8-parameter solutions of fifth order to the Johnson equation

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8-parameter solutions of fifth order to the Johnson equation

Pierre Gaillard

To cite this version:

Pierre Gaillard. 8-parameter solutions of fifth order to the Johnson equation. 2019. �hal-02268910�

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8-parameter solutions of fifth order to the Johnson equation.

Pierre Gaillard, Universit´e de Bourgogne,

Institut de math´ematiques de Bourgogne, 9 avenue Alain Savary BP 47870

21078 Dijon Cedex, France : E-mail : Pierre.Gaillard@u-bourgogne.fr

Abstract

We give different representations of the solutions of the Johnson equation with parameters. First, an expression in terms of Fredholm determinants is given; we give also a representation of the solutions written as a quotient of wronskians of order 2N. These solutions of order N depend on 2N − 1 parameters. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polynomials of degree 2N(N+ 1) inx,tand 4N(N+ 1) inydepending on 2N−2 parameters.

Here, we explicitly construct the expressions of the rational solutions of order 5 depending on 8 real parameters and we study the patterns of their modulus in the plane (x, y) and their evolution according to time and parametersaⁱ andbifor 1≤i≤4.

Key Words : Johnson equation, Fredholm determinants, wronskians, rational solutions, rogue waves.

PACS numbers :

33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd

1 Introduction

We consider the Johnson equation which can be written in the form (ut+ 6uux+uxxx+ u

2t)x−3uyy

t² = 0, (1)

where subscriptsx,y andt denote partial derivatives.

Johnson introduced this equation in a paper written in 1980 [1] to describe waves surfaces in shallow incompressible fluids [2, 3]. This equation was widely accepted, and was later derived for internal waves in a stratified medium [4]. The physical model of this equation have the same degree of universality as the Kadomtsev- Petviashvili (KP) equation [5].

Johnson constructed the first solutions in 1980 [1]. Some time later in 1984, Golinko, Dryuma, and Stepanyants found other types of solutions [6]. Another approach to study this equation was given in 1986 [7] by giving a connection between solutions of the (KP) equation and solutions of the Johnson equation. The use of Darboux transformation gave other type of solutions given in [8]. More re- cently, the extension to the elliptic case has been considered [9] in 2013.

In the following, we recall the representation of the solutions in terms of Fredholm determinants of order 2N depending on 2N −1 parameters. We also recall the expression in terms of wronskians of order 2N with 2N −1 parameters. These

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representations allow to obtain an infinite hierarchy of solutions to the Johnson equation, depending on 2N−1 real parameters and rational solutions to the equation, when a parameter tends towards 0.

Here we construct rational solutions of order 5 depending on 8 parameters, and the representations of their modulus in the plane of the coordinates (x, y) according to the real parametersai andbi for 1≤i≤4 and timet.

The solutions are given without initial conditions nor boundary conditions.

We give three methods to construct solutions to the Johnson equation. The more efficient method to construct solutions of the Johnson equation is that correspond- ing to the representation in terms of degenerate determinants (the third one in the text, without limit) followed by that given in terms of wronskians. The less efficient is that given in terms of Fredholm determinants.

The method used to construct the figures given in the third section is that using the degenerate determinants (without limit, the third one).

2 Rational solutions to the Johnson equation of order N depending on 2N −2 parameters

2.1 Families of rational solutions of order N depending on 2N −2 parameters

We define real numbers λj such that −1< λν <1,ν = 1, . . . ,2N which depend on a parameterǫwhich will be intended to tend towards 0; they can be written as λj= 1−2ǫ²j², λN+j=−λj, 1≤j≤N, (2) The termsκν, δν, γν andxr,ν are functions ofλν,1≤ν ≤2N; they are defined by the formulas :

κj= 2q

1−λ²_j, δj=κjλj, γj=q_1−λ

j

1+λj,;

xr,j= (r−1) ln^γ_γ^j⁻ⁱ

j+i, r= 1,3, τj=−12iλ²_jq

1−λ²_j−4i(1−λ²_j)q 1−λ²_j, κN+j =κj, δN+j =−δj, γN+j =γ_j⁻¹,

x_r,N+j =−xr,j, , τ_N_+j =τ_j j= 1, . . . , N.

