Rational solutions to the Gardner
equation from particular polynomials
PIERRE GAILLARD
Institut de math´ematiques de Bourgogne,
Universit´e de Bourgogne Franche Comt´e,
9 avenue Alain Savary BP 47870
21078 Dijon Cedex, France :
FRANCE
Abstract Rational solutions to the Gardner (G) equation are constructed in
terms of a quotient of determinants involving certain particular polynomials.
This gives a very efficient method to construct solutions to this equation. We
construct very easily explicit expressions of these rational solutions for the first
orders for n= 1 until 8.
Key-Words:Gardner equation, rational solutions.
Received: October 23, 2022. Revised: February 20, 2023. Accepted: March 17, 2023. Published: May 17, 2023.
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DOI: 10.37394/232021.2023.3.2
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E-ISSN: 2732-9976
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1 Introduction
We consider the Gardner equation in
the following normalization
ut+6u(u−2)ux+uxxx +6ux= 0,(1)
where the subscripts xand tstand for
partial derivatives.
This equation was introduced by Gard-
ner [1] in 1968. He considered this equa-
tion as an auxiliary mathematical tool
in the derivation of the infinite set of
local conservation laws of the Korteweg
de Vries equation. This equation is
used to describe nonlinear wave effects
in several physical contexts: for exam-
ple, in plasma physics [2, 3], fluid flows
[4], quantum fluid dynamics [5], in ocean
and atmosphere [6].
A lot of research has been done to solve
this equation. We can quote many meth-
ods as the Hirota method [8], the se-
ries expansion method [9], the map-
ping method [10] or the method of leading-
order analysis [11].
Here, we used particular polynomials
to construct rational solutions to the
Gardner equation. We obtain rational
solutions as a quotient of polynomials
in xand t. We give explicit solutions
for the first orders.
2 Rational solutions to
the Gardner equation
We consider the following polynomials
pn(x, t) defined by
pn(x, t) = Pn
l=0
xk
k!
t
n−k
3
n−k
3!1−1
2(n−k+ 1 −3n−k
3,k≥0,
pn(x, t) = 0, n < 0,
(2)
with [x] denoting the largest integer
less or equal to x.
We denote An(x, t) the determinant de-
fined by
An(x, t) = det(pn+1−2i+j(x, t)){1≤i≤n, 1≤j≤n}(3)
With these notations we have the fol-
lowing result
Theorem 2.1 The function vn(x, t)de-
fined by
vn(x, t) = 1 −∂Xln An+1(X, T )
An(X, T )|X=ix,T =4it
(4)
is a rational to the Gardner equation
(1)
ut+ 6u(u−2)ux+uxxx = 0,
Proof : We have proven in [12] that
the function defined by
un(X, T ) = ∂Xln An+1(X, T )
An(X, T )
is a solution the equation defined by
4uT+ 6u2uX−uXXX = 0.(5)
It is easy to verify that the function
defined by ˜unwhere Tis replaced by
4it and Xreplaced by ix is a solution
of the equation
ut+ 6u2ux+uxxx = 0.(6)
Then it is easy to check that the func-
tion defined by vn(x, t)=1−˜
un(x, t)
is a solution to the Gardner equation
and we get the result.
2
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3 Explicit construction
of rational solutions
to the Gardner equa-
tion for the first or-
ders
The solutions we present in the follow-
ing are singular. There is therefore no
interest in constructing figures in the
plan (x, t) of the coordinates.
3.1 First order rational so-
lution
The function v1(x, t) defined by
v1(x, t) = −−x4−12 xt −2ix3+ 12 it
x(x3+ 12 t)(7)
is a rational to the Gardner equation
(1).
3.2 Second order rational so-
lution
The function v2(x, t) defined by
v2(x, t) = −i−72 ix6t+ 8640 it3−ix9+ 72 x5t+ 4320 x2t2+ 3 x8
(−60 tx3+ 720 t2−x6) (x3+ 12 t)(8)
is a rational to the Gardner equation
(1).
