7] + 117773106967986435753246720[t.sup.8]y + 2662333328088524390400[t.sup.6][y.sup.3] - 227360561430528sss00[t.sup.4][y.sup.5] + 357817599295097678069760[t.sup.6]y - 7575118940838297600[t.sup.4][y.sup.3] + 332791666011065548800[t.sup.4]y + 1504591950643200[t.sup.2][y.sup.3] + 818498021149900800[t.sup.2]y

[b.sub.2] = 22394880[t.sup.9][y.sup.17] + 11466178560[t.sup.9][y.sup.15] + 7430083706880[t.sup.9][y.sup.13] + 2853152143441920[t.sup.9][y.sup.11] + 7524679680[t.sup.7][y.sup.13] + 924421294475182080[t.sup.9][y.sup.9] - 8668430991360[t.sup.7][y.sup.11] + 236651851385646612480[t.sup.9][y.sup.7] - 3358397835509760[t.sup.7][y.sup.9] + 51116799899299668295680[t.sup.9][y.sup.5] - 616280862983454720[t.sup.7][y.sup.7] - 1451188224000[t.sup.5][y.sup.9] + 6542950387110357541847040[t.sup.9][y.sup.3] + 36976851779007283200[t.sup.7][y.sup.5] - 1560317578444800[t.sup.5][y.sup.7] + 1059957962711877921779220480[t.sup.9]y + 53956622115927427645440[t.sup.7][y.sup.3] - 805658836504412160[t.sup.5][y.sup.5] + 4293811191541172136837120[t.sup.7]y - 80116512187849113600[t.sup.5][y.sup.3] + 188073993830400[t.sup.3][y.sup.5] + 32613583269084423782400[t.sup.5]y + 120367356051456000[t.sup.3][y.sup.3] + 34088035233772339200[t.sup.3]y + 3949553870438400ty,

[b.sub.1] = 414720[t.sup.10][y.sup.19] + 119439360[t.sup.10][y.sup.17] + 137594142720[t.sup.10][y.sup.15] + 39627113103360[t.sup.10][y.sup.13] + 179159040[t.sup.8][y.sup.15] + 17118912860651520[t.sup.10][y.sup.11] - 361184624640[t.sup.8][y.sup.13] + 4930246903867637760[t.sup.10][y.sup.9] - 113927950172160[t.sup.8][y.sup.11] + 946607405542586449920[t.sup.10][y.sup.7] - 32811249649582080[t.sup.8][y.sup.9] - 48372940800[t.sup.6][y.sup.11] + 272622932796264897576960[t.sup.10][y.sup.5] - 2875977360589455360[t.sup.8][y.sup.7] - 51081825484800[t.sup.6][y.sup.9] + 19628851161331072625541120[t.sup.10][y.sup.3] + 118325925692823306240[t.sup.8][y.sup.5] - 16583946833756160[t.sup.6][y.sup.7] + 5653109134463348916155842560[t.sup.10]y + 647479465391129131745280[t.sup.8][y.sup.3] - 19977771308380323840[t.sup.6][y.sup.5] + 10448555212800[t.sup.4][y.sup.7] + 29443276741996608938311680[t.sup.8]y + 2033726847845400576000[t.sup.6][y.sup.3] + 3009183901286400[t.sup.4][y.sup.5] + 670907998678308146380800[t.sup.6]y - 674057193888153600[t.sup.4][y.sup.3] + 1063084488646459392000[t.sup.4]y + 658258978406400[t.sup.2][y.sup.3] - 243743896004198400[t.sup.2]y,