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eν 1≤ν ≤2N are defined in the following way : ej= 2i

P1/2M−1

k=1 ak(je)²^k+1−iP1/2M−1

k=1 bK(je)²^k+1 , eN+j= 2i

P1/2M−1

k=1 ak(je)²^k+1+iP1/2M−1

k=1 bk(je)²^k+1

, 1≤j≤N, ak, bk ∈R, 1≤k≤N.

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ǫν, 1≤ν≤2N are real numbers defined by :

ǫj= 1, ǫN+j= 0 1≤j≤N. (5)

LetI be the unit matrix andDr= (djk)1≤j,k≤2N the matrix defined by : dνµ= (−1)^ǫ^ν Y

η6=µ

γη+γν

γη−γµ

exp(κνx+ (κνy

12 −2δν)yt+ 4iτνt+xr,ν+eν). (6) Then we have the following result :

Theorem 2.1 The functionv defined by

v(x, y, t) =−2|n(x, y, t)|²

d(x, y, t)² (7)

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where

n(x, y, t) = det(I+D3(x, y, t)), (8) d(x, y, t) = det(I+D1(x, y, t)), (9) andDr= (djk)1≤j,k≤2N the matrix

dνµ= (−1)^ǫ^ν Y

η6=µ

γη+γν

γη−γµ

exp(κνx+ (κνy

12 −2δν)yt+ 4iτνt+xr,ν+eν). (10) is a solution to the Johnson equation (1), depending on2N−1parametersak,bh, 1≤k≤N−1 andǫ.

We give now the expressions of the solutions to the Johnson equation in terms of wronskians. For this, we define the following notations :

φr,ν= sin Θr,ν, 1≤ν≤N, φr,ν = cos Θr,ν, N+ 1≤ν≤2N, r= 1,3, (11) with the arguments

Θr,ν= ^−iκ₂^ν^x+i(^−κ₂₄^ν^y +δν)yt−i^x^r,ν₂ + 2τνt+γνw−i^e₂^ν, 1≤ν≤2N. (12) We denoteWr(w) the wronskian of the functionsφr,1, . . . , φr,2N defined by

Wr(w) = det[(∂_w^µ−1φr,ν)ν, µ∈[1,...,2N]]. (13) We consider the matrixDr= (dνµ)ν, µ∈[1,...,2N] defined in (10).

Then we have the following statement : Theorem 2.2 The functionv defined by

v(x, y, t) =−2|W3(φ3,1, . . . , φ3,2N)(0)|² (W1(φ1,1, . . . , φ1,2N)(0))²

is a solution to the Johnson equation depending on2N−1 real parametersak,bk

andǫ, withφ^r_ν defined in (11)

φr,ν= sin(^−iκ₂^ν^x+i(^−κ₂₄^ν^y+δν)yt−i^x^r,ν₂ + 2τνt+γνw−i^e₂^ν), 1≤ν≤N,

φr,ν= cos(^−iκ₂^ν^x+i(^−κ₂₄^ν^y +δν)yt−i^x^r,ν₂ + 2τνt+γνw−i^e₂^ν), N+ 1≤ν ≤2N, r= 1,3, κν,δν,xr,ν,γν,eν being defined in(3), (2) and (4).

We can deduce rational solutions to the Johnson equation as a quotient of two determinants.

We use the following notations : Xν= −iκνx

2 +i(−κνy

24 +δν)yt−ix3,ν

2 + 2τνt+γνw−ieν

2 , Yν= −iκνx

2 +i(−κνy

24 +δν)yt−ix_1,ν

2 + 2τνt+γνw−ie_ν 2,

for 1≤ν≤2N, withκ_ν, δ_ν,x_r,ν defined in (3) and parameters e_ν defined by (4).