3.3 Third order rational so-
lution
The function v3(x, t) defined by
v3(x, t) = n(x, t)
d(x, t)(9)
with
n(x, t) = −x16 −4ix15 −240 tx13 −
600 ix12t−10080 t2x10 −36000 ix9t2−
172800 t3x7+2419200 it3x6−18144000 t4x4+
36288000 it4x3+217728000 t5x+217728000 it5
and
d(x, t) = x9+ 180 tx6+ 302400 t3−x6−60 tx3+ 720 t2x
is a rational to the Gardner equation
(1).
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3.4 Rational solution of or-
der four
The function v4(x, t) defined by
v4(x, t) = n(x, t)
d(x, t)(10)
with
n(x, t) = −i(−ix25+5 x24−600 ix22t+
2280 tx21−100800 ix19t2+352800 t2x18
−6955200 ix16t3+4838400 t3x15−254016000 ix13t4+
2794176000 t4x12
+39626496000 it5x10+259096320000 t5x9−
365783040000 ix7t6+5120962560000 t6x6+
76814438400000 it7x4−153628876800000 t7x3+
460886630400000 it8x
+ 460886630400000 t8)
and
d(x, t) = (−x15−420 tx12−25200 t2x9−
2116800 t3x6+254016000 t4x3+1524096000 t5)
(x9+ 180 tx6+ 302400 t3)x
is a rational to the Gardner equation
(1).
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3.5 Rational solution of or-
der five
The function v5(x, t) defined by
v5(x, t) = n(x, t)
d(x, t)(11)
with
n(x, t) = −i(ix36−6x35+1260 itx33−
6300 tx32+544320 it2x30−2419200 t2x29+
110073600 it3x27 −381024000 t3x26 +
11430720000 it4x24−50295168000 t4x23+
329204736000 it5x21+192036096000 t5x20+
119062379520000 it6x18+1371137725440000 t6x17+
7189831434240000 it7x15+129970029772800000 t7x14+
1393721170329600000 it8x12−116143430860800000 t8x11−
59620294508544000000 it9x9+319394434867200000000 t9x8−
4292661204615168000000 it10x6+2146330602307584000000 t10x5
+128779836138455040000000 t11x2+128779836138455040000000 it12)
and
d(x, t) = (−x21−840 tx18−166320 t2x15−
16934400 t3x12+1397088000 t4x9−352066176000 t5x6−
14082647040000 t6x3+84495882240000 t7)(−x15−
420 tx12 −25200 t2x9−2116800 t3x6+
254016000 t4x3+ 1524096000 t5)
is a rational to the Gardner equation
(1).
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3.6 Rational solution of or-
der six
The function v6(x, t) defined by
v6(x, t) = n(x, t)
d(x, t)(12)
with
n(x, t) = −i(ix49−7x48+2352 itx46−
14448 tx45+2116800 it2x43−11733120 t2x42+
979655040 it3x40 −4792435200 t3x39 +
259756761600 it4x37−1190674598400 t4x36+
40391592192000 it5x34−148315878144000 t5x33+
4855696699392000 it6x31−8843902341120000 t6x30+
61513002270720000 it7x28−7215948343296000000 t7x27−
86555891849011200000 ix25t8−2329982402356224000000 t8x24
−29124513867423744000000 ix22t9−196587700520008089600000 t9x21
+125918062002044928000000 it10x19−
19426796003252994048000000 t10x18 −
26702499023308652544000000 ix16t11+
573430854357312602112000000 t11x15−
48092829805906034688000000000 ix13t12+
89978213950266259537920000000 t12x12−
1395306859308999990312960000000 ix10t13+
6295896804199146297753600000000 t13x9−
12251474862225365768601600000000 ix7t14−
171520648071155120760422400000000 t14x6−
857603240355775603802112000000000 ix4t15−
1715206480711551207604224000000000 t15x3+
5145619442134653622812672000000000 it16x−
5145619442134653622812672000000000 t16)
and
d(x, t) = (−x27−1512 tx24−680400 t2x21−
139708800 t3x18 −5029516800 t4x15 −
3168595584000 t5x12+604145558016000 t6x9+
108746200442880000 t7x6
+60897872248012800000 t9)(−x21−840 tx18−
166320 t2x15−16934400 t3x12+1397088000 t4x9−
352066176000 t5x6−14082647040000 t6x3+
84495882240000 t7)x
is a rational to the Gardner equation
(1).