[b.sub.0] = 3456[t.sup.11][y.sup.21] + 1433272320[t.sup.11][y.sup.17] + 1866240[t.sup.9][y.sup.17] + 237762678620160[t.sup.11][y.sup.13] - 5733089280[t.sup.9][y.sup.15] - 1031956070400[t.sup.9][y.sup.13] + 19720987615470551040[t.sup.11][y.sup.9] - 475525357240320[t.sup.9][y.sup.11] - 671846400[t.sup.7][y.sup.13] - 196867497897492480[t.sup.9][y.sup.9] - 619173642240[t.sup.7][y.sup.11] + 817868798388794692730880[t.sup.11][y.sup.5] + 39441975230941102080[t.sup.9][y.sup.7] + 664992491765760[t.sup.7][y.sup.9] - 7099555541569398374400[t.sup.9][y.sup.5] - 171189128606515200[t.sup.7][y.sup.7] + 217678233600[t.sup.5][y.sup.9] + 13567461922712037398774022144[t.sup.11]y + 3271475193555178770923520[t.sup.9][y.sup.3] - 122331751302215761920[t.sup.7][y.sup.5] - 111451255603200[t.sup.5][y.sup.7] + 88329830225989826814935040[t.sup.9]y + 35497777707846991872000[t.sup.7][y.sup.3] + 222278384175022080[t.sup.5][y.sup.5] + 4523836791088020644167680[t.sup.7]y - 77035107872931840000[t.sup.5][y.sup.3] + 27427457433600[t.sup.3][y.sup.5] + 15696673580188591718400[t.sup.5]y - 12036735605145600[t.sup.3][y.sup.3] - 16213482860131123200[t.sup.3]y - 987388467609600ty (A.4)

[c.sub.12] = 8916100448256,

[c.sub.11] = 8916100448256[ty.sup.2] + 2567836929097728t,

[c.sub.10] = 4086546038784[t.sup.2][y.sup.4] + 2139864107581440[t.sup.2][y.sup.2] + 338954474640900096[t.sup.2] - 13374150672384,

[c.sub.9] = 1135151677440[t.sup.3][y.sup.6] + 802449040343040[t.sup.3][y.sup.4] + 231105323618795520[t.sup.3][y.sup.2] + 27116357971272007680[t.sup.3] - 11145125560320[ty.sup.2] + 1069932053790720t,

[c.sub.8] = 212840939520[t.sup.4][y.sup.8] + 178322008965120[t.sup.4][y.sup.6] + 69545583496396800[t.sup.4][y.sup.4] + 14790740711602913280[t.sup.4][y.sup.2] - 4179422085120[t.sup.2][y.sup.4] + 1464283330448688414720[t.sup.4] + 577763309046988800[t.sup.2] + 75229597532160,

[c.sub.7] = 28378791936[t.sup.5][y.sup.10] + 26005292974080[t.sup.5][y.sup.8] + 12125896609628160[t.sup.5][y.sup.6] + 3492258223572910080[t.sup.5][y.sup.4] - 928760463360[t.sup.3][y.sup.6] + 621211109887322357760[t.sup.5][y.sup.2] - 267483013447680[t.sup.3][y.sup.4] + 56228479889229635125248[t.sup.5] + 231105323618795520[t.sup.3][y.sup.2] + 66558333202213109760[t.sup.3] + 50153065021440[ty.sup.2] + 14444082726174720t,

[c.sub.6] = 2759049216[t.sup.6][y.sup.12] + 2600529297408[t.sup.6][y.sup.10] + 1352275234652160[t.sup.6][y.sup.8] + 467916951524474880[t.sup.6][y.sup.6] - 135444234240[t.sup.4][y.sup.8] + 112163117062988759040[t.sup.6][y.sup.4] - 104021171896320[t.sup.4][y.sup.6] + 17890879964754883903488[t.sup.6][y.sup.2] + 28888165452349440[t.sup.4][y.sup.4] + 1574397436898429783506944[t.sup.6] + 25883796245305098240[t.sup.4][y.sup.2] + 14627977297920[t.sup.2][y.sup.4] + 4037872214267595325440[t.sup.4] + 7222041363087360[t.sup.2][y.sup.2] + 1521443380490403840[t.sup.2] - 325994922639360

