aa r X i v : . [ m a t h . A P ] A p r SELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV
HERBERT KOCH
Abstract.
We construct self similar finite energy solutions to the slightlysuper-critical generalized KdV equation. These self similar solutions bifurcateas a function of p from the soliton at the L critical exponent p = 4. Introduction
Let p ≥ ∂ t u + ∂ xxx u ± ∂ x ( | u | p u ) = 0(1) u (0 , x ) = u ( x )or, for integer exponents p , ∂ t u + ∂ xxx u ± ∂ x ( u p +1 ) = 0(2) u (0 , x ) = u ( x )Both the KdV equation ( p = 1,(2)) and the mKdV equation ( p = 2,(2)) are inte-grable and in these cases a remarkable amount of information can be obtained bythe inverse scattering machinery. Both cases (1), (with +) and (2) (either p odd or+) admit soliton solutions u ( x, t ) = Q p ( x − t ) where(3) Q p ( x ) = (cid:18) p + 22 (cid:19) p sech p (cid:16) px (cid:17) . The quantities Z udx, Z u dx and Z u x − p + 2 | u | p +2 dx are conserved. The equations are invariant under translations in space and time,and under the scaling u λ ( x, t ) = λ /p u ( λx, λ t ) , which shows that the homogeneous Sobolev spaces ˙ H s with index s = 1 / − /p are scale invariant; the quintic gKdV equation ( p = 4) has L as critical space.Kenig, Ponce and Vega [4] prove local existence to (2) in the scaling criticalSobolev space for all integers p ≥ p = 1 , ,
3. Thishas been extended to critical Besov spaces by Molinet and Ribaud in [15] andby Strunk [18] to (1) for real p >
4. This raises the question concerning globalexistence and blow up in the critical and the supercritical case p ≥
4. Numericalsimulations by Dix and McKinney [3] suggest that there is self similar blow upin the supercritical case. In contrast to the situation for NLS there is neither avirial identity argument in the style of Glassey nor the explicit formula given bythe pseudo-conformal transformation. Nonetheless, Martel and Merle, and Martel,Merle and Rapha¨el showed in a series of papers [10, 13, 11, 12, 8, 7, 9] that in the L critical case there are solutions which blow up along the soliton manifold, i.e.the spatial scale of the solution tends to zero in finite time. Probably one of theearliest and most prominent prediction of blow-up respectively wave collapse is dueto Zacharov, Kuznetsov and Musher [21] for the (super critical) cubic focussing NLS in three dimensions who write Numerical simulations indicate that for d = 3 thereis self-similar and spherically symmetric blow-up, even from non-symmetric initialdata . The blow-up mechanism for the nonlinear Schr¨odinger equation is describedin detail in the book by Sulem and Sulem [19]. Indeed, Zakharov [20] predictedblow-up of the form 1 L ( t ) exp (cid:0) τ ( t ) (cid:1) Q (cid:18) | x | L ( t ) ; a (cid:19) for some selfsimilar profile Q and the scaling parameters L ( t ) = (2 a ( t ∗ − t )) / and τ ( t ) = 12 a log (cid:18) t ∗ t ∗ − t (cid:19) , where a > t ∗ is the time at which blow-up occurs.There seem to be solutions for each dimension 2 < d ≤ a ( d ) andheuristic arguments in [6] derive a relationship of the form d − ∼ a exp (cid:16) − πa (cid:17) . It seems that the first fully rigorous construction of self-similar blow-up solutionsis due to Kopell and Landman [5] for the cubic NLS in R ε (which has to beunderstood in the sense of existence of a solution to the nonlinear ODE into whichthe dimension enters merely as a parameter). It is crucial that these solutions arein ˙ H ∩ L p +2 and hence their energy vanishes.Self similar solutions for gKdV have been constructed by Bona and Weissler[1]. Their solutions are not in ˙ H and their relation to the blow-up observed insimulations is not clear. Such solutions can be obtained by evolving small self-similar initial data, like for Navier-Stokes, or wave maps (Shatah et al).In 2009, Merle, Raphael and Szeftel [14] established blow up from smooth initialdata for NLS in the slightly super critical case in low dimensions, heuristically bifurcating from the soliton.Here we construct selfsimilar solutions to the generalized KdV equation for p slightly larger than 4 (Theorem 2). Moreover in Theorem 3 we construct an almostinvariant manifold containing the solitons and the selfsimilar solutions, which