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3.7 Rational solution of or-
der seven
The function v7(x, t) defined by
v7(x, t) = n(x, t)
d(x, t)(13)
with
n(x, t) = x64 + 8 ix63 + 4032 tx61 +
29232 itx60+6652800 t2x58+44331840 it2x57+
6008325120 t3x55+36696844800 it3x54+
3329540121600 t4x52+18820451865600 it4x51+
1197306651340800 t5x49+6162707171174400 it5x48+
293930755795353600 t6x46+1373247357176217600 it6x45+
46643754897653760000 t7x43+260442131005292544000 it7x42+
5156738132927496192000 t8x40+7389254129539645440000 it8x39
+ 1123297792337944903680000 t9x37 −
28613498613600193413120000 it9x36
+621308836469927647641600000 t10x34−
9639939765218166551347200000 it10x33+
96034709039084017916313600000 t11x31−
1768604112551541393063936000000 it11x30+
12747529948908855776772096000000 t12x28−
103264051547071763510722560000000 it12x27+
3002223914988325881595822080000000 t13x25−
3265018050783059977332326400000000 it13x24−
498353242970741203369407283200000000 t14x22
−2066613083889788928871869972480000000 it14x21
−104928442540121429745671444889600000000 t15x19
−464599008554226302534480717414400000000 it15x18
−5587991432946635075558051099443200000000 t16x16
+1181438340409670179520687741337600000000 it16x15
−211387398127111390864912290938880000000000 t17x13
−853337549281549667333619616579584000000000 it17x12
+12515617389462728454226421043167232000000000 t18x10
−77427124528032133657502435267051520000000000 it18x9
−76366478986552241415618840263393280000000000 t19x7
−1069130705811731379818663763687505920000000000 it19x6
+8018480293587985348639978227656294400000000000 t20x4
+16036960587175970697279956455312588800000000000 it20x3
+32073921174351941394559912910625177600000000000 t21x
−32073921174351941394559912910625177600000000000 it21
and
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d(x, t) = (−x36−2520 tx33−2162160 t2x30−
884822400 t3x27−163459296000 t4x24−
30207277900800 t5x21+4833164464128000 t6x18−
891718843631616000 t7x15−831255956185374720000 t8x12−
34833582925863321600000 t9x9
−2194515724329389260800000 t10x6+
131670943459763355648000000 t11x3+
526683773839053422592000000 t12)(−x27−
1512 tx24−680400 t2x21−139708800 t3x18−
5029516800 t4x15−3168595584000 t5x12+
604145558016000 t6x9+108746200442880000 t7x6+
60897872248012800000 t9)x
is a rational to the Gardner equation
(1).
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3.8 Rational solution of or-
der eight
The function v8(x, t) defined by
v8(x, t) = n(x, t)
d(x, t)(14)
with
n(x, t) = −i(−ix81+9 x80−6480 itx78+
54000 tx77−17962560 it2x75+139708800 t2x74−
28440720000 it3x72+206689190400 t3x71−
28894574016000 it4x69+196729549632000 t4x68−
20027877904742400 it5x66+127548405221529600 t5x65−
9845156013428736000 it6x63+58650450772193280000 t6x62−
3496233842718769152000 it7x60+19742220213202206720000 t7x59−
917522577057504337920000 it8x57+4584204344380840058880000 t8x56
−186047506160225456947200000 it9x54+
705970885405282482585600000 t9x53 −
25933156618443791880683520000 it10x51+
379336840256361840468295680000 t10x50+
6395652736671085473890304000000 it11x48+
297650067947402651243642880000000 t11x47+
5207428862069036677958467584000000 it12x45+
129905635334268499051254644736000000 t12x44+
1702586892784983334684867952640000000 it13x42
+29620639540599344304800019775488000000 t13x41
+396994691532759366938073508085760000000 it14x39
+4232360525213696990146612942602240000000 t14x38
−4349155069043839525357532208955392000000 it15x36
+583120649740812817456115098224427008000000 t15x35
−202466214174573975549688586510008320000000 it16x33
+28159381522332939098979530493552230400000000 t16x32
