On the smooth periodic traveling waves of the Camassa-Holm equation - PowerPoint PPT Presentation

On the smooth periodic traveling waves of the Camassa-Holm equation Anna Geyer a and Jordi Villadelprat b a Universitat Autonòma de Barcelona, Spain. b Universitat Rovira i Virgili, Tarragona, Spain. Workshop on non-local dispersive equations NTNU Trondheim, Norway September 2015

The Camassa-Holm equation 1 u t + 2 k u x − u txx + 3 u u x = 2 u x u xx + u u xxx (CH) arises as shallow water approximations of the Euler equations. It may be written in non-local form as � � 2 ku + u 2 + 1 x ) − 1 ∂ x u t + uu x + ( 1 − ∂ 2 2 u 2 = 0 . x 1 R. Camassa and D. Holm. An integrable shallow water equation with peaked solitons . Phys. Rev. Lett. (71) 1993.

Camassa-Holm equation u t + 2 k u x − u txx + 3 u u x = 2 u x u xx + u u xxx (CH) Traveling Wave Solutions 2 u ( x , t ) = ϕ ( x − c t ) : ϕ ′′ ( ϕ − c ) + ( ϕ ′ ) 2 + r + ( c − 2 k ) ϕ − 3 2 ϕ 2 = 0 , (1) 2 where c is the wave speed and r ∈ R is an integration constant. We will concentrate on smooth periodic TWS . λ . . . wave length a . . . wave height 2 J. Lenells. Traveling wave solutions of the Camassa-Holm equation . J. Differ. Equ. (217) 2005.

Proposition (Waves ← → Orbits) ◮ ϕ is a smooth periodic solution of (1) if and only if ( w , v ) = ( ϕ − c , ϕ ′ ) is a periodic orbit of the planar system  w ′ = v ,  v ′ = − A ′ ( w ) + 1 2 v 2 (2) ,  w where A ( w ):= α w + β w 2 − 1 2 w 3 . ◮ System (2) has the first integral H ( w , v ):= wv 2 + A ( w ) . 2 ◮ Every periodic orbit of (2) belongs to the period annulus P of a center, which exists if and only if − 2 β 2 < 3 α < 0 .

Observations: periodic solution ϕ ← → periodic orbit γ ϕ wave lenght λ of ϕ = period T of γ ϕ wave height a of ϕ = ℓ ( h ϕ ) , where ℓ is an analytic diffeo with ℓ ( h 0 ) = 0 . { ϕ a } a ∈ ( 0 , a M ) ← → { γ h } h ∈ ( h 0 , h 1 ) Consequence: λ ( a ) is a well-defined function → R + T : ( h 0 , h 1 ) → R + λ : ( 0 , a M ) − λ ( a ) = wave length of ϕ T ( h ) = period of γ ϕ Deduce qualitative properties of the function λ from those of the period function T . Result: λ ( a ) is either unimodal or monotonous.

Theorem (A.G. & J. Villadelprat, 2015) Given c , k, c � = − k, there exist real numbers r 1 < r b 1 < r b 2 < r 2 such that the Camassa-Holm equation u t + 2 k u x − u txx + 3 u u x = 2 u x u xx + u u xxx (CH) has smooth periodic TWS ϕ ( x − c t ) satisfying ϕ ′′ ( ϕ − c ) + ( ϕ ′ ) 2 + r + ( c − 2 k ) ϕ − 3 2 ϕ 2 = 0 , 2 if and only if the integration constant r ∈ ( r 1 , r 2 ) . The set of smooth periodic TWS form a continous family { ϕ a } a parametrized by the wave height a. The wave length λ = λ ( a ) of ϕ a satisfies the following: ◮ If r ∈ ( r 1 , r b 1 ] , then λ ( a ) is monotonous increasing. ◮ If r ∈ ( r b 1 , r b 2 ) , then λ ( a ) has a unique critical point (maximum). ◮ If r ∈ [ r b 2 , r 2 ) , then λ ( a ) is monotonous decreasing.

