A non-local approach to waves of maximal height for the Degasperis-Procesi equation

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

A non-local approach to waves of maximal height for the Degasperis-Procesi equation

MathiasNikolai Arnesen

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received20November2018 Availableonline7June2019 Submittedby T.Yang

Keywords:

Degasperis-Procesi Globalbifurcation Peakedwaves

Weconsiderthenon-localformulationoftheDegasperis-Procesiequationut+uux+ L(³

2u²)x = 0,where Lis thenon-local Fouriermultiplier operator with symbol m(ξ)= (1+ξ²)⁻¹.Weshow thatall L^∞,pointwisetravelling-wavesolutionsare boundedabovebythewave-speedandthatifthemaximalheightisachievedthey arepeakedatthosepoints,otherwisetheyaresmooth.Forsuﬃcientlysmallperiods weﬁndthehighest,peaked,travelling-wavesolutionasthelimitingcaseattheend ofthemainbifurcationcurveofP-periodicsolutions.Theresultsimplythatthere arenoL^∞travellingcusponsolutionstotheDegasperis-Procesiequation.

1. Introduction

Weconsidertheequation

ut+uux+ (L(3

2u²))x= 0, x∈R, t∈R, (1.1)

whereuisascalarfunctionandListhenonlocaloperatorL= (1−∂_x²)⁻¹.Thatis, Lf=K∗f, K=F⁻¹m,

wherem(ξ)= (1+ξ²)⁻¹ and F denotes theFouriertransform.Equation(1.1) isthenonlocalformulation oftheDegasperis–Procesiequation [5]

ut−uxxt+ 4uux−3uxuxx−uuxxx= 0, (1.2) which caneasily be seenby applying the inverseoperator of L, 1−∂_x², to (1.1). The Degasperis–Procesi equation wasdiscoveredas oneofthree equationswithinacertainclass ofthirdorder PDEssatisfyingan asymptoticintegrabilityconditionupto thirdorder, theother twobeingtheKdVandtheCamassa–Holm

E-mailaddress:[email protected].

https://doi.org/10.1016/j.jmaa.2019.06.014

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equations [5].Likethese twoequations, theDegasperis–Procesi equationhasaLaxpair,abi-Hamiltonian structure,andaninﬁnitenumberofconservationlaws[4].Whileitwasdiscoveredsolelyforitsmathematical properties, it has later been rigorously derived as a model for the propagation of shallow water waves, having the same asymptotic accuracy as the Camassa–Holm equation [3]. The Degasperis–Procesi and Camassa–HolmequationsfeaturestrongernonlineareﬀectsthantheKdVequation(orrather,thedispersion is much weaker), making them better suited to modelling nonlinear phenomena like wave breaking and solutionswithsingularities,whilemaintainingtherichmathematicalstructurementionedabovethatother weakly-dispersivemodels liketheWhithamequation[11] lack.

Shortly after its discovery, the well-posedness of (1.1) was extensively studied, establishing that it is locally well-posed in H^s both on R and S for s > 3/2, and admitting both global classical and weak solutions and classical solutions that blow up in ﬁnite time [13], [12], [14]. Moreover, the blow-up only occurs as wave-breaking. Thatis,thesolution remainsbounded, butit’sslope goesto −∞; foradetailed study oftheblow-upfor(1.1),see[8] andreferencestherein.

The weak dispersion allows not only for wave-breaking, but also for waves with singularities in the form of sharp crestsat thewave-peaks. Indeed, explicitpeaked soliton solutionsto (1.2),as well as mul- tipeakon solutions which arenot travelling waves,are known[4]. These are of thesame form as the ones forCamassa-Holm equation[2],andindeedeveryequation intheso-called‘b-family’of equationsthatthe Degasperis-Procesiand Camassa-Holmequationsbelongtohassuchsolutions[4].

In this paper we will focus on travelling-wave solutions to (1.1). Assuming u(x,t) = ϕ(x−μt) is a travelling wave,whereμ∈R isthewave-speed,(1.1) takestheform

−μϕ+1 2ϕ²+3

2L(ϕ²) =a, (1.3)

where a ∈ R is a constantof integration.By a Galilean change of variables this is equivalent to −μϕ+

1

2ϕ²+ ³₂L(ϕ²+kϕ)= 0, where k depends on μ and a; inparticular, k = 0 fora = 0.Hence there is no Galilean changeof variables thatremoves awhile preserving theform of theequation. Wewill workwith theequation intheform(1.3).

From the structure of the equation it is readily deducible that all non-constant solutions to (1.3) are smoothexceptpotentially atpointswherethewave-heightequals thewave-speed(cf.Theorem3.3or [10]) andsingularitiescanonlyoccurintheformofsharpcrestswithheightequaltothewave-speed.Wetherefore call suchsolutionsforwavesofmaximal height.Inthis paperwewill studytheregularity andexistenceof travelling wavesof maximalheightto (1.3) fromanonlocalperspective.

Themotivationofthispaperistwo-fold:toprovidenovelinformationaboutwavesofmaximalheightfor theDPequationspecificallyandtobetterunderstandtheformationofhighestwavesandtheirsingularities fornonlineardispersiveequationsmoregenerally.Wethereforeconsiderthenon-localformulationandfollow thegeneralframeworkof[7] and[6].Weshow firstlythatany non-constantL^∞solutionof(1.3) ispeaked wherever the maximal height is achieved.That is, it is Lipschitz continuous at the crest, but not C¹. In particular this means that there are no cuspon solutions of (1.3) in L^∞. This is due to the smoothing effectof L, whichforces anysolutionto be at leastLipschitz; seeRemark3.6. Therestrictionto bounded solutions is quite natural. While equation (1.3) makes sense for any ϕ ∈ H⁻²(R), if we exclude purely distributional solutions, any function solving (1.3) a.e. clearly belongs to L^∞. Secondly, for sufficiently small periods,peaked solutions of (1.3) arefound as the limitingcase at the end of themain bifurcation curve of C_even^α (SP) solutionsfor α ∈(1,2). While it has been established thatthere are peaked periodic travelling-wavesolutionsto(1.2) forallnon-zerowavespeedsin[10],theapproachofthatpaperworksonly for thelocalformulationand cannotbe extendedtoagenuinelynon-localequation. Moreoverthemethod in[10] andtheoneusedinthispaperareentirelydifferent andgivedifferentinsightandinformation.