+5518612694466199647008064688942649180160000000 it17x30
−31710925844235863073697671356573181542400000000 t17x29
+1253795797863202772049566358144828624076800000000 it18x27
−3779312133537331858253773128621109542912000000000 t18x26
+127205858675790678540997355173101824901120000000000 it19x24
−454698326162344677609910772685294048116736000000000 t19x23
+1234128242496442531104836094950737862197248000000000 it20x21
−28547550061967276275702055938476943173746688000000000 t20x20
+446177931156674484955283193242186295080386560000000000 it21x18
+3908572835159623913132810908575137703584071680000000000 t21x17
+13662139312759241693421665342506948154118635520000000000 it22x15
+370790999888692634519688589966263523867203993600000000000 t22x14
+2648353159088714543647892051009039180644535500800000000000 it23x12
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−147130731060484141313771780611613287813585305600000000000 t23x11
−42483998593714795804351601651603336856172756992000000000000 it24x9
+333802846093473395605619727262597646727071662080000000000000 t24x8
−3058847898747465297913315318915440253644438503424000000000000 it25x6
+1019615966249155099304438439638480084548146167808000000000000 t25x5
+61176957974949305958266306378308805072888770068480000000000000 t26x2
+40784638649966203972177537585539203381925846712320000000000000 it27)
and
d(x, t) = (x45+3960 tx42+5821200 t2x39+
4324320000 t3x36+1743565824000 t4x33+
455986052121600 t5x30+33630769396224000 t6x27+
18378107874846720000 t7x24+3172344159319695360000 t8x21−
6261336530923932057600000 t9x18
−1088291697919577411420160000 t10x15−
98937546915666185433907200000 t11x12+
5121473016810955481284608000000 t12x9−
645305600118180390641860608000000 t13x6−
19359168003545411719255818240000000 t14x3+
77436672014181646877023272960000000 t15)
(−x36−2520 tx33−2162160 t2x30−884822400 t3x27−
163459296000 t4x24−30207277900800 t5x21+
4833164464128000 t6x18−891718843631616000 t7x15−
831255956185374720000 t8x12−34833582925863321600000 t9x9
−2194515724329389260800000 t10x6+
131670943459763355648000000 t11x3+
526683773839053422592000000 t12)
is a rational to the Gardner equation
(1).
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3.9 Rational solution of or-
der nine
The function v9(x, t) defined by
v9(x, t) = n(x, t)
d(x, t)(15)
with
n(x, t) = −i(−ix100+10 x99−9900 ix97t+
93060 tx96−43243200 ix94t2+383961600 t2x93−
111198700800 ix91t3+934226092800 t3x90−
189254729664000 ix88t4+1506596547456000 t4x87−
227051874137088000 ix85t5+1713219123103488000 t5x84−
199652557934269440000 ix82t6+1428145956200291328000 t6x81−
131959599374218936320000 ix79t7
+895581632952719032320000 t7x78−66760303356563349012480000 ix76t8
+428022377616371217039360000 t8x75−
26201771314573067845632000000 ix73t9+
158789776023912561028300800000 t9x72−
7982051135095428620156928000000 ix70t10+
48867778365532304849672601600000 t10x69−
1845794510242697463355146240000000 ix67t11+
10500563590228982713863477657600000 t11x66−
436938674857771455558389071872000000 ix64t12−
3509125889298278795399059734528000000 t12x63
−229767750578469269555278604402688000000 ix61t13
−6153349357243245450710701786005504000000 t13x60
−144981592655468437732757578564239360000000 ix58t14
−3665293540829092709903367569229742080000000 t14x57
−65471826215870206689389630766270382080000000 ix55t15
−1304243412541510083295718221838737735680000000 t15x54
−15948204848317269786001172326369984512000000000 ix52t16
−321470231876680131350631334233741176340480000000 t16x51
−3263409627924212769117367114275659710464000000000 ix49t17
−50226666131933164197071036030534503366656000000000 t17x48
−1109034109153602138195454830018333524361216000000000 ix46t18