willplay a central role in resolving the dynamic bifurcation at p = 4, together with fairlyprecise estimates in Theorem 1 for the constructed functions and their derivativeswith respect to all parameters.In Section 2 we formulate the bifurcation problem and state the main technicalresult, Theorem 1, and the main consequences, the existence result of Theorem 2and Theorem 3. In Section 3 we deduce Theorem 2 and Theorem 3 from Theorem1. This reduction is elementary but conceptually interesting.The generalized Airy and Scorers functions are studied in Section 4 with standardarguments: Stationary phase, contour integrals, and explicit formulas of Fouriertransforms of homogeneous functions. The next Section 5 derives explicit formulasfor a unique Green’s function for the linear part.The next step (Section 6) consists in a study of estimates for integral operatorswith integral kernel related to the Green’s function. After this preparation we setup the inverse function theorem in Section 6. Due to the weigths differentiabilitywith respect to a is not immediate. We approach it after establishing a fairlyprecise asymptotic expansion (Section 7) for the solution constructed by the inversefunction theorem.The final section shows plots of numerically computed self similar solutions forvarious values of a and p due to Strunk [17]. I want to thank Nils Strunk for allowingme to include this data and S. Steinerberger for many discussions. ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 3 The bifurcation problem
We search for self-similar solutions ψ ( t, x ) of the form(4) ψ ( t, x ) = (3 t ) − p v (cid:18) x (3 t ) / (cid:19) , for which the self similar profile v has to satisfy(5) 2 p v + yv y − v yyy − ( | v | p v ) y = 0 . A change of coordinates leads to a formulation in which the bifurcation from thesoliton equation becomes visible: let a > y = a / ( x + a − ) , u ( x ) = a p v ( a / ( x + a − ))then (5) is equivalent to(6) a (cid:18) p u + xu x (cid:19) − u xxx + u x − ( | u | p u ) x = 0 . Reversing the derivation, if u satisfies (6) then(7) v ( y ) = a − p u ( a − / y − a − )is a solution for (5) and we thus get via (4) a self-similar solution for (1). We willconstruct self similar solutions in L p +2 with derivative in L . Since for any solutionof gKdV the quantity Z R u x − p + 2 | u | p +2 dx is formally conserved, plugging the ansatz into gKdV one sees that the existence ofthe integral already implies it being 0 for all times.For a = 0 the equation simplifies to the derivative of the soliton equation(8) (cid:16) − u xx + u − ( | u | p u ) (cid:17) x = 0 , which motivates searching for a branch of solutions bifurcating from the soliton Q p using a as bifurcation parameter (see also Sulem and Sulem [19]).This is not yet the complete picture and complications arise from the linearizationaround the soliton(9) Lψ := − ψ xx + ψ − ( p + 1) Q pp ψ being elliptic but not invertible. Its spectrum, however, is explicitly known: there isa ground state Q p +1 p and the second eigenvalue is 0 with an eigenspace spanned by Q ′ p . We search p and u as functions of a . This requires an additional normalizationwhich we choose to be(10) h u, Q ′ p i = 0 . Our considerations lead to the bifurcation formulation(11) a (cid:18) p u + xu x (cid:19) − u xxx + u x − ( | u | p u ) x + h Q ′ p , u i Q ′′ p = 0 . It will be useful to consider a generalization which will give an approximate invariantmanifold which contains both, the solitons, and the selfsimilar blow up solutions.We consider(12) a ((1 + γ ) u + xu x ) − u xxx + u x − ( | u | p u ) x + h Q ′ p , u i Q ′′ p = 0 . HERBERT KOCH
Theorem 1.
Let q > . There exists ε > and a unique map u ∈ C ∞ (cid:18) [0 , ε ) × ( − − ε, −
12 + ε ) × (3 , q ) × R (cid:19) with the following properties: (13) u a,γ,p ( x ) satisfies (12) for ≤ a < ε, |
12 + γ | < ε, ≤ p ≤ q (14) sup a,γ,p,x (1 + a | x | ) γ | u a,γ,p ( x ) | < ∞ , (15) sup a,γ,p,x (1 + x + ) − k | ∂ nγ ∂ mp ∂ ka u a,γ,p ( x ) | < ∞ for k ≥ , (16) u ,γ,p ( x ) = Q p ( x )(17) u a,γ,p ( x ) > , ∂ x u a,γ,p ∈ L ( R ) The solution u a,γ,p is the unique solution to (12) satisfying (14) and (17) in a smallneighborhood of the soliton. The main results are consequences.