AstravellingL^∞cusponsolutionsto(1.2) havebeenclaimedbyseveralauthors,ourclaimthattheydo notexists requires somecomment.Thecuspons areinvariablyfoundstudying thelocalequation andthey

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are strong solutionsin allpoints except the cusps. The exclusionof the cusps makes acrucial diﬀerence, however.Consider forinstance the stationary cuspedsoliton u(x)=√

1−e⁻²^|^x^|discovered in[15], which isapointwisesolutionto (1.2) atallpointsexcept 0,where thefunctionhasacusp. Foranytest function ϕ∈C₀^∞(R),treatingtheleft-handside of(1.2) as adistribution(note thatuisindependentof time),one canwithbasiccalculusshowthat

4uu_x−3u_xu_xx−uu_xxx, ϕ=u²,1

2ϕ_xxx−2ϕ_x=

R

u² 1

2ϕ_xxx−2ϕ_x

dx= 2ϕ_x(0)

andhenceitisnotaweaksolutionto (1.2),butratherto

u_t−u_xxt+ 4uu_x−3u_xu_xx−uu_xxx=−2δ,

where δ is the usual delta-distribution.This is the case with all cuspons of the DP equation - thereare pointmassdistributionsatthecusps. Toacceptanyfunctionthatsolvestheequation pointwiseatallbut a countable number of points as a solution is equivalent to claiming that the sawtooth function u(x) = x−ﬂoor(x),orindeedanypiece-wiselinearfunction,isasolutionto theequation

u(x) = 0, x∈R.

Hencewe thinkitmorecorrect to callthe cuspons solutionsnotof(1.2) with 0 right-hand side,butwith somepointmassdistributions.

The paper is structured as follows: ﬁrst some essential properties of the operator L and its kernel K are recounted in Section 2. In Section 3 we establish some general results about solutions to (1.3) and, in particular, using the properties of K, study the behaviour around points of critical height and prove Theorem 3.5, stating that any even, nonconstant solution is peaked at points where ϕ = μ. Lastly, in Section 4 we use the bifurcation Theory of [1] to construct a global bifurcation curve of even, periodic solutions in C^α for α ∈ (1,2). Using the properties of solutions established in Section 3, we show that forsuﬃcientlysmall periodsthesolutionsalong thecurveconvergeto aneven,non-constantsolutionthat achievesthemaximalheight andmustthereforebeapeakon.

2. TheoperatorLanditskernel

AsLf(ξ) = (1+ξ²)⁻¹f(ξ),Lf canformallybeexpressedas aconvolution Lf(x) =K∗f(x) =

R

K(x−y)f(y) dy,

whereK(x) istheinverseFouriertransformofm(ξ).Inthiscase,anexplicitexpressioniswellknownfrom virtuallyanytextbook onFourieranalysis:

K(x) =F⁻¹((1 +ξ²)⁻¹) = 1

2e^−|x|. (2.1)

Inparticular,wenotethatKiscompletelymonotoneon(0,∞);itispositive,strictlydecreasingandstrictly convexforx>0.

Theperiodickernel is

KP(x) =

n∈Z

K(x+nP),

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for P ∈(0,∞).Forx∈(−P/2,P/2),K(x+nP)= ¹₂e^−|x+nP^|= ¹₂e^−xe^−nP forn≥1,andK(x+nP)=

1

2e^xe^nP forn≤ −1.Thus

K_P(x) =

n∈Z

K(x+nP)

=1

2e^−|^x^|+1

2(e^x+ e⁻^x) ∞ n=1

e⁻^nP

=1

2e^−|^x^|+ cosh(x) 1

e^P−1. (2.2)

Forperiodicfunctions,theoperatorLisgiven byLf(x)=P /2

−P /2K_P(x−y)f(y)dy.

Weconcludethissectionwitharatherobvious,butcruciallemma:

Lemma 2.1.L isstrictly monotone:Lf > Lgif f andg are boundedandcontinuousfunctions withf g.

Proof. Letf andgbeasinthestatementofthelemma.AsKisstrictlypositive,wegetthatforallx∈R, K(x− ·)(f−g)0 andbycontinuitystrictlypositiveonaset ofnon-zeromeasure.Hence

Lf(x)−Lg(x) =

R

K(x−y)(f(y)−g(y)) dy >0.

Clearly, thesameargumentholdsforK_p. 2 3. Periodictravellingwaves

Notethatifϕ(x) isatravellingwavesolutionto(1.1) withwave-speedμ,then−ϕ(−x) isalsoatravelling solutionto (1.1) withwave-speed−μ.Wewill thereforeonlyconsiderμ>0.

Firstweinvestigatehowtheparameterain(1.3) inﬂuencesthebehaviour/existenceofsolutions.

Theorem3.1. Fix μ>0andP <∞.Forallvaluesofa∈R,non-constantP-periodicsolutions to(1.3)(if they exist)satisfy

minϕ < μ+ μ²+ 8a

4 <maxϕ.