−1648734229738630070957192326433181511188480000000000 t18x45
+27768030541570534401038610181385410880471040000000000 it19x43
−1744391947969980802122124052001220299533058048000000000 t19x42
+176353619079396164221683485553881739201481801728000000000 it20x40
−984738559457980653135258698718009747572885815296000000000 t20x39
+60347123414229766954512942201852809682107215380480000000000 it21x37
−276308694142269115155541508033547011849948428763136000000000 t21x36
+7917139362968738240500030294776989889075583679201280000000000 it22x34
−39520943879481443272459668542895966001157408009748480000000000 t22x33
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+861918461622108393759032387159223091252121997064273920000000000 it23x31
−201812703802412530614295675464043415116483567382691840000000000 t23x30
+37114684499860893874540947963358435349516821785162547200000000000 it24x28
−111009714025675678910786955318908309101987512458202316800000000000 t24x27
−790292787560605227039974112295286693718824812235383111680000000000 it25x25
−128446740342524481121949490167970826891662327175606330982400000000000 t25x24
−1402972306391365342605254033889010694950503223584607961088000000000000 ix22t26
−8451785841866764569439496907554855022602870712303714985574400000000000 t26x21
−18140732513551310434501648952412631403775273259595804442624000000000000 ix19t27
−696721788790016910954151103576762981348806810236689080909824000000000000 t27x18
+2034153160898066379192666232190869484574785313405769869164544000000000000 it28x16
+4912219331191612575642775040874437032348812776784915131793408000000000000 t28x15
−607396012532772883555258342267793701093434874630017594666516480000000000000 ix13t29
+901409771150358737265998828097675983701800352846971264533790720000000000000 t29x12
−13516110758478004753195200305107052180199062273973580301724549120000000000000 ix10t30
+65364797930344449216271870327976727756700383128232888344405606400000000000000 t30x9
−79767211033640683789348723112107193194617416698860473911817011200000000000000 ix7t31
−1116740954470969573050882123569500704724643833784046634765438156800000000000000 t31x6
−4187778579266135898940807963385627642717414376690174880370393088000000000000000 ix4t32
−8375557158532271797881615926771255285434828753380349760740786176000000000000000 t32x3
+16751114317064543595763231853542510570869657506760699521481572352000000000000000 it33x
−16751114317064543595763231853542510570869657506760699521481572352000000000000000 t33)
and
d(x, t) = (−x54−5940 tx51−13899600 t2x48−
17254036800 t3x45−12586365792000 t4x42−
5851581261926400 t5x39−1623511727405568000 t6x36−
400077271429355520000 t7x33−28551097433877258240000 t8x30+
50926698237612176179200000 t9x27
−42318665025272343181393920000 t10x24−
24094202601815176923316224000000 t11x21−
4866679734224610446090698752000000 t12x18−
89328732359216685504566132736000000 t13x15−
56277101386306511867876663623680000000 t14x12
+4292066932395643305123393545699328000000 t15x9
+772572047831215794922210838225879040000000 t16x6
+216320173392740422578219034703246131200000000 t18)(x45+
3960 tx42+5821200 t2x39+4324320000 t3x36+
1743565824000 t4x33+455986052121600 t5x30+
33630769396224000 t6x27+18378107874846720000 t7x24+
3172344159319695360000 t8x21−6261336530923932057600000 t9x18−
1088291697919577411420160000 t10x15−
98937546915666185433907200000 t11x12+
5121473016810955481284608000000 t12x9−
645305600118180390641860608000000 t13x6−
19359168003545411719255818240000000 t14x3+
77436672014181646877023272960000000 t15)x
EQUATIONS
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is a rational to the Gardner equation
(1).