Theorem 2.
There exists ε > and a unique function p ∈ C ∞ ([0 , ε )) with p (0) = 4 , dpda (0) = k Q k L k Q k L = Γ(1 / π ∼ . . . . such that x → u a, p ( a ) − ,p ( a ) ( x ) is a solution to (6) with (18) ∂ x u a, p ( a ) − ,p ( a ) ∈ L , sup(1 + a | x | ) γ | u a,p ( a ) | < ∞ and (19) E ( u a, p ( a ) − ,p ( a ) ) := Z
12 ( ∂ x u a, p ( a ) − ,p ( a ) ) − p + 2 | u a, p ( a ) − ,p ( a ) | p +2 dx = 0These solutions are contained in a family of solutions which contains the solitonsand the selfsimilar solution. Theorem 3.
Let q > . There exists ε > and a unique function γ ( a, p ) ∈ C ∞ ([0 , ε ) × [3 , q ]) with γ (0 ,
4) = − , γ ( a, p ( a )) = 2 p − ,∂γ∂a (0 ,
4) = 18 k Q k L k Q k L = 18 Γ(1 / π ∼
18 4 . . . . ,∂γ∂p (0 ,
4) = 0 such that x → u a,γ ( a,p ) ,p ( x ) is a solution to (20) a ((1 + γ ( a, p )) u + xu x ) − ( u xx − u + | u | p u ) x = 0 with (21) ∂ x u a,γ ( a,p ) ,p ∈ L , sup(1 + a | x | ) γ | u a,γ ( a,p ) ,p | ≤ c. Moreover u ,γ (0 ,p ) ,p = Q p . In the process of proving Theorem 1 we obtain fairly precise asymptotics for theconstructed solutions. This asymptotics can be expressed concisely in terms of thespecial functions Hi γ and Gi γ constructed in Section 4. ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 5 Theorem 1 implies Theorem 2 and Theorem 3
The soliton.
We recall that solitons Q satisfy, possibly after rescaling,(22) − Q xx + Q − | Q | p Q = 0 . There is a unique solution, up to the choice of sign and a translation parameter. Wedenote by Q (or Q p ) the unique symmetric and nonnegative solution. We multiplyby Q and xQ x , respectively, and integrate to obtain the identities(23) Z Q x + Q − Q p +2 dx = 0 = Z Q x − Q + 1 p + 2 Q p +2 dx. This implies(24) k Q k p +2 L p +2 = 2( p + 2) p + 4 k Q k L , k Q x k L = pp + 4 k Q k L and hence(25) E ( Q ) = Z Q x − p + 2 Q p +2 dx = p − p + 4) k Q k L from which we see that the energy vanishes if p = 4. Let Q c ( x ) = c − /p Q ( x/c ),which is a rescaling of the soliton so that(26) u ( x, t ) = Q c ( x − c t )is a traveling wave solution to the gKdV equation with speed c . Then˜ Q c := − c ∂∂c Q c = 2 p Q c + xc Q ′ c satisfies k Q c k L = c − p k Q k L , hence, using the notation ˜ Q = ˜ Q ,(27) h ˜ Q, Q i = − ddc k Q c k L (cid:12)(cid:12)(cid:12)(cid:12) c =1 = (cid:18) p − (cid:19) k Q k L which changes sign as p passes through 4. We differentiate (26) with respect to c ,evaluate at c = 1 and obtain a solution to the linearized equation, hence ddx ( − Q − L ˜ Q ) = 0and(28) L ˜ Q = − Q. An integration by parts gives(29) Z ˜ Qdx = Z p Q + xQ x dx = (cid:18) p − (cid:19) Z Qdx.