We define the following functions :

ϕ4j+1,k=γ^4j−1_k sinXk, ϕ4j+2,k=γ^4j_k cosXk,

ϕ4j+3,k=−γ_k^4j+1sinXk, ϕ4j+4,k=−γ_k^4j+2cosXk, (14)

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for 1≤k≤N, and

ϕ4j+1,N+k=γ^2N_k ^−4j−2cosXN+k, ϕ4j+2,N+k=−γ_k^2N^−4j−3sinXN+k,

ϕ4j+3,N+k=−γ_k^2N^−4j−4cosXN+k, ϕ4j+4,N+k =γ_k^2N^−4j−5sinXN+k, (15) for 1≤k≤N.

We define the functionsψj,k for 1 ≤j ≤2N, 1≤ k≤ 2N in the same way, the termXk is only replaced byYk.

ψ4j+1,k=γ_k^4j−1sinYk, ψ4j+2,k=γ^4j_k cosYk,

ψ4j+3,k=−γ_k^4j+1sinYk, ψ4j+4,k =−γ^4j+2_k cosYk, (16) for 1≤k≤N, and

ψ4j+1,N+k =γ_k^2N^−4j−2cosYN+k, ψ4j+2,N+k=−γ_k^2N^−4j−3sinYN+k,

ψ4j+3,N+k =−γ_k^2N−4j−4cosYN+k, ψ4j+4,N+k =γ_k^2N^−4j−5sinYN+k, (17) for 1≤k≤N.

The following ratio

q(x, t) := W3(0) W1(0) can be written as

q(x, t) =∆3

∆1

= det(ϕj,k)j, k∈[1,2N]

det(ψj,k)j, k∈[1,2N]

. (18)

The termsλj depending onǫare defined byλj = 1−2jǫ². All the functionsϕj,k

andψj,kand their derivatives depend onǫ. They can all be prolonged by continuity whenǫ= 0.

We use the following expansions

ϕj,k(x, y, t, ǫ) =

N−1

X

l=0

1

(2l)!ϕj,1[l]k^2lǫ^2l+O(ǫ^2N), ϕj,1[l] = ∂^2lϕj,1

∂ǫ^2l (x, y, t,0), ϕ_j,1[0] =ϕ_j,1(x, y, t,0), 1≤j≤2N, 1≤k≤N, 1≤l≤N−1, ϕj,N+k(x, y, t, ǫ) =

N−1

X

l=0

1

(2l)!ϕj,N+1[l]k^2lǫ^2l+O(ǫ^2N), ϕj,N+1[l] = ∂^2lϕj,N+1

∂ǫ^2l (x, y, t,0), ϕj,N+1[0] =ϕj,N+1(x, y, t,0), 1≤j ≤2N, 1≤k≤N, 1≤l≤N−1.

We have the same expansions for the functionsψj,k. ψj,k(x, y, t, ǫ) =

N−1

X

l=0

1

(2l)!ψj,1[l]k^2lǫ^2l+O(ǫ^2N), ψj,1[l] = ∂^2lψ_j,1

∂ǫ^2l (x, y, t,0), ψj,1[0] =ψj,1(x, y, t,0), 1≤j≤2N, 1≤k≤N, 1≤l≤N−1,

ψj,N+k(x, t, ǫ) =

N−1

X

l=0

1

(2l)!ψj,N+1[l]k^2lǫ^2l+O(ǫ^2N), ψj,N+1[l] = ∂^2lψj,N+1

∂ǫ^2l (x, y, t,0), ψ_j,N+1[0] =ψ_j,N+1(x, t,0), 1≤j≤2N, 1≤k≤N, N+ 1≤k≤2N..

Then we get the following result :

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Theorem 2.3 The functionv defined by

v(x, y, t) =−2|det((njk)_j,k∈[1,2N])|²

det((djk)_j,k∈[1,2N])² (19) is a rational solution to the Johnson equation (1), where

nj1=ϕj,1(x, y, t,0),1≤j≤2N njk=^∂^2k_∂ǫ⁻²2k−2^ϕ^j,1(x, y, t,0),

njN+1=ϕj,N+1(x, y, t,0),1≤j≤2N njN+k = ^∂^2k−2_∂ǫ2k^ϕ^j,N+1−2 (x, y, t,0), d_j1=ψ_j,1(x, y, t,0),1≤j≤2N d_jk= ^∂^2k−2_∂ǫ2k−2^ψ^j,1(x, y, t,0),

djN+1=ψj,N+1(x, y, t,0),1≤j≤2N djN+k =^∂^2k_∂ǫ⁻²2k−2^ψ^j,N+1(x, y, t,0), 2≤k≤N,1≤j≤2N

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The functionsϕandψ are defined in (14),(15), (16), (17).