3.10 Rational solution of or-
der ten
The function v10(x, t) defined by
v10(x, t) = n(x, t)
d(x, t)(16)
with
n(x, t) = −i(−ix121 +11 x120 −14520 ix118 t+
151800 tx117 −95135040 ix115 t2+948261600 t2x114 −
375171825600 ix112 t3+ 3570850483200 t3x111 −
1001486566080000 ix109 t4+9111786780096000 t4x108 −
1928677978260633600 ix106 t5+16783754184791193600 t5x105 −
2788145214059132928000 ix103 t6+23214265448682147840000 t6x102 −
3107387740090288951296000 ix100 t7+24758895105773203415040000 t7x99 −
2720957040436266106675200000 ix97 t8+20739568173400178020638720000 t8x96
−1897998004600345592443699200000 ix94 t9+13838512826996381618660966400000 t9x93
−1065031427670194137372455075840000 ix91 t10 +
7441832235420948554463447613440000 t10 x90 −
484359826252919742513781250457600000 ix88 t11
+3216285223633520093028577443840000000 t11 x87
−180973477671370782253997564834611200000 ix85 t12
+1094007551086362182820948800962560000000 t12 x84
−55662624659826717571311231389663232000000 ix82 t13
+366409302919024594307060726369353728000000 t13 x81
−10629210508837371167598914804499087360000000 ix79 t14
+287719612082605280136521050805250293760000000 t14 x78
+4280030807361509169541614389644146769920000000 it15 x76
+304538976982963877702699431050813489807360000000 t15 x75
+6885670109355945389014278608796839313408000000000 it16 x73
+228041011956866466611552032321683792592896000000000 t16 x72
+4453721520274708640987473009279000243077120000000000 it17 x70
+117762805364914466257694778298876385933918208000000000 t17 x69
+1928767694961190051291693412300125238030499840000000000 it18 x67
+42939910195222368442373794245841509282389950464000000000 t18 x66
+654735071078952292481291250564426986112984023040000000000 it19 x64
+11100640725265289005976333178685141575591650131968000000000 t19 x63
+118983885100292683006832864372628739280595937394688000000000 it20 x61
+2548058689427082387186572761563839262072167014072320000000000 t20 x60
−7153367620935353165234889584513031133610669475102720000000000 ix58 t21
+501342812165200991524948146064415704363708955523809280000000000 t21 x57
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+11220981111161432183874791372551241736238612859961999360000000000 it22 x55
−53280631666435839568505888704723585563633395222917939200000000000 t22x54
+18666731499151103582697435263135280572769467169281723596800000000000 it23 x52
−111900720524349972607748089158539708346975177833478881280000000000000 t23 x51
+8211604920987036524942436185081467249913411096884077920256000000000000 it24 x49
−47235626798516616642910711000941244534600220241264883689062400000000000 t24 x48
+2074576890338308833679718042344039076057947005326331519455723520000000000 it25 x46
−8871281053769860654870984551097381234761018804136244822535045120000000000 t25 x45
+310982628871002671709704390044044172790555050597378687653799526400000000000 it26 x43
−1310226464648804607175637470426054827417012493833765156242325504000000000000 t26 x42
+27593018623265107361540465787534690530315439226784893489415769292800000000000 it27 x40
−270396210866183602782620361062750448925713674136174283687987249152000000000000 t27 x39
+346126813079491553416389442370233142246940495924022083393612152832000000000000 it28 x37
+16944349808274198207103097908284912672054542890962731378372411129856000000000000 t28 x36
+867875156870281716096455706687283790071157369998346682307347008716800000000000000 it29 x34
+12152219828755012595427314762121360837004848698644507877113870260633600000000000000 t29 x33
+142716414561877226702889120299857500886877537625672562581184735988416512000000000000 it30 x31
+2468717573591209665318695663836278059479411539959616720916339262025105408000000000000 t30 x30
+7206589624665134114472628375800919394722083677444844476366967637656207360000000000000 it31 x28
+141388375243865028423165470229703381827467394179429146613239329827114188800000000000000 t31 x27
+2779987873793633994314858080971065091655145970397153221949879560965738987520000000000000 it32 x25
+2186347006955808854323160657473222559728478424077556347724728600882262835200000000000000 t32 x24
−90840229532843846042412137018193738079990797936534294316472658867592888320000000000000000 ix22 t33
+401214645216852730269554616818765276029818689646851534038793223244097362329600000000000000 t33 x21
−30685216390419131344239327546082995354047298725127290106307567225118146953216000000000000000 ix19 t34
+157627886390273508530607586958466768093765832388950910966517947023627068112896000000000000000 t34 x18