Moreover,

(i) Fora≤0,allsolutions are non-negative. When a<−^μ₈² thereare no real solutions andfor a=−^μ₈² there isonlytheconstant solution ϕ=^μ₄,

(ii) there areonly constant solutionswhen a≥μ².

Proof. Atany pointxwhereϕ(x)²=L(ϕ²)(x)=:R²,(1.3) reducesto R(2R−μ) =a,

which has the positive solution R = ^μ+ ^μ₄²^+8a. As L(c) = c for constants and L is strictly monotone (Lemma 2.1),there has to exist points where ϕ² < L(ϕ²) andpoints where ϕ² > L(ϕ²) for non-constant P-periodicsolutionsϕ.Thustheﬁrstinequalityhastoholdifmaxϕ>|minϕ|.

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Considerﬁrstthecasea≤0.Thenϕcannotbenegativeinanypointasthentheleft-handsideof(1.3) wouldbestrictlypositiveinthatpoint(Lfisnon-negativeiff isnon-negative). Letm= minϕ≥0.Then L(ϕ²)≥m²withequality ifand onlyifϕ≡m.Hence,ifϕisasolutionto(1.3),weget

m(2m−μ)≤a.

Fora<−^μ₈² this hasnoreal solutions,and for a=−^μ₈² this hasonly theconstantsolutionϕ= ^μ₄. This proves(i).

Now leta>0.Assumethatϕ<0 onsomeintervals. ByTheorem 3.3, ϕissmooth onthese intervals.

Clearly, ϕ is bounded below, so there is a point x0 such that ϕ(x0) = minϕ. Then L(ϕϕ)(x0) = 0 and L(ϕ²) attains it minimum at x0. This implies that ϕ also has to be positive at some point, and M := maxϕ>|minϕ|.Thustheﬁrstinequalityholds andM > ^μ+ ^μ₄²^+8a.Inparticular, thismeansthat maxϕ≥^μ₂ foralla≥0 andM >√

aifa< μ².Wehavethat

(ϕ−μ)²=μ²+ 2a−3L(ϕ²). (3.1)

Assume a ≥ μ². Note that if ϕ = μ at any point, then 3L(ϕ²) = μ²+ 2a ≥ 3μ² at those points. If a = μ², then the constant solution ϕ ≡μ is avalid solution, otherwise Lemma2.1 implies thatϕ must alsotakevaluesaboveμ.Assumeϕμisanon-constantsolution.Thentheleft-handsideof(3.1) attains itsminimum where ϕis attainsitsminimum, while theright-handside attainsits minimumwhere L(ϕ²) attains its maximum. This is acontradiction. As both K and KP are even and completely monotone on (0,∞) and(0,P/2),respectively,L(ϕ²) cannotbe maximalwhereϕ² isminimal.

Assumenowthatϕtakesvaluesbothaboveandbelowμ.ThenL(ϕ²)(x) ismaximalwheneverϕ(x)=μ and 3L(ϕ²)(x)=μ²+ 2a these points.Moreover, 3L(ϕ²)< μ²+ 2a whenϕ> μ. This impliesthatthere are inﬁnitelymanydisjoint intervals, eachofﬁnite length,where ϕ> μ, andthatL(ϕ²) hasitsminimum oneachintervalatthepointswhere ϕismaximal. Thisisagainnotpossible. 2

Henceforthwewillassumethataissuchthatnon-constantsolutionsexists,i.e.that−μ²/8< a< μ². Theorem 3.2. Let P(0,∞]. Any P-periodic, non-constant and even solution ϕ ∈ BC¹(R) (the space of boundedfunctionswithboundedandcontinuousﬁrstderivative)thatisnon-decreasingon(−P/2,0)satisﬁes

ϕ>0, ϕ < μon(−P/2,0).

If ϕ∈BC²(R),then

ϕ(0)<0, and ϕ(0)< μ, andif P <∞,then

ϕ(±P/2)>0.

Proof. Letϕbeanon-constantandevensolutionthatisnon-decreasingon(−P/2,0).Wecanrewrite(1.3) as(μ−ϕ)²=μ²+ 2a−3L(ϕ²),andifϕ∈BC¹(R) wecandiﬀerentiate oneachside toget

(μ−ϕ(x))ϕ(x) =3

2L(ϕ²)(x). (3.2)

Asϕiseven,ϕ will beoddandusingtheevennessofK_P weget

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L(ϕ²)(x) = 2 0

−P /2

(KP(x−y)−KP(x+y))ϕ(y)ϕ(y) dy. (3.3)

WeclaimthatK_P(x−y)> K_P(x+y) foranyx,y∈(−P/2,0).Fixx∈(−P/2,0).AsK_P strictlydecreases from theoriginin(−P/2,P/2) and isP-periodic,theclaimfollowsif,forally∈(−P/2,0),

|x−y|<min{|x+y|, x+y+P}. As xandy aresame-signed,wehave

|x+y|=|x|+|y|>|x−y|.

Moreover, wehave−x< P +xand−y < P+y,so |x−y|= max{x−y,y−x}< P+x+y.Thisproves theclaim.

Now weclaimthatϕϕ0 on(−P/2,0).Byassumption ϕ0 onthisinterval. Ifa≤0,it isplainto seethatϕ>0 astherighthandsideof(1.3) isstrictlypositivewheneverϕ≤0.ForaBC¹(R) solutionthe same istrue when 0< a< μ² too; thisfollows from equation (4.14) in[10].Hence theintegrand in(3.3) isnon-negativeandstrictlypositiveonasetofpositivemeasurein(−P/2,0),anditfollowsthattheright handsideof(3.2) isstrictlypositiveforx∈(−P/2,0).This impliestheﬁrstpartofthestatement.