3 Explicit expression of rational solutions of order 5 depending on 8 parameters

We construct rational solutions to the Johnson equation of order 5 depending on 8 parameters.

We give patterns of the modulus of the solutions in the plane (x, y) of coordinates in function of the parametersaiand bi, for 1≤i≤4 and time t.

The (x;y) plane is the horizontal plane. To shorten the text, one cut certain characters of the figures and one made appear only the letteryof the (x;y) plane.

Figure 1. Solution of order 5 to (1), on the left fort= 0; in the center fort= 0, a1= 10³; on the right fort= 0,a2= 10³; all other parameters not mentioned

equal to 0.

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Figure 2. Solution of order 5 to (1), on the left fort= 0,a3= 10³; in the center fort= 0,a4= 10³; on the right fort= 0, b1= 10³; all other parameters not

mentioned equal to 0.

Figure 3. Solution of order 5 to (1), on the left fort= 0, b2= 10³; in the center fort= 0, b3= 10³; on the right fort= 0,b4= 10³; all other parameters not

mentioned equal to 0.

Figure 4. Solution of order 5 to (1), on the left fort= 0,01,a1= 10³; in the center fort= 0,1,a2= 10³; on the right fort= 1,b1= 10³; all other parameters

not mentioned equal to 0.

Figure 5. Solution of order 5 to (1), on the left fort= 0,01,a2= 10³; in the center fort= 0,1,a2= 10³; on the right fort= 1,a2= 10; all the other

parameters to equal to 0.

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Figure 8. Solution of order 5 to (1), on the left fort= 0,01,b1= 10; in the center fort= 0,1,b4= 10; on the right fort= 1,b1= 10; all the other

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Figure 9. Solution of order 5 to (1), on the left fort= 0,01,b₂= 10³; in the center fort= 0,1,b₂= 10; on the right fort= 1,b₂= 10; all the other

Figure 10. Solution of order 5 to (1), on the left fort= 0,01,b₃= 10³; in the center fort= 0,1,b3= 10³; on the right fort= 1,b3= 10³; all the other

Figure 11. Solution of order 5 to (1), on the left fort= 0,01,b4= 10³; in the center fort= 0,1,b4= 10³; on the right fort= 1,b4= 10; all the other

In these constructions, we note that the initial rectilinear structure becomes deformed very quickly as timet increases. The heights of the peaks also decrease very quickly according to time t and of the various parameters. Because of the structure of the polynomials, one notices that the modulus of these solutions tend towards value 2 when timet and variablesxandy tend towards the infinite.

The preceding solutions depends one parameters aj and bj for 1 ≤ j ≤ 4.

The Johnson equation allows explaining the existence of the horseshoelike solitons

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and multisoliton solutions quite naturally. The horseshoe multisoliton solutions correspond very well to real waves observed in thin films of shallow water being cooled along an inclined plane.

It should be relevant to give a physical meaning of these parameters and to give an explanation of the evolution of the figures according to time in the (x;y) plane.

4 Conclusion

We succeed in obtaining rational solutions to the Johnson equation depending on 2N−2 real parameters. These solutions can be expressed in terms of a ratio of two polynomials of degree 2N(N+ 1) in x,tand 4N(N+ 1) iny. Here we have made the study of rational solutions of order 5 depending on 8 parameters and tried to describe the structure of those rational solutions.

In the (x;y) plane of coordinates, various structures appear. But, contrary to the rational solutions of the NLS or KP equations, there are not well defined structures which appear according to the parametersai or bi. Thus, one cannot carry out a classification of these solutions here, according to the parameters by means of their module in the plan (x, y). It would be important to better understand these structures.