The derivatives with respect to a . Let ˙ v be the derivative of u with respectto a evaluated at a = 0. It decays at −∞ and hence it satisfies(30) L ˙ v + h ˙ v, Q x i Q x = − Z x −∞ (1 + γ ) Q + xQ x dy We multiply by Q x (supressing p in the notation) and, since LQ x = 0, and(31) h ˙ v, Q x i = k Q x k − L h Q, ˜ Q i = (cid:18) γ + 12 (cid:19) k Q k L k Q x k L . Observe that h ˙ v, Q x i = 0 if γ = − . The norms on the right hand side can beevaluated and this gives the derivative of the inner product with respect to a at a = 0 as a function of p and γ . HERBERT KOCH
We set γ = − , multiply (30) by ˙ v and integrate h Q + xQ x , ˙ v i + Z ˙ v x ˙ vdx + ( p + 1) h Q p ˙ v, ˙ v x i = 0 . We rewrite the middle integral as a limit Z ˙ v x ˙ vdx = lim R →∞ Z R −∞ ˙ v x ˙ v dx = lim R →∞
12 ( ˙ v ( R )) This limit can be calculated: the inverse of − ∂ xx + 1 is given by the convolution by e −| x | . It maps the constant function 1 to itself, hencelim R →∞
12 ( ˙ v ( R )) = 12 ( Z ˜ Qdx ) = 18 (cid:18)Z Qdx (cid:19) and(32) h Q + xQ x , ˙ v i + ( p + 1) h Q p ˙ v, ˙ v x i = − (cid:18)Z Qdx (cid:19) . Let ¨ v be the second derivative with respect to a evaluated at a = 0. It satisfies2( 12 ˙ v + x∂ x ˙ v ) + ∂ x (cid:0) L ¨ v − p ( p + 1) Q p − ˙ v + h ¨ v, Q x i Q x (cid:1) = 0We fix p = 4, multiply by Q and integrate. Then, since h ˙ v, Q + xQ x i + h
12 ˙ v + x ˙ v x , Q i = 0 , and using (32), k Q x k L h ¨ v, Q x i =2 h
12 ˙ v + x∂ x ˙ v, Q i + 20 Z Q Q x ˙ v dx = − h ˙ v, Q + xQ x i − Z Q ˙ v ˙ v x dx = 14 k Q k L and hence the second derivative of the inner product with respect to a at a = 0, γ = − and p = 4 is given by(33) h ¨ v, Q x i = 14 k Q k L k Q x k L . We define the smooth function( a, γ, p ) → η ( a, γ, p ) := h u a,γ,p , ∂ x Q p i on [0 , ε ) × ( − − ε, + ε ) × [2 , q ]. The orthogonality h Q, Q x i = 0 implies η (0 , γ, p ) = 0 , the derivative with respect to a is given by (31) ∂η∂a (0 , γ, p ) = (cid:18) γ + 12 (cid:19) k Q k L k Q x k L , hence(34) ∂η∂a (0 , − , p ) = 0and ∂ η∂a∂γ (0 , γ, p ) = k Q k L k Q x k L . ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 7
We read the second derivative with respect to a from (33)(35) ∂ η∂a (0 , − ,
4) = 14 k Q k L k Q x k L . Let(36) g ( a, γ, p ) = η ( a, γ, p ) /a which is a smooth function with (again we suppress p in the notation of Q ) g (0 , γ, p ) = 0 ,∂g∂a (0 , − ,
4) = 18 k Q k L k Q x k L ∂g∂γ (0 , − , p ) = k Q k L k Q x k L ∂g∂p (0 , − , p ) = 0and by the implicit function theorem the equation g ( a, p − , p ) = 0can be solved for p = p ( a ) for a ∈ [0 , ε ), possibly after decreasing ε if necessary.Clearly p (0) = 4 and dpda (0) = − ∂g∂a∂g∂p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a =0 ,p =4 = − ∂ η∂a ∂ η∂a∂p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a =0 ,p =4 = k Q k L k Q k L We recall Q ( x ) = 3 / sech / (2 x )and thus Z Qdx = 3072 √ π Γ (cid:18) (cid:19) , Z Q dx = √ π and hence dpda (0) = 64 π Γ (cid:18) (cid:19) = 14 π Γ(1 / ∼ . . . . . The changes for Theorem 3 are quite obvious: We solve g ( a, γ, p ) = 0for γ ( a, p ) near (0 , − ,
4) and obtain ∂∂a γ (0 ,
4) = 132 π Γ(1 / ∼ . ∂∂p γ (0 ,
4) = 0 . Vanishing energy.