Assumenowthatϕ∈BC²(R).Thenwecandiﬀerentiateeachsideof(3.2) toget (μ−ϕ(x))ϕ(x)−(ϕ(x))²=3

2L(ϕ²)(x) = 3L(ϕϕ+ (ϕ)²)(x). (3.4) Evaluatingthisat x= 0 andusingtheevennessofϕ,ϕ,(ϕ)² andKP,weget

(μ−ϕ(0))ϕ(0) = 6 0

−P /2

KP(y)

ϕ(y)ϕ(y) +ϕ(y)² dy

= 6 [K_P(y)ϕ(y)ϕ(y)]^y=0_{y=−P /2}−6 0

−P /2

K_P (y)ϕ(y)ϕ(y) dy.

The ﬁrst termon thesecond line vanishesas ϕ(0) =ϕ(−P/2) = 0.If P =∞, then lim_y→−∞K(y)= 0 and wegetthesameconclusion.AsKP isstrictlyincreasingon(−P/2,0),wegetthattheﬁnalintegralis strictly positive.Thatis,

(μ−ϕ(0))ϕ(0) =−6 0

−P /2

K_P (y)ϕ(y)ϕ(y) dy <0

As wealreadyprovedthatϕ< μon(−P/2,0),itisnotpossible thatϕ(0)> μ,andthusweconcludethat ϕ(0)< μandϕ(0)<0.

NowweassumethatP <∞.NotethatKP(−P/2−y)=KP(P/2−y)=KP(−P/2+y)=KP(P/2+y).

Evaluating(3.4) atx=−P/2,weget

(μ−ϕ(−P/2))ϕ(−P/2) = 6 0

−P /2

K_P(P/2 +y)

ϕ(y)ϕ(y) +ϕ(y)² dy

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= [KP(P/2 +y)ϕ(y)ϕ(y)]^y=0_y=₋_{P /2}−6 0

−P /2

K_P (P/2 +y)ϕ(y)ϕ(y) dy.

As above,theﬁrst terminthe secondline vanishes. As KP isstrictly decreasingon (0,P/2),we getthat K_P (P/2+y)<0 fory∈(−P/2,0),anditfollowsthatthelastintegralisnegative.Thatis,

(μ−ϕ(−P/2))ϕ(−P/2) =−6 0

−P /2

K_P (P/2 +y)ϕ(y)ϕ(y) dy >0,

andweconcludethatϕ(−P/2)>0. 2 3.1. Singularity atϕ=μ

Nowweinvestigatewhathappensasasolutionapproaches μfrom below.Firstweshow thatasolution issmoothbelowμ:

Theorem3.3. Letϕ≤μbeasolution of (1.3).Then:

(i) If ϕ< μ uniformlyon R,thenϕ∈C^∞(R) andallofitsderivativesare uniformlyboundedon R.

(ii) If ϕ< μ uniformlyon Rand ϕ∈L²(R),thenϕ∈H^∞(R).

(iii) ϕissmoothon anyopen setwhereϕ< μ.

Proof. Assumeﬁrstthatϕ< μuniformlyonR.Notethatasϕ→ −∞,theleft-handsideof(1.3) goesto∞, henceϕmustbeboundedbelowaswell.Clearly,|m⁽ⁿ⁾(ξ)|(1+|ξ|)⁻²⁻ⁿ(thatis,misaS⁻²-multiplier)and ListhereforecontinuousfromtheBesovspaceB_p,q^s (R) toB^s+2_p,q (R) foralls∈Rand1≤p,q≤ ∞.Denoting byC^s(R),s∈RtheZygmundspaceB_∞,∞^s (R),wehaveinparticularthatLmapsL^∞(R)⊂B_∞,∞⁰ (R) into C²(R),andthereforeϕ→L(ϕ²) mapsL^∞(R) intoC²(R).Recallthatifs∈R₊\N,thenC^s(R)=C^s(R), theordinaryHölderspace,and ifs∈N thenW^s,^∞(R)C^s(R).

Asϕsolves(1.3) wehave

(ϕ−μ)²=μ²+ 2a−3L(ϕ²).

The assumption ϕ < μ therefore implies that 3L(ϕ²) < μ² + 2a, and the operator L(ϕ²) → μ − μ²+ 2a−3L(ϕ²) therefore maps B_p,q^s (R)∩L^∞(R) into itself for s >0. Since ϕ< μ, we also get that μ− μ²+ 2a−3L(ϕ²) =ϕ.Combiningthismapwithϕ→L(ϕ²) anditerating,weget(i).Whenp=q= 2, B_p,q^s (R) canbe identiﬁedwith H^s(R).Assumenow thatϕ∈L²(R) inaddition.As ϕisalso bounded,we getthatϕ²∈L²(R)∩L^∞(R),andingeneralϕ²∈H^s(R)∩L^∞(R) ifϕ∈H^s(R)∩L^∞,andthusϕ→L(ϕ²) maps H^s(R)∩L^∞(R) to H^s+2(R)∩L^∞(R),and we canapply the aboveiteration argument again.This proves(ii).

Lastly,toprove(iii),wenotethatifϕ∈L^∞(R) andCloc^s onanopensetU inthesense thatψϕ∈ C^s(R) for any ψ∈ C₀^∞(U), westill get thatL(ϕ) isCloc^s+2 in U (the proof ofthis is the sameas inTheorem 5.1 [7]).Thuswecanapply thesameiterationargumentas aboveagain. 2

The next lemma will be essential for showing that the global bifurcation curves do not converge to a trivialcase.

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Lemma 3.4. Let P < ∞, and let ϕ be an even, non-constant solution of (1.3) that is non-decreasing on (−P/2,0) withϕ≤μ.Then thereexists auniversal constant CK,P,μ>0,depending only on thekernel K and theperiodP andμ>0,suchthat

μ−ϕ(P

2)≥C_K,P,μ.