References

[1] R.E. Johnson, Water waves and Kortewegde Vries equations, J. Fluid Mech., V. 97, N. 4, 701719, 1980

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

[3] M. J. Ablowitz, Nonlinear Dispersive Waves : Asymptotic Analysis and Soli- tons, Cambridge University Press, Cambridge, 2011

[4] V.D. Lipovskii1, On the nonlinear internal wave theory in fluid of finite depth, Izv. Akad. Nauka., V. 21, N. 8, 864871, 1985

[5] B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., V. 15, N. 6, 539-541, 1970

[6] V.I. Golinko, V.S. Dryuma, Yu.A. Stepanyants, Nonlinear quasicylindrical waves: Exact solutions of the cylindrical Kadomtsev- Petviashvili equation, in Proc. 2nd Int. Workshop on Nonlinear and Turbulent Processes in Physics, Kiev, Harwood Acad., Gordon and Breach, 13531360, 1984

[7] V.D. Lipovskii, V.B. Matveev, A.O. Smirnov, Connection between the Kadomtsev-Petvishvili and Johnson equation, Zap. Nau. Sem., V. 150, 7075, 1986

[8] C. Klein, V.B. Matveev, A.O. Smirnov, Cylindrical Kadomtsev-Petviashvili equation: Old and new results, Theor. Math. Phys., V. 152, N. 2, 1132-1145, 2007

[9] K. R. Khusnutdinova, C. Klein, V.B. Matveev, A.O. Smirnov, On the inte- grable elliptic cylindrical K-P equation Chaos, V. 23, 013126-1-15, 2013 [10] P. Gaillard, Families of quasi-rational solutions of the NLS equation and

multi-rogue waves, J. Phys. A : Meth. Theor., V. 44, 1-15, 2010

[11] P. Gaillard, Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves, Jour. Of Math.

Phys., V. 54, 2013, 013504-1-32

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[12] P. Gaillard, V.B. Matveev, Wronskian addition formula and its applications, Max-Planck-Institut f¨ur Mathematik, MPI 02-31, V.161, 2002

[13] P. Gaillard, A new family of deformations of Darboux-P¨oschl-Teller potentials, Lett. Math. Phys., V.68, 77-90, 2004

[14] P. Gaillard, V.B. Matveev, New formulas for the eigenfunctions of the two- particle Calogero-Moser system, Lett. Math. Phys., V.89, 1-12, 2009 [15] P. Gaillard, V.B. Matveev, Wronskian and Casorai determinant represen-

tations for Darboux-P¨oschl-Teller potentials and their difference extensions, RIMS Kyoto, N. 1653, 1-19, 2009

[16] P. Gaillard, V.B. Matveev, Wronskian and Casorai determinant representations for Darboux-P¨oschl-Teller potentials and their difference extensions, J.

Phys A : Math. Theor., V.42, 1-16, 2009

[17] P. Gaillard, From finite-gap solutions of KdV in terms of theta functions to solitons and positons, halshs-00466159, 2010

[18] P. Gaillard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves, J. Phys. A : Meth. Theor., V.44, 1-15, 2011

[19] P. Gaillard, Wronskian representation of solutions of the NLS equation and higher Peregrine breathers, J. Math. Sciences : Adv. Appl., V. 13, N. 2, 71-153, 2012

[20] P. Gaillard, Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves, J. Math. Phys., V. 54, 013504-1-32, 2013

[21] P. Gaillard, Wronskian representation of solutions of NLS equation and seventh order rogue waves, J. Mod. Phys., V.4, N. 4, 246-266, 2013

[22] P. Gaillard, V.B. Matveev, Wronskian addition formula and Darboux-P¨oschl- Teller potentials, J. Math., V. 2013, ID 645752, 1-10, 2013

[23] P. Gaillard, Two parameters deformations of ninth Peregrine breather solution of the NLS equation and multi rogue waves, J. Math., V.2013, 1-111, 2013 [24] P. Gaillard, Two-parameters determinant representation of seventh order

rogue waves solutions of the NLS equation, J. Theor. Appl. Phys., V. 7, N. 45, 1-6, 2013