The function u a, p ( a ) − ,p ( a ) satisfies (11) and (10), hence(6). Moreover | u a, p ( a ) − ,p ( a ) | ≤ c (1 + | x | ) − /p ∈ L p +2 for p ≥
1. By Theorem 1 the derivative with respect to x is in L . We observedabove that then the energy has to vanish.This completes the proof that Theorem 1 implies Theorem 2 and Theorem 3. HERBERT KOCH The Airy function and Scorer’s functions
Definition and first properties.
In this section we study a class of spe-cial functions closely related to the Airy function. The Airy function and Scorersfunctions are discussed in [16], and the notation is motivated by Dix [2] but withdeliberate essential changes. We define for γ ∈ C with real part larger than − γ ( x ) = 12 π real Z ∞−∞ ( σ/i ) γ e i ( σ / xσ ) dσ = 1 π Z ∞ σ γ cos( 13 σ + xσ − γπ dσ = 1 π Z ∞ σ γ (cid:18) cos( γπ σ + xσ ) + sin( γπ σ + xσ ) (cid:19) dσ. Clearly Ai γ depends holomorphically on γ .The first line of the equation defines Ai γ through the Fourier transform. Thesecond line is the corresponding real formulation ( if γ is real) and the last lineconnects the definition to the slightly different ones in [2]. We easily see that(37) Ai ′ γ = − A γ +1 and(38) (1 + γ ) Ai γ + x Ai ′ γ − Ai ′′′ γ = 0 . This identity can be rewritten as(39) (1 + γ ) Ai γ − x Ai γ +1 + Ai γ +3 = 0 , moreover, Ai = Ai . It is not hard to evaluate the function Ai γ at x = 0Ai γ (0) = 1 π real Z ∞ ( σ/i ) γ e iσ / dσ = 1 π γ − e − ( γ − π Im Z ∞ µ γ − e − µ dµ = − π sin( 13 π ( γ − γ − Γ(( γ + 1) / . (40)We work out the asymptotic behavior using the standard approach via contourintegration and stationary phase. If x < ξ = ( − x ) / . We obtain the leading term(41) Ai γ ∼ √ π | x | − + γ cos( 23 | x | / − π − γπ x → −∞ . More precisely(42) Ai γ ( x ) = real (cid:26)(cid:18) √ π | x | − + γ + O ( | x | − + γ ) (cid:19) e i ( | x | / − π − γπ ) (cid:27) as x → −∞ and we can replace O ( | x | − + γ ) by an asymptotic series(43) | x | − + γ ∞ X j =0 c j | x | − j/ . ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 9
We turn to x >
0, shift the contour of integration to R + i √ x and obtain again bystationary phase(44) Ai γ = (cid:18) √ π | x | − + γ + O ( | x | − + γ ) (cid:19) e − x / as x → ∞ . Again the O ( | x | − + γ ) can be replaced by an asymptotic series (43).These series can be differentiated term by term with respect to γ , with the expectedestimates for the difference of Ai γ to the partial sum.Similarly, we set for the same set of γ Gi γ ( x ) = 1 π Im Z ∞ ( σ/i ) γ e i ( σ + xσ ) dσ = 1 π Z ∞ σ γ sin( 13 σ + xσ − γπ dσ π Z ∞ σ γ (cid:18) − sin( γπ σ + xσ ) + cos( γπ σ + xσ ) (cid:19) dσ. Again it is easily seen that Gi ′ γ = − Gi γ +1 and (1 + γ ) Gi γ + x Gi ′ γ − Gi ′′′ γ = 0which we can again rewrite as(1 + γ ) Gi γ − x Gi γ +1 + Gi γ +3 = 0 . Evaluation at zero gives(45) Gi γ (0) = 1 π Im Z ∞ ( σ/i ) γ e i σ dσ = − π cos( π γ − γ − Γ(( γ + 1) / x , one from the integral near zero and asecond one from the oscillatory part. We choose a smooth cutoff function supportedin | σ | ≤