Proof. If ϕ(−P/2) = ϕ(P/2) < 0, the statement is true with C_K,P,μ = μ. Assume therefore that ϕ is non-negative.Fromtheevennessand periodicityofK_P andϕ,wegettheformula

L(ϕ²)(x+h)−L(ϕ²)(x−h)

= 0

−P /2

(KP(x−y)−KP(x+y))(ϕ(y+h)²−ϕ(y−h)²) dy. (3.5)

As ϕ ≥ 0 is non-decreasing, both factors in the integrand are non-negative for x ∈ (−P/2,0) and h ∈ (0,P/2).Wealsohavetheequality

(2μ−ϕ(x)−ϕ(y))(ϕ(x)−ϕ(y)) = 3

L(ϕ²)(x)−L(ϕ²)(y)

, (3.6)

which shows thatL(ϕ²)(x) =L(ϕ²)(y) whenever ϕ(x) =ϕ(y). As ϕ is assumed to be non-constant and non-negative, this identity together with (3.5) implies that ϕ is strictly increasing on (−P/2,0), and it therefore followsfromTheorem 3.3thatϕissmoothawayfromx=kP,k∈Z. Letx∈

−^3P₈ ,−^P₈ .Then forasolutionϕasintheassumptions,

(μ−ϕ(P

2))ϕ(x)≥(μ−ϕ(x))ϕ(x) =3 2 lim

h→0

L(ϕ²)(x+h)−L(ϕ²)(x−h)

4h .

As theintegrandin(3.5) isnon-negative forh∈(0,P/2) andnon-positiveforh∈(−P/2,0),wecanapply Fatou’slemma tothelimitaboveandweget

(μ−ϕ(P

2))ϕ(x)≥3

P /2

−P /2

KP(x−y)ϕ(y)ϕ(y) dy

= 3 0

−P /2

(KP(x−y)−KP(x+y))ϕ(y)ϕ(y) dy.

Assume foracontradiction thatthestatement isnottrue. Thenfor allk < μ theremust exist asolution ϕ satisfyingtheassumptions and suchthatk≤ϕ≤μ. Then μ−ϕ(P/2)< μ−k.On theotherhand, as K_P(x−y)> K_P(x+y) forx,y∈(−P/2,0),wegetthat

(μ−ϕ(P

2))ϕ(x)≥3 0

−P /2

(KP(x−y)−KP(x+y))ϕ(y)ϕ(y) dy

≥3k

−P /8

−3P /8

(KP(x−y)−KP(x+y))ϕ(y) dy.

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Thereisauniversalconstantλ˜_K,P >0 dependingonlyonK_P and P <∞suchthat min{K_P(x−y)−K_P(x+y) :x, y∈

−3P 8 ,−P

8

} ≥˜λ_K,P. Integratingbothsidesaboveoverx∈

−^3P₈ ,−^P₈

,wegetthat (μ−ϕ(P

2))(ϕ(−P/8)−ϕ(−3P/8))≥3kP

8˜λK,P(ϕ(−P/8)−ϕ(−3P/8)).

As shown aboveϕ is strictly increasing on (−P/2,0), so ϕ(−P/8) > ϕ(−3P/8) and we may divide out (ϕ(−P/8)−ϕ(−3P/8)) onboth sidestoget

(μ−ϕ(P

2))≥3kP 8

˜λ_K,P.

Thisimpliesthatμ−k≥3k^P₈λ˜K,P forallk < μ.Takingthelimitkμ,wegetacontradiction. 2 Nowwecome tothemainresultofthissection, concerningtheregularityatthepointwhere ϕ=μ.

Theorem3.5. Letϕ≤μbeasolutionof(1.3)whichiseven,non-constant,andnon-decreasingon(−P/2,0) with ϕ(0)=μ.Then:

(i) ϕissmoothon (−P,0).

(ii) ϕ∈C^0,1(R),i.e.ϕisLipschitz.

(iii) ϕisexactlyLipschitzatx= 0;that is, thereexistsconstants0< c1< c2 such that c1|x| ≤ |μ−ϕ(x)| ≤c2|x|

for|x|1.

Proof. Part(i)willfollowdirectlyfromTheorem3.3ifwecanshowthatϕ< μon(−P/2,0).Assumethat x0∈(−P/2,0] isthesmallestnumbersuchthatϕ(x0)=μ;asϕisassumedto benon-constant,itmustbe thecasethatx0>−P/2.Thenϕ(x)=μand L(ϕ²)(x)= 0 forx∈[x0,0].Thatis,

0

−P /2

(K_P (x−y) +K_P (x+y)) (ϕ(y))²dy= 0, x∈[x₀,0].

Clearly,0

−P /2(K_P (x−y) +K_P (x+y)) dy= 0, andas

K_P (x−y) +K_P (x+y)<0, −P/2< y < x <0, K_P (x−y) +K_P (x+y)>0, −P/2< x < y <0, we get thatx

−P /2(K_P (x−y) +K_P (x+y)) dy =−0

x(K_P (x−y) +K_P (x+y)) dy. Hence,by themean valuetheoremforintegrals,

L(ϕ²)(x₀) = 0

−P /2

(K_P (x₀−y) +K_P (x₀+y)) (ϕ(y))²dy

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=ϕ(c)²

x0

−P /2

(K_P (x0−y) +K_P (x0+y)) dy

+μ² 0

x0

(K_P (x0−y) +K_P (x0+y)) dy

= 0

x0

(K_P(x₀−y) +K_P (x₀+y)) dy(μ²−ϕ(c)²),

for some c ∈ (−P/2,x0). As −μ < ϕ < μ on (−P/2,0),we get (μ²−ϕ(c)²) >0, which is contradiction unless 0

x0(K_P (x0−y) +K_P (x0+y)) dy= 0. Thatcanonlyhappenifx0= 0.Thisproves part(i).