[c.sub.5] = 197074944[t.sup.7][y.sup.14] + 180592312320[t.sup.7][y.sup.12] + 100553799499776[t.sup.7][y.sup.10] + 39230841972326400[t.sup.7][y.sup.8] - 13544423424[t.sup.5][y.sup.10] + 11298482488030003200[t.sup.7][y.sup.6] - 19503969730560[t.sup.5][y.sup.8] + 2402016291564313116672[t.sup.7][y.sup.4] - 534966026895360[t.sup.5][y.sup.6] + 357817599295097678069760[t.sup.7][y.sup.2] + 3749041916482682880[t.sup.5][y.sup.4] + 2437996216320[t.sup.3][y.sup.6] + 32387604416196269832142848[t.sup.7] + 1397724997246475304960[t.sup.5][y.sup.2] + 1504591950643200[t.sup.3][y.sup.4] + 152072479700416513179648[t.sup.5] + 741462913276968960[t.sup.3][y.sup.2] + 102610763686745210880[t.sup.3] - 162997461319680[ty.sup.2] - 61387351586242560t,

[c.sub.4] = 10264320[t.sup.8][y.sup.16] + 8599633920[t.sup.8][y.sup.14] + 5056584744960[t.sup.8][y.sup.12] + 2139864107581440[t.sup.8][y.sup.10] - 940584960[t.sup.6][y.sup.12] + 697595699071549440[t.sup.8][y.sup.8] - 2167107747840[t.sup.6][y.sup.10] + 177488888539234959360[t.sup.8][y.sup.6] - 501530650214400[t.sup.6][y.sup.8] + 34787822153690052034560[t.sup.8][y.sup.4] + 222545867188469760[t.sup.6][y.sup.6] + 253957939200[t.sup.4][y.sup.8] + 4907212790332768156385280[t.sup.8][y.sup.2] + 167936535162991411200[t.sup.6][y.sup.4] + 167176883404800[t.sup.4][y.sup.6] + 485814066242944047482142720[t.sup.8] + 44727199911887209758720[t.sup.6][y.sup.2] + 138422459459174400[t.sup.4][y.sup.4] + 3757084792598525619732480[t.sup.6] + 32354745306631372800[t.sup.4][y.sup.2] - 33957804441600[t.sup.2][y.sup.4] + 4409489574646618521600[t.sup.4] - 18055103407718400[t.sup.2][y.sup.2] - 5705412676839014400[t.sup.2] + 117546246144000,

[c.sub.3] = 380160[t.sup.9][y.sup.18] + 268738560[t.sup.9][y.sup.16] + 171992678400[t.sup.9][y.sup.14] + 75951966781440[t.sup.9][y.sup.12] - 44789760[t.sup.7][y.sup.14] + 27104945362698240[t.sup.9][y.sup.10] - 150493593600[t.sup.7][y.sup.12] + 7806224264457093120[t.sup.9][y.sup.8] - 55725627801600[t.sup.7][y.sup.10] + 1814330860623290695680[t.sup.9][y.sup.6] + 1069932053790720[t.sup.7][y.sup.8] + 16930529280[t.sup.5][y.sup.10] + 340778665995331121971200[t.sup.9][y.sup.4] + 8662169907489669120[t.sup.7][y.sup.6] + 10448555212800[t.sup.5][y.sup.8] + 44164915112994913407467520[t.sup.9][y.sup.2] + 3756848140747139973120[t.sup.7][y.sup.4] + 12259638116352000[t.sup.5][y.sup.6] + 5182016706591403173142855680[t.sup.9] + 894543998237744195174400[t.sup.7][y.sup.2] + 8666449635704832000[t.sup.5][y.sup.4] - 3773089382400[t.sup.3][y.sup.6] + 61340159879159601954816000[t.sup.7] + 249593749508299161600[t.sup.5][y.sup.2] - 1755357275750400[t.sup.3][y.sup.4] + 118740066432748187811840[t.sup.5] - 24073471210291200[t.sup.3][y.sup.2] - 333562017089794867200[t.sup.3] + 39182082048000[ty.sup.2] - 12789031580467200t,