2, identically 1 in | σ | ≤ γ ( x ) = Gi sγ + Gi γ = 1 π Im Z ∞ η ( σ )( σ/i ) γ e i ( σ + xσ ) dσ + 1 π Im Z ∞ (1 − η ( σ ))( σ/i ) γ e i ( σ + xσ ) dσ. Then Gi sγ ( x ) = ∞ X j =0 j ! ( − / j π Im Z ∞ ( σ/i ) γ +3 j e ixσ η ( σ ) dσ = ∞ X j =0 ( − / j πj ! Z ∞ ( σ/i ) γ +3 j e ixσ dσ + O ( | x | −∞ ) . (46)in the sense of oscillatory integrals. Now suppose that x >
0. Then we move thecontour of integration to i R + :(47) Z ∞ ( σ/i ) µ e ixσ dσ = Z i R + ( σ/i ) µ e ixσ dσ = i Z ∞ t µ e − xt dt = ix − − µ Γ(1 + µ )If x < − i R + and obtain for Gi sγ ( x )(48) ∞ X j =0 ( − / j Γ(1 + γ + 3 j ) j ! π | x | − − γ − j (cid:26) − cos( π ( γ + 3 j ) x < x > O ( | x | −∞ ) . The oscillatory part (for x < oγ ∼ − √ π | x | − + γ/ sin( 23 | x | / − π − γπ x → −∞ , again with the same type of asymptotic series, and it is O ( | x | −∞ ) as x → ∞ . Again it can be differentiated term by term with respect to x and γ .Finally, we set for γ > − γ ( x ) = 1 π Z ∞ σ γ e − σ + σx dσ. The derivative is again simple Hi ′ γ = Hi γ +1 and furthermore (1 + γ ) Hi γ + x Hi ′ γ − Hi ′′′ γ = 0which we rewrite as (1 + γ ) Hi γ + x Hi γ +1 − Hi γ +3 = 0The evaluation at x = 0 is given by(50)Hi γ (0) = 1 π Z ∞ σ γ e − σ / dσ = 1 π γ − Z ρ ( γ − / e − ρ dρ = 1 π γ − Γ(( γ + 1) / . It is not hard to see thatHi γ ( x ) = ∞ X j =0 Γ(1 + γ + 3 j )3 j j ! π | x | − − γ − j + O ( | x | −∞ )as x → −∞ and(51) Hi γ ( x ) = (cid:26) √ π x − + γ + O ( x − + γ ) (cid:27) e x / as x → ∞ , where again the O ( | x | − + γ ) terms can be sharpened to an asymptoticseries. The functions H γ and all their x derivatives are nonnegative. Derivativeswith respect to x and γ can be handled as above.4.2. Wronskian determinant.
The three functions Ai γ , Gi γ and Hi γ satisfy thesame differential equation. Here we will collect properties of the Wronskian matrixdefined by those functions.The Wronskian determinant W is independent of x since there is no secondderivative in the ODE and we evaluate it at x = 0 W = det Ai γ (0) Gi γ (0) Hi γ (0) − Ai γ +1 (0) − Gi γ +1 (0) Hi γ +1 (0)Ai γ +2 (0) Gi γ +2 (0) Hi γ +2 (0) = π − γ − Γ( γ + 13 )Γ( γ + 23 )Γ( γ + 33 ) det sin( ( γ − π ) cos( ( γ − π ) 1 − sin( ( γ − π ) − cos( ( γ − π ) 1sin( γπ ) cos( γπ ) 1 The Gaussian multiplication formula simplifies the product of the Γ functionsΓ( γ + 13 )Γ( γ + 23 )Γ( γ + 33 ) = 2 π − / − γ Γ(1 + γ ) . The remaining determinant can be expanded and simplified via addition theo-rems and evaluates to 3 √ / W = Γ( γ + 1) π . ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 11
In particular, the functions Ai γ , Gi γ and Hi γ are a fundamental system for thedifferential equation(53) (1 + γ ) u + xu x − u xxx = 0 . Subdeterminants.