Atanypoint x₀ whereϕ(x₀)=μ,(3.6) reduces to (ϕ(x0)−ϕ(x))²= 3

(L(ϕ²)(x0)−L(ϕ²)(x)

. (3.7)

From (3.7),wegetinthereal linecasethat (ϕ(0)−ϕ(x))²=3

2

R

(2K(y)−K(x+y)−K(x−y))(ϕ(y))²dy

≤3 2

|y|<|x|

(2K(y)−K(x+y)−K(x−y))(ϕ(y))²dy

≤3

2ϕ²L^∞(R)

|y|<|x|

|2K(y)−K(x+y)−K(x−y)|dy, (3.8)

where we used thattheﬁrstintegral ontheright-hand sideis clearlynon-negative,while 2K(y)−K(x+ y)−K(x−y)<0 when|y|≥ |x|.Indeed,for|y|>|x|wecanexpandK(y+x) andK(y−x) aroundy and usetheLagrange remainderto get

2K(y)−K(x+y)−K(x−y) =−x²

2(K(ξ1) +K(ξ2))<0,

where ξ1∈(y,y+x),ξ2∈(y−x,y) andthelast inequalityfollows fromthestrictconvexityofK.

Similarly,expandingto onelessorder,weget

2K(y)−K(x+y)−K(x−y) =x(K(ξ1)−K(ξ2)).

As K isuniformlybounded, thereisaconstantC thatcanbechosenindependentlyofxsuchthat

|2K(y)−K(x+y)−K(x−y)| ≤C|x|, (3.9) forally∈R.Takingthesquarerootoneachsideof(3.8) wethengetthat

|ϕ(0)−ϕ(x)| ≤CϕL^∞(R)|x|=Cμ|x|.

ThisprovesthatϕisLipschitzat0.Fortheperiodickernel,wehavethat2K_P(y)−K_P(x+y)−K_P(x−y)<0 when|x|≤ |y|≤P/2− |x|(weareonlyinterestedinxcloseto0,sowecanassume|x|< P/2− |x|).Inthe intervals |y|<|x|andP/2− |x|<|y|≤P/2,(3.9) holdsforK_P andwethereforegetthesameresult.

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Itremainstoshowtheoppositeinequality,i.e.that|μ−ϕ(x)||x|nearx= 0;inparticularthisimplies thatϕ∈/C¹.Asϕissmoothon(−P/2,0) and(atleast)Lipschitzin0,wecanuseintegrationbypartsfor x∈(−P/2,0) toget

(μ−ϕ(x))ϕ(x) = 3

2L(ϕ²)(x)

= 3 2

0

−P /2

(K_P(x−y) +K_P (x+y)) (ϕ(y))²dy

= 3 0

−P /2

(K_P(x−y)−K_P(x+y))ϕ(y)ϕ(y) dy.

Asμ−ϕ(x)≤Cμ|x| forx∈(−P/2,0) asshownabove,we divideoutμ−ϕ(x):

ϕ(x)≥C 0

−P /2

KP(x−y)−KP(x+y)

|x| ϕ(y)ϕ(y) dy,

forsomeconstantC >0 independentof x.Letx∈(−P/2,0).Bythemeanvaluetheorem,

|μ−ϕ(x)|

|x| =ϕ(ξ)≥C 0

−P /2

KP(ξ−y)−KP(ξ+y)

|ξ| ϕ(y)ϕ(y) dy (3.10)

forsomeξ∈(x,0).Itsufficestoshowthatthisisboundedbelowbyapositiveconstantasx0,butwhile ϕ isdefinedforallx∈(−P/2,0),thelimitmaynotexist.Wethereforeconsiderthelimitinfimum.Onthe otherhand,thelimitoftheintegralontherighthandsideexists.Indeed,wehavethat

ξ0lim

KP(ξ−y)−KP(ξ+y)

|ξ| = 2K_P (y)

Thisfunctionisnon-negativeandstrictlymonotonicallyincreasingon(−P/2,0),andasϕisnon-decreasing on this interval, we get by Lebesgue’s dominated convergence theorem that for any sequence {ξ_n}n ⊂ (−P/2,0) suchthatξn →0,

n→∞limC 0

−P /2

KP(ξn−y)−KP(ξn+y)

|ξn| ϕ(y)ϕ(y) dy

=C 0

−P /2 nlim→∞

KP(ξn−y)−KP(ξn+y)

|ξ_n| ϕ(y)ϕ(y) dy

≥C 0

−P /2

ϕ(y)ϕ(y) dy

= C

2 (μ²−(ϕ(−P/2))²)>0.

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Inparticularthelimitexistsand thereforeequalsthelimitinﬁmumand from(3.10) itfollowsthatforany sequence {x_n}n ⊂(−P/2,0),andbysymmetryindeedanysequencein(−P/2,P/2),suchthatx_n→0,

lim inf

n→∞

|μ−ϕ(xn)|

|x_n| 1.

As thesequencewasarbitrarythis proves(iii).