Criteria to bound the number of critical periods of a center for planar systems which have first integrals of a certain type: ◮ potential systems: H ( x , y ) = V ( x ) + 1 2 y 2 [F. Mañosas, J. Villadelprat, JDE (2009)] ◮ systems with quadratic-like centers: H ( x , y ) = A ( x ) + B ( x ) y + C ( x ) y 2 [A. Garijo, J. Villadelprat, JDE (2014)]

Consider an analytic differential system ˙ x = p ( x , y ) , ˙ y = q ( x , y ) satisfying these hypotheses : The system has a center at the origin, an analytic first integral of the form (H) H ( x , y ) = A ( x ) + B ( x ) y + C ( x ) y 2 with A ( 0 ) = 0 , and its integrating factor K depends only on x . The function V := 4 AC − B 2 defines an involution σ which satisfies 4 | C | V ◦ σ = V . For any analytic function f one can define its σ -balance as B σ ( f )( x ):= f ( x ) − f ( σ ( x )) . 2

Recall the σ -balance B σ ( f )( x ) = f ( x ) − f ( σ ( x )) of the involution σ 2 satisfying V ◦ σ = V for the function V := 4 AC − B 2 on ( x ℓ , x r ) . 4 | C | Theorem ([ GaVi14 ] 3 , THM A ( b ) ) Under hypotheses (H) let µ 0 = − 1 and define for i � 1 √ � � ′ � � | C | V K µ i − 1 1 1 √ √ K µ i := 2 + µ i − 1 + and ℓ i := | C | V ′ µ i 2 i − 3 ( 2 i − 3 ) K | C | V ′ If the number of zeros of B σ ( ℓ i ) on ( 0 , x r ) , counted with multiplicities, is n � 0 and it holds that i > n, then the number of critical periods of the center at the origin, counted with multiplicities, is at most n. 3 A. Garijo and J. Villadelprat, Algebraic and analytical tools for the study of the period function , J. Differential Equations 257 (2014) 2464–2484.

The system  w ′ = v ,  v ′ = − A ′ ( w ) + 1 2 v 2 (2) ,  w has the first integral H ( w , v ):= A ( w ) + w 2 v 2 . To apply the criterion of [ GaVi14 ], our system has to satisfy the following hypotheses : The system has a center at the origin, an analytic first integral of the form (H) H ( x , y ) = A ( x ) + B ( x ) y + C ( x ) y 2 with A ( 0 ) = 0 , and its integrating factor K depends only on x .

Lemma (Move center to origin) Let α and β satisfy − 2 β 2 < 3 α < 0 . Then the transformation � � x = w − w c v with ∆:= 4 + 6 α √ ∆ , y = √ β 2 , brings system (2) to 2 β 2 β ∆ x ′ = y ,   y ′ = − x − 3 x 2 + y 2 (3) ,  2 ( x + ϑ ) � � where ϑ := 1 2 ∆ − 1 is positive. √ 6 System (3) is analytic for x � = − ϑ and satisfies hypotheses (H) 2 x 2 − x 3 , B ( x ) = 0, C ( x ) = x + ϑ, K ( x ) = 2 ( x + ϑ ) . with A ( x ) = 1

Period Annuli  x ′ = y , 2 x 2 − x 3 , A ( x ) = 1  � � y ′ = − A ′ ( x ) + y 2 where ϑ := 1 2 − 1 . 2 ( x + ϑ ) ,  6 � 4 + 6 α β 2

Proposition (1) If ϑ � 1 6 , then the period function of the center of system (3) is monotonous increasing. Proof. ◮ Apply the criterion in [ GaVi14 ] for n = 0 to deduce monotonicity. � � � 0 , 1 We study the number of roots of B σ ℓ 1 ) on . 3

Proof. We compute ℓ 1 ( x ) = 1 ( 6 ϑ + 1 ) x − 4 ϑ − 1 √ x + ϑ ( 3 x − 1 ) 3 . 2 We are interested in the number of roots of = 1 � � ℓ 1 2 ( ℓ 1 ( x ) − ℓ 1 ( σ ( x ))) . B σ There exist L , S ∈ R [ x , y ] such that � � L x , ℓ 1 ( x ) ≡ 0 , � � S x , σ ( x ) ≡ 0 . Let z = σ ( x ) and R = Res z ( L ( x , z ) , S ( x , z )) . � � We show that R � = 0 on ( 0 , 1 3 ) which implies that B σ ℓ 1 � = 0. From ([ GaVi14 ], THM A) we conclude that the period function is monotonous.

