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First and second order rational solutions to the Johnson equation and rogue waves
P Gaillard
To cite this version:
P Gaillard. First and second order rational solutions to the Johnson equation and rogue waves. 2018.
�hal-01806297�
First and second order rational solutions to the Johnson equation and
rogue waves.
P. 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
Rational solutions to the Johnson equation are constructed as a quo- tient of two polynomials inx,yandtdepending on several real parame- ters. We obtain an infinite hierarchy of rational solutions written in terms of polynomials of degrees 2N(N+ 1) inx, andt, 4N(N+ 1) iny, depend- ing on 2N−2 real parameters for each positive integerN. We construct explicit expressions of the solutions in the casesN = 1 andN = 2 which are given in the following. We study the evolution of the solutions by constructing the patterns of their modulus in the (x, y) plane, and this for different values of parameters.
Key Words : Johnson equation, Fredholm determinants, wronskians, ratio- nal solutions, rogue waves.
PACS numbers :
33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd
1 Introduction
We consider the Johnson equation (J) which can be written in the form (ut+ 6uux+uxxx+ u
2t)x−3uyy
t2 = 0, where subscriptsx, yandt denote partial derivatives.
This equation first appears first in 1980, in a paper written by Johnson [1]. It gives the description of 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 equation [5].
The first solutions were constructed in 1980 by Johnson [1]. Various research were led and more general solutions of this equation were obtained, for example, some special solutions were found in [6]. In 2007, some important classes of so- lutions were obtained by using the Darboux transformation [7]. This equation is dissipative and there is no soliton-like solution with a linear front localized along straight lines in the (x, y) plane. The Johnson equation allows explaining the existence of the horseshoe like solitons and multisoliton solutions quite na- turally. Other extensions have been considered as for example the elliptic case [8].
In this paper, we study rational solutions of the Johnson equation. We presents multi parametric families of rational solutions as a quotient of two polynomials in x, y and tdepending on several real parameters. These solutions presented belong to an infinite hierarchy of rational solutions written in terms of polyno- mials of degrees 2N(N+ 1) inx,t and 4N(N+ 1) in y, depending on 2N −2 real parameters for each positive integer N. The study here is limited to the simplest cases whereN= 1 and N = 2.
The patterns of the modulus of the solutions in the (x, y) plane for different values of timet and parameters are studied.
2 First order rational solutions
We consider the Johnson equation
(ut+ 6uux+uxxx+ u
2t)x−3uyy
t2 = 0, (1)
We have the following result at orderN= 1 : Theorem 2.1 The functionv defined by
v(x, y, t) =−2|n(x, y, t)|2
d(x, y, t)2 (2)
where
n(x, y, t) = 144x2+ (6912t+ 24y2t)x+ 108 + 576iyt+y4t2+ 82944t2 (3) and
d(x, y, t) = 144x2+ (6912t+ 24y2t)x−36 +y4t2+ 82944t2, (4) is a rational solution to the Johnson equation (1), quotient of two polynomials
|n(x, y, t)|2 andd(x, y, t)2 of degree4 inx,t and8in y.
ProofWe have to replace the expression of the solution given by (2) and check that (1) is verified.
2
The modulus of the solution for different values oftare represented in the plane (x, y). It can be seen that as t tends to 0, the front of the solution contracts and is concentrated in a small strip along they axis.
When the timetgrows, we get peaks on special curves.
Figure 1. Solution of order 1 to (1), on the left fort= 0; in the center for t= 0,01; on the right fort= 1.
3 Second order rational solutions depending on 2 parameters
The Johnson equation defined by (1) is always considered. For this second order, we get the rational solutions given by :
Theorem 3.1 The functionv defined by
v(x, y, t) =−2|n(x, y, t)|2
d(x, y, t)2 (5)
is a rational solution to the Johnson equation (1), quotient of two polynomials
|n(x, y, t)|2 andd(x, y, t)2 of degree 12 in x, t and24 in y depending on 2 real parametersa1,b1.