[25] P. Gaillard, Six-parameters deformations of fourth order Peregrine breather solutions of the NLS equation, J. Math. Phys., V.54, 073519-1-22, 2013 [26] P. Gaillard, Deformations of third order Peregrine breather solutions of the

NLS equation with four parameters, Phys. Rev. E, V. 88, 042903-1-9, 2013 [27] P. Gaillard, Ten parameters deformations of the sixth order Peregrine breather

solutions of the NLS equation, Phys. Scripta, V.89, 015004-1-7, 2014 [28] P. Gaillard, The fifth order Peregrine breather and its eight-parameters de-

formations solutions of the NLS equation, Commun. Theor. Phys., V.61, 365-369, 2014

[29] P. Gaillard, Higher order Peregrine breathers, their deformations and multi- rogue waves, J. Of Phys. : Conf. Ser., V. 482, 012016-1-7, 2014

[30] P. Gaillard, M. Gastineau, Eighteen parameter deformations of the Peregrine breather of order ten solutions of the NLS equation, Int. J. Mod. Phys. C, V.

26, N. 2, 1550016-1-14, 2014

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[31] P. Gaillard, Two parameters wronskian representation of solutions of nonlinear Schr¨odinger equation, eight Peregrine breather and multi-rogue waves, J.

Math. Phys., V.5, 093506-1-12, 2014

[32] P. Gaillard, Hierarchy of solutions to the NLS equation and multi-rogue waves, J. Phys. : Conf. Ser., V. 574, 012031-1-5, 2015

[33] P. Gaillard, Tenth Peregrine breather solution of the NLS, Ann. Phys., V.

355, 293-298, 2015

[34] P. Gaillard, M. Gastineau, The Peregrine breather of order nine and its deformations with sixteen parameters solutions of the NLS equation, Phys. Lett.

A., V.379, 13091313, 2015

[35] P. Gaillard, Other 2N-2 parameters solutions to the NLS equation and 2N+1 highest amplitude of the modulus of the N-th order AP breather, J. Phys.

A: Math. Theor., V. 48, 145203-1-23, 2015

[36] P. Gaillard, Multi-parametric deformations of the Peregrine breather of order N solutions to the NLS equation and multi-rogue waves, Adv. Res., V. 4, 346-364, 2015

[37] P. Gaillard, Higher order Peregrine breathers solutions to the NLS equation, Jour. Phys. : Conf. Ser., V.633, 012106-1-6, 2016

[38] P. Gaillard, M. Gastineau Patterns of deformations of Peregrine breather of order 3 and 4, solutions to the NLS equation with multi-parameters, Journal of Theoretical and Applied Physics, V.10,1-7, 2016

[39] P. Gaillard, M. Gastineau Twenty parameters families of solutions to the NLS equation and the eleventh Peregrine breather, Commun. Theor. Phys, V.65, 136-144, 2016

[40] P. Gaillard, Rational solutions to the KPI equation and multi rogue waves, Annals Of Physics, V. 367, 1-5, 2016

[41] P. Gaillard, M. Gastineau Twenty two parameters deformations of the twelfth Peregrine breather solutions to the NLS equation, Adv. Res., V. , 1-11, 2016 [42] P. Gaillard, Towards a classification of the quasi rational solutions to the NLS

equation, Theor. And Math. Phys., V.189, 1440-1449, 2016

[43] P. Gaillard, Fredholm and Wronskian representations of solutions to the KPI equation and multi-rogue waves, Jour. of Math. Phys., V.57, 063505-1-13, doi: 10.1063/1.4953383, 2016

[44] P. Gaillard, M. Gastineau Families of deformations of the thirteenth Peregrine breather solutions to the NLS equation depending on twenty four parameters, Jour. Of Bas. And Appl. Res. Int., V.21, N. 3, 130-139, 2017

[45] P. Gaillard, From Fredholm and Wronskian representations to rational solutions to the KPI equation depending on 2N2 parameters, Int. Jour. of Appl.