−1319763410247538789741875435615067271281418483444308067430020702793061740262195200000000000000 ix16 t35
−591801242565108195890928339048751084245952648706547487191551445014106805685452800000000000000 t35 x15
−44775481636106741230848667473679733960126499145132566887017527747258209311653888000000000000000 ix13 t36
+109248305855313099384476832466836725986968112280927016425655782661754772112015360000000000000000 t36 x12
+1639196163250654676807661020092335796535400683259722197534875541865868724581433344000000000000000 it37 x10
+9855926298026088246628341576504550675371080057574279035810960536535286635141529600000000000000000 t37 x9
−7469754667977666881655164142192922617123344885740506216614622722426743555054632960000000000000000 ix7t38
+104576565351687336343172297990700916639726828400367087032604718113974409770764861440000000000000000 t38 x6
+522882826758436681715861489953504583198634142001835435163023590569872048853824307200000000000000000 it39 x4
−1045765653516873363431722979907009166397268284003670870326047181139744097707648614400000000000000000 t39 x3
+1568648480275310045147584469860513749595902426005506305489070771709616146561472921600000000000000000 it40 x
+1568648480275310045147584469860513749595902426005506305489070771709616146561472921600000000000000000 t40 )
and
d(x, t)=(x66 + 8580 tx63 + 30270240 t2x60 +
58853995200 t3x57 +70523605152000 t4x54 +55594337313715200 t5x51 +
29373557030737920000 t6x48 +11136392862316001280000 t7x45 +
2669301014056143667200000 t8x42
+278616413032517777817600000 t9x39 +593719639698671917622231040000 t10 x36
−143134362516421637127025459200000 t11 x33 −
437557180951607550386188949913600000 t12 x30
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DOI: 10.37394/232021.2023.3.2
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E-ISSN: 2732-9976
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−208542888399944642126271001657344000000 t13 x27 −
30762135562543449987207439130296320000000 t14x24
−5486487844439459470606303635273940992000000 t15 x21
+384097069780086119375492488404632862720000000 t16 x18
+11977957029575169684473956835854028636160000000 t17 x15
−37241464731120798410643610795476150701260800000000 t18 x12
−1198993498660474485415843079268988266479616000000000 t19 x9
−60429272332487914064958491195157008630572646400000000 t20 x6
+2417170893299516562598339647806280345222905856000000000 t21 x3
+7251512679898549687795018943418841035668717568000000000 t22 )
(−x54 −5940 tx51 −13899600 t2x48 −17254036800 t3x45 −
12586365792000 t4x42 −5851581261926400 t5x39 −
1623511727405568000 t6x36 −400077271429355520000 t7x33 −
28551097433877258240000 t8x30 +50926698237612176179200000 t9x27 −
42318665025272343181393920000 t10 x24 −24094202601815176923316224000000 t11 x21
−4866679734224610446090698752000000 t12 x18 −
89328732359216685504566132736000000 t13 x15 −
56277101386306511867876663623680000000 t14 x12 +
4292066932395643305123393545699328000000 t15 x9
+772572047831215794922210838225879040000000 t16 x6+
216320173392740422578219034703246131200000000 t18 )x
is a rational to the Gardner equation
(1).
We could go on for greater orders but
the expressions of the solutions become
too long to be given in this text.
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DOI: 10.37394/232021.2023.3.2
Pierre Gaillard
E-ISSN: 2732-9976
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4 Conclusion
We have constructed rational solutions
to the Gardner equation and we obtain
an infinite hierarchy of families of ra-
tional solutions of this equation as a
quotient of a polynomials in xand t.
At order kthe numerator is a poly-
nomial of degree (k+ 1)2en xand of
degree (k+ 1)2
3in t, where as usual,
[x]denotes the bigger integer less or
equal to x.
The denominator is also a polynomial
of degree (k+ 1)2en xand of degree
(k+ 1)2
3in tat order k, with the
same notation as previously.
It will relevant to study in details the
structure of these polynomials.
References
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M.D. Kruskal, Korteweg-deVries
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[2] S. Watanabe, Ion Acoustic Soli-
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11, 2021
EQUATIONS
DOI: 10.37394/232021.2023.3.2
Pierre Gaillard
E-ISSN: 2732-9976
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Volume 3, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflicts of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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