Sinceϕ∈L^∞(R) issymmetricandϕ≥0,andthereforealsoL(ϕ²) ≥0,on(−P/2,0),wehavethatfor x<0

L(ϕ²)

(x) =

R

K(x−y)(ϕ(y))²dy

= 0

−∞

(K(x−y) +K(x+y))(ϕ(y))²dy

≤ 0

x

(K(x−y) +K(x+y))(ϕ(y))²dy

≤C|x|,

forsomeconstantC >0,whereweusedthatKiscompletelymonotoneon(0,∞) andthattheintegrandis L^∞.Theresultsaboveimplythat(μ−ϕ(x))≥C|x|forsomeconstantC independentofxwhenϕ(x)> ^μ₄ and fromtheequation

(μ−ϕ(x))ϕ(x) = 3 L(ϕ²)

(x)≤min(L(ϕ²)(0), C|x|),

which holds for x ≤ 0, we then see that ϕ is uniformly bounded on the closed interval [−P/2,0] and therefore Lipschitz.Thisproves(ii). 2

Remark 3.6 (On cuspons).Theequality (3.7) holds when ϕ(x0) =μ for any solutionof (1.3), regardless of the integrationconstanta. Theproof aboveused thatϕis even, butthis isnotneeded to showthat a solutionisat leastLipschitz,as(3.7) holdsinanycase.Ifϕ∈L^∞(R),thenL(ϕ²)∈ C²(R),and(3.7) then impliesthatϕ∈C^1/2(R),andhenceL(ϕ²)∈C^5/2(R).Diﬀerentiatingboth sidesof(3.7),wethenget

|(ϕ(x₀)−ϕ(x))ϕ(x)||x|,

whichprovesthatϕisatleastLipschitzatanypointx0whereϕ(x0)=μ,andwehavethereforeshownthat there areno cuspedtravelling L^∞ solutionsfor the Degasperis-Procesiequation. We havenot yetproved thatasolutionthattouchestheline μexists,butanythatdowillbeLipschitz.

4. Globalbifurcation

In this section we will show that there are non-constantperiodic solutionswhich achieve the maximal height; i.e. periodic peakons. These will be obtained by constructing a curve of even, periodic smooth solutionsusing“standard”bifurcation theoryandshowingthatinthelimitofthecurvewegetapeakon.

Wethereforefixα∈(1,2) andconsiderC_even^α (S_P),thespaceofeven,real-valuedfunctionsonthecircleS_P offinitecircumferenceP >0 thatareα-timesdifferentiablewiththeαderivativebeingα− α-Hölder continuous. Themain pointis toworkwith regularitystrictly higherthan Lipschitz,i.e.α >1,andavoid

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integervaluesofαinorderto avoidthe Zygmundspaces whichdonotcoincidewith C^αwhen α∈Z(see theproofofTheorem 3.3).

From[10] weknowthattherearenoperiodicpeakonswhena= 0 in(1.3),onlyaone-parameterfamilyof smoothperiodicsolutionsandapeakedsolitarywave,andfora∈(−^μ₈²,0) thereareonlysmoothsolutions.

Asourﬁnalgoalistoﬁndabifurcationcurveofperiodicsolutionsthatconvergestoapeakedsolution,the casea≤0 isnotrelevantandhenceforthwewillonlyconsider a>0.

Remark4.1.Asonecaneasilycheck(followingtheprocedurebelow),fora= 0 onecandolocalbifurcation from thecurve(ϕ,μ)= (μ/2,μ) ofconstantsolutionsonlywhen theperiodis√

2π, butthis curvecannot beextendedtoaglobalone.Whena∈(−^μ₈²,0) alltheresultsregardingbifurcationbelowholdsforperiods 0< P <√

2πandwegetglobalbifurcationcurves.However,inthiscase√

−8a < μ<∞andtheequivalent of Lemma 4.8 does not hold.That is, we cannot preclude that alternative (ii) in Theorem 4.5 occurs by μ(s) approaching√

−8a.

Fixa>0 andletF :C_even^α (S_P)×R→C_even^α (S_P) betheoperatordeﬁnedby F(ϕ, μ) =μϕ−3

2L(ϕ²)−1

2ϕ²+a. (4.1)

Thenϕisasolutionto(1.3) withwave-speedμifandonlyifF(ϕ,μ)= 0.Therearetwocurvesofconstant solutionsF(ϕ(s),μ(s))= 0,namely(ϕ(s),μ(s))= (^s₄±^√^s²₄^+8a,s) foralls∈R.Thenegativeone,however, is not interesting as (1.3) has no non-positive solutions and therefore no curve of non-trivial solutions intersects it.Wetherefore takethecurve(ϕ(s),μ(s))= (^s₄+^√^s²₄^+8a,s) asourstartingpoint. Set

λ(μ) := μ

4+ μ²+ 8a 4 anddeﬁne

F(φ, μ) =˜ F(λ(μ)−φ, μ) = (λ−μ)φ+ 3λL(φ)−3

2L(φ²)−1

2φ². (4.2)

ThenF˜(0,μ)= 0 forallμ∈R,andletting

ϕ:=λ(μ)−φ, (4.3)

wehavethat

F˜(φ, μ) = 0⇔F(ϕ, μ) = 0.

Henceacurve(φ(s),μ(s)) alongwhichF˜= 0 givesrisetoacurveofsolutions(ϕ(s),μ(s)) to(1.3). Inthe sequel,ϕwillalwaysbedeﬁnedthrough(4.3).

Notethat

D_φF˜[0, μ] = (λ(μ)−μ) id +3λ(μ)L.

Whenμ²> awe havethat4λ> μwhileμ> λ,andas L(cos(p·)(x)=^cos(px)_1+p2 wegetthat

ker DφF[0, μ] =˜ {Ccos

4λ−μ μ−λ x

:C∈R}.

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Restricting to P-periodic functions, the kernel is one-dimensional if and only if

4λ−μμ−λ = ^2kπ_P for some k∈N.Clearly,

4λ(μ)−μ

μ−λ(μ) iscontinuous inμforμ∈(√

a,∞),strictly monotoneon thisinterval, bounded below by √

2,the bound being achieved in the limitas μ → ∞, and unboundedabove as μ² a. This means thatforeveryP >0 andeachk∈N such that ^2kπ_P >√

2,thereexists auniqueμ>√

asuchthat cos

4λ−μ μ−λx

∈C_even^α (SP).WhenP ≥√

2π,wegetthatk >1.