Sci. And Math., V.4, N. 3, 60-70, 2017

[46] P. Gaillard, Families of Rational Solutions of Order 5 to the KPI Equation Depending on 8 Parameters, New Hor. in Math. Phys., V. 1, N. 1, 26-31, 2017

[47] P. Gaillard, 6-th order rational solutions to the KPI Equation depending on 10 parameters, Jour. Of Bas. And Appl. Res. Int., V.21, N. 2, 92-98, 2017

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[48] P. Gaillard,N-Order rational solutions to the Johnson equation depending on 2N−2 parameters, Int. Jour. of Adv. Res. in Phys. Sci., V. 4, N. 9, 19-37, 2017

[49] P. Gaillard, Families of rational solutions to the KPI equation of order 7 depending on 12 parameters, Int. Jour. of Adv. Res. in Phys. Sci., V.4, N.

11, 24-30, 2017

[50] P. Gaillard, Rational solutions to the Johnson equation and rogue waves, Int.

Jour. of Inn. In Sci. and Math., V. 6, N. 1, 14-19, 2018

[51] P. Gaillard, Multiparametric families of solutions of the KPI equation, the structure of their rational representations and multi-rogue waves, Theo. And Mat. Phys., V.196, N. 2, 1174-1199, 2018

[52] P. Gaillard, The Johnson Equation, Fredholm and Wronskian representations of solutions and the case of order three, Adv. In Math. Phys., V.2018, 1-18, 2018

Because of the length of the complete expression, we only give the explicit expression of the rational solution of order 5 to the Johnson equation without parameters.

It can be written as

v(x, y, t) =−2|n(x, y, t)|² (d(x, y, t))² with

n(x, y, t) =A(x, y, t) +iB(x, y, t), d(x, y, t) =C(x, y, t) +iD(x, y, t), A(x, y, t) =

30

X

k=0

ak(y, t)x^k, B(x, y, t) =

30

X

k=0

bk(y, t)x^k, C(x, y, t) =

30

X

k=0

ck(y, t)x^k, D(x, y, t) = 0.

a30 = 237376313799769806328950291431424, a29 = (593440784499424515822375728578560y2 +170910945935834260556844209830625280)t, a28 = (717074281270137956618704005365760y4 +398792207183613274632636489604792320y2 +59477009185670322673781785021057597440)t2+

2670483530247410321200690778603520, a27 = (557724440987885077370103115284480y6 +448641233081564933961716050805391360y4 + 129208675127490700980974222631952711680y2 +13322850057590152278927119844716901826560)t3+(6231128237243957416134945150074880y2 + 2592149346693486285112137182431150080)t, a26 = (313719998055685356020683002347520y8 +324018668336685785639017147803893760y6 + 134991162131653093463147451731222200320y4 +26875404426518065804042638307446164029440y2 +2158301709329604669186193414844138095902720)t4+

(7010019266899452093151813293834240y4 +5084600641591069251566115242461102080y2 +1098273738583670958338280892371598049280)t2− 23366730889664840310506044312780800, a25 = (135945332490796987608962634350592y10 +168759723092023846686988097814528000y8 +

90293202243156438931406111854685061120y6 +26004442246029054412244960214149297602560y4 +4031310663977709870606395746116924604416000y2 + 269356053324334662714436938172548434368659456)t5+(5062791692760715400609642934435840y6 +4698270690881943891765748643156459520y4 + 1884293178942572722639207413382643712000y2 +282191746478439690942447702228184722309120)t3+(−50627916927607154006096429344358400y2− 9197145278172081146215179041510522880)t, a24 = (47203240448193398475334248038400y12 +67503889236809538674795239125811200y10 + 43427502075680803214118270504271872000y8 +16064413551241765670934951498586647429120y6 +3602050732165268541791825828706326151168000y4 + 464406988490232177093856789952669714428723200y2 +26935605332433466271443693817254843436865945600)t6+(2636870673312872604484189028352000y8 + 2700155569472381546991809565032448000y6 +1480266824039674523777017569851788492800y4 +430695583758302336603247408773175705600000y2 + 50391383299721373382579946826461557555200000)t4+(−52737413466257452089683780567040000y4−23179797042547521588021995958278553600y2− 282643976841385908395881112007396556800)t2+297925818843226713958952064987955200, a23 = (13486640128055256707238356582400y14 + 21563742395091935965559590276300800y12 +15990921316986437383851494424025497600y10 +7114585906891388690537779186712696586240y8 +