[c.sub.2] = 9504[t.sup.10][y.sup.20] + 4976640[t.sup.10][y.sup.18] + 3941498880[t.sup.10][y.sup.16] + 1651129712640[t.sup.10][y.sup.14] - 1399680[t.sup.8][y.sup.16] + 653847366205440[t.sup.10][y.sup.12] - 6449725440[t.sup.8][y.sup.14] + 205426954327818240[t.sup.10][y.sup.10] - 2941074800640[t.sup.8][y.sup.12] + 54232715942544015360[t.sup.10][y.sup.8] - 416084687585280[t.sup.8][y.sup.10] + 705438720[t.sup.6][y.sup.12] + 11359288866511037399040[t.sup.10][y.sup.6] + 181888449144422400[t.sup.8][y.sup.8] + 348285173760[t.sup.6][y.sup.10] + 2249139195569185405009920[t.sup.10][y.sup.4] + 142977160212161495040[t.sup.8][y.sup.6] + 487599243264000[t.sup.6][y.sup.8] + 235546213935972871506493440[t.sup.10][y.sup.2] + 42242355472337920327680[t.sup.8][y.sup.4] + 930840886797926400[t.sup.6][y.sup.6] - 235818086400[t.sup.4][y.sup.8] + 37310520287458102846628560896[t.sup.10] + 11041228778248728351866880[t.sup.8][y.sup.2] + 256141733677498368000[t.sup.6][y.sup.4] - 41794220851200[t.sup.4][y.sup.6] + 640391269138426244408279040[t.sup.8] - 18902566629428523171840[t.sup.6][y.sup.2] + 57174494124441600[t.sup.4][y.sup.4] + 1936048796185974936698880[t.sup.6] + 35050974082183987200[t.sup.4][y.sup.2] + 4897760256000[t.sup.2][y.sup.4] - 11162387130787823616000[t.sup.4] + 1880739938304000[t.sup.2][y.sup.2] - 2193695064037785600[t.sup.2] - 105791621529600,

[c.sub.1] = 144[t.sup.11][y.sup.22] + 41472[t.sup.11][y.sup.20] + 59719680[t.sup.11][y.sup.18] + 17199267840[t.sup.11][y.sup.16] - 25920[t.sup.9][y.sup.18] + 9906778275840[t.sup.11][y.sup.14] - 156764160[t.sup.9][y.sup.16] + 2853152143441920[t.sup.11][y.sup.12] - 77396705280[t.sup.9][y.sup.14] + 821707817311272960[t.sup.11][y.sup.10] - 12383472844800[t.sup.9][y.sup.12] + 16796160[t.sup.7][y.sup.14] + 236651851385646612480[t.sup.11][y.sup.8] - 1069932053790720[t.sup.9][y.sup.10] + 4837294080[t.sup.7][y.sup.12] + 34077866599533112197120[t.sup.11][y.sup.6] + 1746129111786455040[t.sup.9][y.sup.8] + 4179422085120[t.sup.7][y.sup.10] + 9814425580665536312770560[t.sup.11][y.sup.4] + 1360748145467468021760[t.sup.9][y.sup.6] + 24742178743910400[t.sup.7][y.sup.8] - 7860602880[t.sup.5][y.sup.10] + 565310913446334891615584256[t.sup.11][y.sup.2] + 187428266297432117084160[t.sup.9][y.sup.4] + 23765330778799472640[t.sup.7][y.sup.6] + 2612138803200[t.sup.5][y.sup.8] + 162809543072544448785288265728[t.sup.11] + 77288601447741098463068160[t.sup.9][y.sup.2] + 3176311567816725626880[t.sup.7][y.sup.4] - 3176360784691200[t.sup.5][y.sup.6] + 3886512529943552379857141760[t.sup.9] - 501583599011877995151360[t.sup.7][y.sup.2] + 5568996339980697600[t.sup.5][y.sup.4] + 272097792000[t.sup.3][y.sup.6] + 17481945565560486557122560[t.sup.7] + 404434316332892160000[t.sup.5][y.sup.2] + 579894814310400[t.sup.3][y.sup.4] - 188626316295071953059840[t.sup.5] - 844076084310835200[t.sup.3][y.sup.2] - 113674931054995046400[t.sup.3] - 17631936921600[ty.sup.2] - 16362437463244800t,