Polynomialsnanddare respectively given by
n(x, y, t) =A(x, y, t) +iB(x, y, t),d(x, y, t) =C(x, y, t) +iD(x, y, t) with A(x, y, t) =P6
k=0ak(y, t)xk, B(x, y, t) =P6
k=0bk(y, t)xk, C(x, y, t) =P6
k=0ck(y, t)xk, D(x, y, t) =P6
k=0dk(y, t)xk,
a6 = −2985984, a5 = (−1492992y2 −429981696)t, a4 = 17915904tyb1+ (−311040y4− 143327232y2−25798901760)t2+ 11197440a1
2+ 2239488b1
2−6718464, a3 = (−34560y6− 17915904y4−5159780352y2−825564856320)t3+(5971968y3b1+1719926784yb1)t2−17915904a1b1+ (3732480y2a1
2+ 746496y2b1
2−2239488y2−71663616ya1+ 1074954240a1
2+ 214990848b1 2− 931627008)t, a2= (−2160y8−995328y6−358318080y4−82556485632y2−14860167413760)t4−
2799360a1
4−3359232a1 2b1
2−559872b1
4+ (746496y5b1+ 286654464y3b1+ 61917364224yb1)t3+ (466560y4a1
2+93312y4b1
2−279936y4−17915904y3a1+214990848y2a1
2+35831808y2−5159780352ya1+ 38698352640a1
2+7739670528b1
2−43858132992)t2+8398080a1
2+8398080b1
2+(−4478976y2a1b1− 26873856ya1
2b1−8957952yb1
3+ 100776960yb1−1289945088a1b1)t+ 8398080, a1= (−72y10− 20736y8−11943936y6−3439853568y4−495338913792y2−142657607172096)t5+ (41472y7b1+ 11943936y5b1+3439853568y3b1+990677827584yb1)t4+4478976a1
3b1+4478976a1b1
3+(25920y6a1 2+ 5184y6b1
2−15552y6−1492992y5a1+13436928y4a1
2−4478976y4b1
2+25380864y4−573308928y3a1+ 3869835264y2a1
2−1289945088y2b1
2+ 2149908480y2−123834728448ya1+ 619173642240a1 2+ 123834728448b1
2−866843099136)t3+(−373248y4a1b1−4478976y3a1
2b1−1492992y3b1
3+16796160y3b1− 1289945088ya1
2b1−429981696yb1
3+ 3117367296yb1−30958682112a1b1)t2+ (−466560y2a1 4− 559872y2a1
2b1
2−93312y2b1
4+1399680y2a1
2+1399680y2b1
2+17915904ya1
3+53747712ya1b1 2− 134369280a1
4−161243136a1 2b1
2−26873856b1
4+ 1399680y2−67184640ya1+ 618098688a1 2+ 188116992b1
2+ 1048080384)t, a0= (−y12−248832y8−20639121408y4−570630428688384)t6+ 46656a1
6+139968a1 4b1
2+139968a1 2b1
4+46656b1
6+(864y9b1+143327232y5b1+5944066965504yb1)t5+ (540y8a1
2+108y8b1
2−324y8−41472y7a1+248832y6a1
2−248832y6b1
2+1327104y6−11943936y5a1+ 89579520y4a1
2+17915904y4b1
2+89579520y4−3439853568y3a1+20639121408y2a1
2−20639121408y2b1 2− 990677827584ya1+3715041853440a1
2+743008370688b1
2−6191736422400)t4−279936a1
4−2519424a1 2b1
2− 2239488b1
4+ (−10368y6a1b1−186624y5a1
2b1−62208y5b1
3+ 699840y5b1+ 8957952y4a1b1− 71663616y3a1
2b1+ 23887872y3b1
3−179159040y3b1+ 2579890176y2a1b1−15479341056ya1 2b1− 5159780352yb1
3+ 16769286144yb1−247669456896a1b1)t3+ (−19440y4a1
4−23328y4a1 2b1
2− 3888y4b1
4+ 58320y4a1
2+ 58320y4b1
2+ 1492992y3a1
3+ 4478976y3a1b1
2 −8957952y2a1 4+ 8957952y2b1
4+ 58320y4−5598720y3a1−22394880y2a1
2−165722112y2b1
2+ 429981696ya1 3+ 1289945088ya1b1
2−1612431360a1
4−1934917632a1 2b1
2−322486272b1
4+107495424ya1+9997074432a1 2− 322486272b1
2+ 13436928000)t2−734832a1
2−734832b1
2+ (373248y2a1
3b1+ 373248y2a1b1 3+ 1119744ya1
4b1+2239488ya1 2b1
3+1119744yb1
5−28553472ya1
2b1−33032448yb1
3+107495424a1 3b1+ 107495424a1b1
3−35271936yb1+ 429981696a1b1)t+ 2099520
b6= 5971968, b5= (2985984y2+859963392)t, b4=,(622080y4+286654464y2+51597803520)t2− 11197440a1
2−2239488b1
2+(−17915904yb1+35831808y)t+13436928, b3= (69120y6+35831808y4+ 10319560704y2+1651129712640)t3+(−5971968y3b1+11943936y3−1719926784yb1+3439853568y)t2+ 17915904a1b1+ (−3732480y2a1