Theorem 4.2 (Local bifurcation). Fix a>0and P >0, andlet F andF˜ be deﬁned as in (4.1) and(4.2), respectively.Thenforeachk∈N suchthat ^2kπ_P >√

2,thereexistsauniqueμ_k∈(√

a,∞)suchthat (0,μ_k) is abifurcation pointfor F,˜ andhence (λ(μ_k),μ_k)is abifurcation pointforF.That is, there existsε>0 and ananalytic curve

s→(ϕ(s), μ(s))⊂C_even^α (SP)×(√

a,∞), |s|< ε, of nontrivial P/k-periodicsolutions, whereμ(0)=μ_k and

D_sφ(0) =−D_sϕ(0) = cos

4λ(μ_k)−μ_k μk−λ(μk)x

.

Proof. Itissuﬃcient toconsider k= 1 andP <√

2π.Asshownabove,thereexists auniqueμ∈(√ a,∞) such that ker D_φF˜[0,μ] is one-dimensional. The space C_even^α (S_P) has basis {cos(^2π_Pk·) : k ∈ N} and by straightforward calculationoneﬁndsthatD_φF˜[0,μ] mapsthebasiselement k= 1 tozerowhile allothers arepreservedmoduloaconstant.Thuscodim range DφF˜[0,μ]= 1 andDφF˜[0,μ] isFriedholmofindexzero.

TheresultnowfollowsfromTheorem8.3.1in[1].NotethatDsφ(0)=−Dsϕ(0) becauseDsμ(0)= ˙μ(0)= 0 (see (4.8) below). 2

Wewanttoextendthesebifurcation curvesglobally.Let U :={(ϕ, μ)∈C_even^α (S_P)×(√

a,∞) :ϕ < μ}, and

S :={(ϕ, μ)∈U :F(ϕ, μ) = 0}.

Inorder to establishTheorem4.5below;thatis,to extendthecurvesglobally, itsuﬃcesto establishthat

¨

μ(0)= 0 andthefollowing Lemma:

Lemma4.3.Whenever(ϕ,μ)∈S thefunctionϕissmooth,andboundedandclosedsubsetsofS arecompact in C_even^α (S_P)×(√

a,∞).

Proof. ThesmoothnesspartwasprovedinTheorem3.3.Recallfromtheproofofthattheoremthat(ϕ,μ)∈ S implies3L(ϕ²)< μ²+ 2aandhence

ϕ=μ− μ²+ 2a−3L(ϕ²)∈C_even^α+2(SP), asL:C^α→C^α+2and√

xisrealanalyticforx>0.LetE⊂SbeboundedandclosedintheC_even^α (S_P)×R topology. Then, as shown above,{ϕ: (ϕ,μ) ∈E}⊂C_even^α+2(S_P) is abounded subset. Bounded subsets of C_even^α+2(S_P) are pre-compact in C_even^α (S_P), hence any sequence {(ϕ_n,μ_n)}n ⊂ E has a subsequence that converges in theC_even^α (S_P)×R topology.As E is closed, thelimit must itselflie inE, proving thatE is compact. 2

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In order to establish the bifurcation formulas we will apply the Lyapunov-Schmidt reduction [9]. For simplicitywe considerthecaseP <√

2πand k= 1.Letμ^∗:=μ₁and φ^∗(x) := cos

2π P x

, (4.4)

andletfurthermore

M:={

k =1

a_kcos 2πkx

P

∈C_even^α (S_P)},

and

N := ker D_φF[0, μ˜ ^∗] = span(φ^∗).

ThenC_even^α (SP)=M⊕N andwecanusethecanonicalembeddingC^α(SP)→L²(SP) todeﬁneacontinuous projection

Πφ=φ, φ^∗L²(SP)φ^∗, (4.5)

whereu,vL²(SP)= _P² P /2

−P /2uvdx.

Theorem4.4(Lyapunov-Schmidtreduction[9]). ThereexistsaneighbourhoodO×Y ⊂U around (0,μ^∗)in whichtheproblem

F˜(φ, μ) = 0 (4.6)

isequivalent to

Φ(εφ^∗, μ) := Π ˜F(εφ^∗+ψ(εφ^∗, μ), μ) = 0 (4.7) forfunctionsψ∈C^∞(ON×Y,M),Φ∈C^∞(ON×Y,N),andON⊂N an openneighbourhood ofthezero function in N. One has Φ(0,μ^∗) = 0, ψ(0,μ^∗) = 0, D_φψ(0,μ^∗) = 0, and solving the ﬁnite dimensional problem(4.7)provides asolution φ=εφ^∗+ψ(εφ^∗,μ) totheinﬁnitedimensional problem(4.6).

Wewanttoshowthatμ(ε) isnotconstantaround0.Wecalculate D²_φφF˜[0, μ^∗](φ^∗, φ^∗) =−(φ^∗)²−3L((φ^∗)²),

D²_μφF˜[0, μ^∗]φ^∗= (λ(μ^∗)−1)φ^∗+ 3λ(μ^∗)L(φ^∗).

AsL(cos(p·))(x)= _1+p¹2cos(px) forp= 0,wegetthat D²_μφF˜[0, μ^∗]φ^∗=

λ(μ^∗)(1 + 3

1 + (2π/P)²)−1

φ^∗.

Bychoice,

4λ(μ^∗)−μ^∗

μ^∗−λ(μ^∗) = ^2π_P,sothatthecoeﬃcientofφ^∗ aboveiszeroifandonlyif λ(μ^∗) = λ(μ^∗)

μ^∗ .