Let f ( x ) = Ai γ Gi ′ γ − Ai ′ γ Gi γ = − Ai γ Gi γ +1 + Ai γ +1 Gi γ =: [Ai γ , Gi γ ]and calculate xf ′ − f ′′′ = x (cid:16) Ai γ Gi γ +2 − Ai γ +2 Gi γ (cid:17) − Ai γ Gi γ +4 − γ +1 Gi γ +3 +2 Ai γ +3 Gi γ +1 + Ai γ +4 Gi γ = Ai γ ( x Gi γ +2 − Gi γ +4 ) − Gi γ ( x Ai γ +2 − Ai γ +4 )+ 2 Ai γ +1 ( x Gi γ +1 − Ai γ +3 ) − γ +1 ( x Ai γ +1 − Ai γ +3 )=(2 + γ ) (Ai γ Gi γ +1 − Ai γ +1 Gi γ ) + 2(1 + γ ) (Ai γ +1 Gi γ − Ai γ Gi γ +1 )= − γ (Ai γ Gi γ +1 − Ai γ +1 Gi γ )= − (1 + ˜ γ ) f with ˜ γ = − − γ. Hence f = c Ai ˜ γ + c Gi ˜ γ + c Hi ˜ γ . The function f heritates the faster than polynomial decay for x >> γ .Thus c = c = 0. The leading term to the right is12 √ π x + γ Γ(1 + γ ) π x − − γ e − x We compare this with the asymptotic of Ai − − γ which gives(54) [Ai γ , Gi γ ] = Γ(1 + γ ) π Ai − − γ Similarly [Ai γ , Hi γ ] = c Ai ˜ γ + c Gi ˜ γ + c Hi ˜ γ The leading term for x >> π x γ and hence c = 0 , c = 1Γ( − γ ) . We recall that Γ(1 − s )Γ( s ) = π sin( sπ ) . to rewrite c = − Γ(1 + γ ) π sin( γπ )The leading term for x << − γ )2 π / | x | − − γ cos (cid:18) | x | / − π − ( γ + 1) π (cid:19) wherecos (cid:18) | x | / − π − ( γ + 1) π (cid:19) = cos (cid:18) | x | / − π − ˜ γπ (cid:19) cos(( γ + 1) π )+ sin (cid:18) | x | / − π − ˜ γπ (cid:19) sin(( γ + 1) π ) hence c = − Γ(1 + γ ) π cos( γπ )(55) [Ai γ , Hi γ ] = − Γ(1 + γ ) π (cid:16) cos( γπ ) Ai ˜ γ + sin( γπ ) Gi ˜ γ (cid:17) . Finally [Gi γ , Hi γ ] = c Ai ˜ γ + c Gi ˜ γ + c Hi ˜ γ . The leading term for x >> γ ) π / | x | − − γ e x hence(56) c = Γ(1 + γ ) π . The leading oscillatory term for x << − γ )2 π / | x | − − γ sin( 23 | x | / − π − ( γ + 1) π | x | / − π − ( γ + 1) π | x | / − π − ˜ γπ γ + 1) π ) − cos( 23 | x | / − π − ˜ γπ γ + 1) π )hence c = Γ(1 + γ ) π sin( γπ ) , c = − Γ(1 + γ ) π cos( γπ )and(57) [Gi γ , Hi γ ] = Γ(1 + γ ) π (cid:16) sin( γπ ) Ai ˜ γ − cos( γπ ) Gi ˜ γ + Hi ˜ γ (cid:17) We collect all the formulas in a proposition.
Proposition 4.
The following identities hold [Ai γ , Gi γ ] = Γ(1 + γ ) π Ai − − γ [Ai γ , Hi γ ] = Γ(1 + γ ) π ( − cos( πγ ) Ai − − γ − sin( πγ ) Gi − − γ )[Gi γ , Hi γ ] = Γ(1 + γ ) π (sin( πγ ) Ai − − γ − cos( πγ ) Gi − − γ + Hi − − γ ) . (58) 5. Green’s functions
The Green’s function for (59) . We consider the linear problem(59) L γ u := (1 + γ ) u + xu x − u xxx = f. The identities of Propositon 4 and (52) imply explicit formulas for Greens functionsin terms of generalized Airy and Scorer’s functions. There is a unique right inverse
ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 13 T L with integral kernel K L ( x, y ) supported on the left of the diagonal. It is for x ≥ y K Lγ ( x, y ) π = π Γ( γ + 1) n [Ai γ , Gi γ ]( y ) Hi γ ( x )+ [Gi γ , Hi γ ]( y ) Ai γ ( x ) + [Hi γ , Ai γ ]( y )] Gi γ ( x ) o = Hi − − γ ( y ) Ai γ ( x ) + Ai − − γ ( y ) Hi γ ( x )+ sin( γπ ) (cid:16) Gi − − γ ( y ) Gi γ ( x ) + Ai − − γ ( y ) Ai γ ( x ) (cid:17) + cos( γπ ) (cid:16) Ai − − γ ( y ) Gi γ ( x ) − Gi − − γ ( y ) Ai γ ( x ) (cid:17) . (60)It is easy to read off the leading terms of K Lγ in various asymptotic regimes. Let x, y >>
1. The leading term of the second line is given by the product of the Gifunctions. It is(61) sin( πγ ) | x | − − γ | y | γ . The third line decays fast as x ∼ y → ∞ .For x, y << − sin πγ cos ( πγ ) | x | − − γ | y | γ , We recall that we will set γ = p − u x ∈ L and u ∈ L p +2 . Let X ⊂ C be the Banach space of functions such that the norm(63) k u k X = sup | (1 + | x | ) γ u | + | (1 + | x | ) γ u x | The decay of the generalized Airy functions and of Scorer’s function determineuniquely the right inverse which maps compactly supported functions to X . Theorem 5.