[c.sub.0] = [t.sup.12][y.sup.24] + 497664[t.sup.12][y.sup.20] - 216[t.sup.10][y.sup.20] + 103195607040[t.sup.12][y.sup.16] - 1658880[t.sup.10][y.sup.18] - 806215680[t.sup.10][y.sup.16] + 11412608573767680[t.sup.12][y.sup.12] + 174960[t.sup.8][y.sup.16] - 173368619827200[t.sup.10][y.sup.12] + 709955554156939837440[t.sup.12][y.sup.8] + 68475651442606080[t.sup.10][y.sup.10] - 148343685120[t.sup.8][y.sup.12] - 11093055533702184960[t.sup.10][y.sup.8] - 534966026895360[t.sup.8][y.sup.10] - 109175040[t.sup.6][y.sup.12] + 23554621393597287150649344[t.sup.12][y.sup.4] + 7572859244340691599360[t.sup.10][y.sup.6] + 666300186498170880[t.sup.8][y.sup.8] + 116095057920[t.sup.6][y.sup.10] - 51116799899299668295680[t.sup.10][y.sup.4] + 121612756962068398080[t.sup.8][y.sup.6] - 400527949824000[t.sup.6][y.sup.8] + 325619086145088897570576531456[t.sup.12] + 235546213935972871506493440[t.sup.10][y.sup.2] + 11758638865724316057600[t.sup.8][y.sup.4] + 164056248247910400[t.sup.6][y.sup.6] + 5668704000[t.sup.4][y.sup.8] + 10458251898757195494888308736[t.sup.10] - 3680409592749576117288960[t.sup.8][y.sup.2] + 19932834162121113600[t.sup.6][y.sup.4] + 27862813900800[t.sup.4][y.sup.6] + 67167475067679764140523520[t.sup.8] - 6123366654603606097920[t.sup.6][y.sup.2] + 25013841179443200[t.sup.4][y.sup.4] - 1242244730886105480560640[t.sup.6] - 34280623003454668800[t.sup.4][y.sup.2] - 734664038400[t.sup.2][y.sup.4] - 497165327434933862400[t.sup.4] + 1880739938304000[t.sup.2][y.sup.2] - 1126187075056435200[t.sup.2] + 4407984230400 (A.5)

https://doi.org/10.1155/2018/1642139

Conflicts of Interest

The author declares that there are no conflicts of interest.

References

[1] R. S. Johnson, "Water waves and Kortewegde Vries equations," Journal of Fluid Mechanics, vol. 97, no. 4, pp. 701-719, 1980.

[2] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, UK, 1997.

[3] M. J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Cambridge University Press, Cambridge, UK, 2011.

[4] V. D. Lipovskii, "On the nonlinear internal wave theory in fluid of finite depth," Izv. Akad. Nauka, Phys. of Atmosphere and Ocean, vol. 21, no. 8, pp. 864-871, 1985.

[5] V. I. Golinko, V. S. Dryuma, and Y. A. Stepanyants, "Nonlinear quasicylindrical waves: exact solutions of the cylindrical Kadomtsev-Petviashvili equation," in Nonlinear and Turbulent Processes in Physics: Proceedings of the Second International Workshop on Nonlinear and Turbulent Processes in Physics, Kiev, USSR, 10-25 October 1983, pp. 1353-1360, Harwood Academic Publishers; Gordon and Breach, 1984.

[6] V. D. Lipovskii, V. B. Matveev, and A. O. Smirnov, "On a connection between the Kadomtsev-Petviashvili equation and the Johnson equation," Zapiski Nauchnykh Seminarov LOMI, vol. 150, pp. 70-75, 1986.

[7] B. B. Kadomtsev and W. I. Petviashvili, "On the stability of solitary waves in weakly dispersing media," Soviet Physics, Doklady, vol. 15, no. 6, pp. 539-541, 1970.

[8] K. Klein, V. B. Matveev, and A. O. Smirnov, "Cylindrical Kadomtsev-Petviashvili equation: old and new results," Theoretical and Mathematical Physics, vol. 152, no. 2, pp. 1132-1145, 2007.

[9] K. R. Khusnutdinova, C. Klein, V. B. Matveev, and A. O. Smirnov, "On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation," Chaos, vol. 23, no. 1, Article ID 013126, 15 pages, 2013.

[10] M. J. Ablowitz and H. Segur, "On the evolution of packets of water waves," Journal of Fluid Mechanics, vol. 92, no. 4, pp. 691-715, 1979.