Proposition (2) For ϑ < 1 6 the period function of the center of (3) is either monotonous decreasing for ϑ ∈ ( 0 , ϑ 1 ] or unimodal √ ϑ ∈ ( ϑ 1 , 1 / 6 ) , where ϑ 1 = − 1 10 + 1 6 . 15 Proof. ◮ Apply criterion in [ GaVi14 ] to obtain an upper bound for the critical periods. ◮ Compute the first period constants. ◮ Determine the sign of T ′ ( h ) for h ≈ h m .

Proposition (2) For ϑ < 1 6 the period function of the center of (3) is either monotonous decreasing for ϑ ∈ ( 0 , ϑ 1 ] or unimodal √ ϑ ∈ ( ϑ 1 , 1 / 6 ) , where ϑ 1 = − 1 10 + 1 6 . 15 � � � We study the number of roots of B σ ℓ i ) on − ϑ, 0 . Intervals ( 0 , ϑ 0 ) ( ϑ 0 , ϑ 1 ) ( ϑ 1 , 1 / 6 ) � � # roots of B σ ℓ 3 0 1 2 ⇓ [ GaVi14 ], THM A Period function T ( h ) monot. ≤ 1 crit. per. ≤ 2 crit. per.

Lemma Let an analytic differential system ˙ x = p ( x , y ) , ˙ y = q ( x , y ) satisfy the hypothesis (H) with H ( x , y ) = A ( x ) + C ( x ) y 2 . Let T ( h ) be the period of the periodic orbit γ h ⊂ { H = h } . Then � T ′ ( h ) = 1 R ( x ) dx y , h γ h � KA � ′ − K ( AC ) ′ 1 where R = 4 A ′ C 2 . 2 C A ′ Proof. dt = H y ( x , y ) = 2 C ( x ) y We have dx K ( x ) , and so K ( x ) � K � � � ( x ) dx T ( h ) = dt = y . 2 C γ h γ h

Proof of Lemma cont’. We have � K � � ( x ) dx T ( h ) = y . 2 C γ h Recalling that A ( x ) + C ( x ) y 2 = h on γ h we get � � KA � � � ( x ) dx � � 2 hT ( h ) = y + K ( x ) ydx = G + K ( x ) ydx , C γ h γ h γ h � KA � ′ − KAC ′ A ′ C , in view of [ GrMaVi11 ] 4 , Lemma 4.1. with G := 2 A ′ Now we apply Gelfand-Leray and obtain � � G + K � ( x ) dx � ′ = 2 hT ′ ( h ) + 2 T ( h ) = � 2 hT ( h ) y . 2 C γ h � G − K � � 1 ( x ) dx This implies that T ′ ( h ) = y , which proves the result. 2 h 2 C γ h 4 M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for abelian integrals , Trans. Amer. Math. Soc 363 (2011) 109–129.

Lemma (Gelfand-Leray derivative 5 ) Let ω and η be two rational 1 -forms such that d ω = dH ∧ η and let γ h ∈ H 1 ( L h , Z ) be a continuous family of cycles on non-critical level curves L h = { H = h } not passing through poles of neither ω nor η . Then � � d ω = η. dh γ h γ h 5 Y. Ilyashenko and S. Yakovenko, Lectures on analytic differential equations , Graduate Studies in Mathematics, AMS, Vol. 86 (2007)

Lemma ([ GrMaVi11 ] 6 , Lemma 4.1) Let γ h be an oval inside the level curve { A ( x ) + C ( x ) y 2 = h } and consider a function F such that F / A ′ is analytic at x = 0 . Then, for any k ∈ N , � � F ( x ) y k − 2 dx = G ( x ) y k dx , γ h γ h � � ′ � C ′ F � where G = 2 CF − . k A ′ A ′ 6 M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for abelian integrals , Trans. Amer. Math. Soc 363 (2011) 109–129.