2049000741184719942874880405773256616837120y6 +381989081582243441961459365809834811405107200y4 +42725442941101360292634824675645613727442534400y2 + 2216415524497382367478795376962684259947826380800)t7+(1054748269325149041793675611340800y10 +1068811579582817695684257952825344000y8 +

687418067132753382148007148956260761600y6 +293112272279955756854987819859522355200000y4 +68945749048029038043447845196409966952448000y2 + 6733901333108366567860923454313710859216486400)t5+(−35158275644171634726455853711360000y6−27198874755569874121429035580076851200y4− 3598102689314150770372962727459238707200y2 +586607385078807782453622970749065311027200)t3+(595851637686453427917904129975910400y2 + 149173209999620340542270586892792627200)t, a22 = (3231174197346571919442522931200y16 +5647646817762173705265606977126400y14 + 4686520013866647416514950953382707200y12+2404650543949324110489025926235296890880y10 +833085694235126170199619544787182461911040y8 +

199451334717332739020401766425660465317150720y6 +32241885182405193183793872324677014062838579200y4 +3222719124700216890644455346962983435441379737600y2 + 152932671190319383356036881010425213936400020275200)t8+(336933474923311499461868598067200y12 +301892393531287103517834263868211200y10 +

196581710671318116550030627909612339200y8 +111168910843332770889692278388190326292480y6 +41083477473926115485468299326328199656243200y4 + 8284901595950744543924381066912050097789337600y2 +704969808528172444828474607148152626502801817600)t6+(−16846673746165574973093429903360000y8− 19944802966045033696145270267864678400y6−5893276464220590045032108424037820006400y4 +61589468477437234134264379454564125900800y2 +

192215993466660266407336492556198132003635200)t4+(571024486116184535087991457893580800y4 +306384575425285386151355253029181849600y2 + 40148903948469245940233969386574472806400)t2−3220227600731935805291614231855104000, a21 = (658202151311338724330884300800y18 + 1235422741385475498026851526246400y16+1121999167795418509446100586122444800y14 +644681143306279233325551142681769410560y12 +

257668285559154848361096809724264421785600y10 +74208466241036596327995881200588153474252800y8 +15400060632114055121938900025835844412972728320y6 + 2223079396205242206972332623599465425373914726400y4 +203031304856113664110600686858667956432806923468800y2 +

8972050043165403823554163685944945884268801189478400)t9+(88244481527533964144775109017600y14 +59300291586502823905288873259827200y12 + 27296380373356376788403739814369689600y10 +21148715098677447212545092263680545914880y8 +12821383213576217196154466233952501256683520y6 +

4348178573295697999273444103720255249134387200y4 +780287095764088304318696253210989445573613977600y2 +59620303806954012476922423918815193555665525145600)t7+

(−6177113706927377490134257631232000y10−10318250736051491359520263947209932800y8−4813733497118074422686729176787347046400y6−

664461593726228683993779429432160891699200y4 +120781685335675757462995083831896755824230400y2 +34457658910431505716776403073747844724896563200)t5+

(348959408182112771442661446490521600y6 +299616012377579137408078668859913011200y4 +82170885045054025479758103285261454540800y2 +

8181708656864590337283589400759632291430400)t3+(−5903750601341882309701292758401024000y2−1569052612510104362010170369558917939200)t, a20 = (115185376479484276757904752640y20 +228781989145458425560528060416000y18 +223474246997283790087523809414348800y16 +

140676105054274432389719951071846072320y14 +63000101751078125517920820552925477601280y12 +20936039586792486291222563547568367135096832y10 +