[11] D. E. Pelinovsky, Y. A. Stepanyants, and Y. S. Kivshar, "Self-focusing of plane dark solitons in nonlinear defocusing media," Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 51, no. 5, pp. 5016-5026, 1995.

[12] P. Gaillard, "Families of rational solutions of order 5 to the KPI equation depending on 8 parameters," New Horizons in Mathematical Physics, vol. 1, no. 1, pp. 26-31, 2017.

[13] P. Gaillard, "Families of quasi-rational solutions of the NLS equation and multi-rogue waves," Journal of Physics A: Mathematical and Theoretical, vol. 44, pp. 1-15, 2010.

[14] P. Gaillard, "Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves," Journal of Mathematical Physics, vol. 54, Article ID 013504, 32 pages, 2013.

Pierre Gaillard (iD)

Universite de Bourgogne, Institut de Mathematiques de Bourgogne, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France

Correspondence should be addressed to Pierre Gaillard; pierre.gaillard@u-bourgogne.fr

Received 10 November 2017; Accepted 8 May 2018; Published 1 August 2018

Academic Editor: Giampaolo Cristadoro

Caption: Figure 1: Solution of order 3 to (1), on the left for t = 0, [a.sub.1] = 0, [b.sub.1] = 0, [a.sub.2] = 0, and [b.sub.2] = 0; in the center for t = 0, [a.sub.1] = 103, [b.sub.1] = 0, [a.sub.2] = 0, and [b.sub.2] = 0; and on the right for t = 0, [a.sub.1] = 0, [b.sub.1] = 103, [a.sub.2] = 0, and [b.sub.2] = 0.

Caption: Figure 2: Solution of order 3 to (1), on the left for t = 0, [a.sub.1] = 103, [b.sub.1] = 0, [a.sub.2] = 0, and [b.sub.2] = 0; in the center for t = 0, [a.sub.1] = 0, [b.sub.1] = 0, [a.sub.2] = 0, and [b.sub.2] = 106; and on the right for t = 0,01, [a.sub.1] = 0, [b.sub.1] = 103, [a.sub.2] = 0, and [b.sub.2] = 0.

Caption: Figure 3: Solution of order 3 to (1), on the left for t = 0,01, [a.sub.1] = 0, [b.sub.1] = 0, [a.sub.2] = [10.sup.6], and [b.sub.2] = 0; in the center for t = 0, 1, [a.sub.1] = [10.sup.3], [b.sub.1] = 0, [a.sub.2] = 0, and [b.sub.2] = 0; and on the right for t = 1, [a.sub.1] = [10.sup.3], [b.sub.1] = 0, [a.sub.2] = 0, and [b.sub.2] = 0.

Caption: Figure 4: Solution of order 3 to (1), on the left for t = 0, 1, [a.sub.1] = 0, [b.sub.1] = 0, [a.sub.2] = [10.sup.6], and [b.sub.2] = 0; in the center for t = 1, [a.sub.1] = 0, [b.sub.1] = 0, [a.sub.2] = [10.sup.6], and [b.sub.2] = 0; and on the right for t = 10, [a.sub.1] = [10.sup.3], [b.sub.1] = 0, [a.sub.2] = 0, and [b.sub.2] = 0.

Caption: Figure 5: Solution of order 3 to (1), on the left for t = 10, [a.sub.1] = 0, [b.sub.1] = 0, [a.sub.2] = [10.sup.6], and [b.sub.2] = 0; in the center for t = 100, [a.sub.1] = [10.sup.6], [b.sub.1] = 0, [a.sub.2] = 0, and [b.sub.2] = 0; and on the right for t = [10.sup.3], [a.sub.1] = [10.sup.5], [b.sub.1] = [10.sup.3], [a.sub.2] = 0, and [b.sub.2] = 0.
COPYRIGHT 2018 Hindawi Limited
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2018 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Title Annotation:Research Article
Author:Gaillard, Pierre
Publication:Advances in Mathematical Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
Words:8147
Previous Article:Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method.
Next Article:The Perturbed Riemann Problem with Delta Shock for a Hyperbolic System.
Topics:

Terms of use | Privacy policy | Copyright © 2020 Farlex, Inc. | Feedback | For webmasters