2−746496y2b1
2+ 4478976y2+ 71663616ya1−1074954240a1 2− 214990848b1
2+ 1863254016)t, b2= (4320y8+ 1990656y6+ 716636160y4+ 165112971264y2+ 29720334827520)t4+ 2799360a1
4+ 3359232a1 2b1
2+ 559872b1
4+ (−746496y5b1+ 1492992y5− 286654464y3b1+573308928y3−61917364224yb1+123834728448y)t3+(−466560y4a1
2−93312y4b1 2+ 559872y4+ 17915904y3a1−214990848y2a1
2−71663616y2+ 5159780352ya1−38698352640a1 2− 7739670528b1
2+87716265984)t2−8398080a1
2−8398080b1
2+(4478976y2a1b1+26873856ya1 2b1+ 8957952yb1
3−100776960yb1+ 1289945088a1b1+ 53747712y)t−16796160, b1 = (144y10+ 41472y8+ 23887872y6+ 6879707136y4+ 990677827584y2+ 285315214344192)t5+ (−41472y7b1+ 82944y7−11943936y5b1+ 23887872y5−3439853568y3b1+ 6879707136y3−990677827584yb1+ 1981355655168y)t4−4478976a1
3b1−4478976a1b1
3+ (−25920y6a1
2−5184y6b1
2+ 31104y6+ 1492992y5a1−13436928y4a1
2+4478976y4b1
2−50761728y4+573308928y3a1−3869835264y2a1 2+ 1289945088y2b1
2−4299816960y2+ 123834728448ya1−619173642240a1
2−123834728448b1 2+ 1733686198272)t3+(373248y4a1b1+4478976y3a1
2b1+1492992y3b1
3−16796160y3b1+1289945088ya1 2b1+ 429981696yb1
3+8957952y3−3117367296yb1+30958682112a1b1−859963392y)t2+(466560y2a1 4+ 559872y2a1
2b1
2+93312y2b1
4−1399680y2a1
2−1399680y2b1
2−17915904ya1
3−53747712ya1b1 2+ 134369280a1
4+ 161243136a1 2b1
2+ 26873856b1
4−2799360y2+ 67184640ya1−618098688a1 2− 188116992b1
2−2096160768)t, b0= (2y12+ 497664y8+ 41278242816y4+ 1141260857376768)t6− 46656a1
6−139968a1 4b1
2−139968a1 2b1
4−46656b1
6+ (−864y9b1+ 1728y9−143327232y5b1+ 286654464y5−5944066965504yb1+ 11888133931008y)t5+ (−540y8a1
2−108y8b1
2+ 648y8+ 41472y7a1−248832y6a1
2+248832y6b1
2−2654208y6+11943936y5a1−89579520y4a1
2−17915904y4b1 2− 179159040y4+ 3439853568y3a1−20639121408y2a1
2+ 20639121408y2b1
2+ 990677827584ya1− 3715041853440a1
2−743008370688b1
2+12383472844800)t4+279936a1
4+2519424a1 2b1
2+2239488b1 4+ (10368y6a1b1+ 186624y5a1
2b1+ 62208y5b1
3−699840y5b1−8957952y4a1b1+ 71663616y3a1 2b1− 23887872y3b1
3+373248y5+179159040y3b1−2579890176y2a1b1+15479341056ya1
2b1+5159780352yb1 3− 16769286144yb1+247669456896a1b1−51597803520y)t3+(19440y4a1
4+23328y4a1 2b1
2+3888y4b1 4− 58320y4a1
2−58320y4b1
2−1492992y3a1
3−4478976y3a1b1
2+ 8957952y2a1
4−8957952y2b1 4− 116640y4+5598720y3a1+22394880y2a1
2+165722112y2b1
2−429981696ya1
3−1289945088ya1b1 2+ 1612431360a1
4+1934917632a1 2b1
2+322486272b1
4−107495424ya1−9997074432a1
2+322486272b1 2− 26873856000)t2+ 734832a1
2+ 734832b1
2+ (−373248y2a1
3b1−373248y2a1b1
3−1119744ya1 4b1− 2239488ya1
2b1
3−1119744yb1
5+28553472ya1
2b1+33032448yb1
3−107495424a1
3b1−107495424a1b1 3+ 35271936yb1−429981696a1b1−33592320y)t−4199040
c6=−2985984, c5= (−1492992y2−429981696)t, c4= 17915904tyb1+(−311040y4−143327232y2− 25798901760)t2+11197440a1
2+2239488b1
2+2239488, c3= (−34560y6−17915904y4−5159780352y2− 825564856320)t3+ (5971968y3b1+ 1719926784yb1)t2+ (3732480y2a1
2+ 746496y2b1
2+ 746496y2+ 1074954240a1
2+ 214990848b1
2−71663616)t, c2 = (−2160y8−995328y6−358318080y4− 82556485632y2−14860167413760)t4−2799360a1
4−3359232a1 2b1
2−559872b1
4+ (746496y5b1+ 286654464y3b1+ 61917364224yb1)t3+ (466560y4a1
2+ 93312y4b1
2+ 93312y4+ 214990848y2a1 2+