Let − < γ < . Then there exists a unique right inverse T aγ : C ( R ) → X with the integral kernel K γ ( x, y ) = π (cid:16) Hi − − γ ( y ) Ai γ ( x ) χ y
We will use the Green’s function for the trans-formed problem. The equations(1 + γ ) v + yv y − v yyy = f and a ((1 + γ ) u + xu x ) − u xxx + u x = g are equivalent via(65) x = a − / y − a − , v ( y ) = au ( a − / y − a − ) f ( y ) = g ( x ) , Then, u ( x ) = a − v ( a / ( x + a − ))= a − Z K γ ( a / ( x + a − ) , z ) f ( z ) dz = a − Z K γ ( a / ( x + a − ) , z ) g ( a − / z − a − ) dz = a − / Z K γ ( a / ( x + a − ) , a / ( y + a − )) g ( y ) dy Thus u ( x ) = Z ˜ K aγ ( x, y ) g ( y ) dy where ˜ K a ( x, y ) = a − / K γ ( a / ( x + a − ) , a / ( y + a − ))We apply it to g = ∂ x F , where one integration by parts yields(66) u ( x ) = − Z ∂ y ˜ K a ( x, y ) F ( y ) dy = Z K a ( x, y ) F ( y ) dy =: T aγ F with a new kernel a / π K a ( x, y ) = − Ai − γ ( a / ( y + a − )) Hi γ ( a / ( x + a − ) χ x
It depends analytically on x , y , a ∈ R and γ away from the diagonal x = y . Weclaim that K a is smooth with respect to a , γ , x and y . To see this we have to showthat a − / Ai γ ( a − / (1 + ax ) Hi − γ ( a − / (1 + ay )is smooth in a and γ . It suffices to consider this at x = y = 0, since solutions toanalytic ODEs are analytic. We claim that a − / Ai γ ( a − / ) Hi − γ ( a − / )is smooth with respect to a ∈ R . Analyticity with respect to γ follows from analyt-icity of Ai γ and Hi − γ for fixed a . Smoothness in a and even analyticity is obviousfor a = 0. At a = 0 smoothness follows from the asymptotics of Ai γ and Hi − γ in(44) and (51).The following Lemma quantifies the dependence on a in a crucial region. It isan immediate consequence of the asymptotics of the Airy and Scorers functions. Lemma 6.
The following estimate (cid:12)(cid:12)(cid:12) K a ( x, y ) − (cid:16) e − | x − y | + aχ x>y (1 + ax ) − − γ (1 + ay ) − γ (cid:17)(cid:12)(cid:12)(cid:12) ≤ c δ (cid:16) a + | a | e − | x − y | (cid:17) holds for | x | , | y | ≤ a − / .Proof. First we observe (cid:12)(cid:12)(cid:12) a − / Gi γ ( a − / (1 + ax )) Gi − γ ( a − / (1 + ay ) − a (1 + ax ) − − γ (1 + ay ) − γ (cid:12)(cid:12)(cid:12) ≤ a ((1 + ax ) − The terms Ai − γ Gi γ , Gi − γ Ai γ and Ai − γ Ai γ are much smaller. To be precise weassume x ≤ y and estimate π − a − / (cid:12)(cid:12)(cid:12)(cid:12) Ai − γ ( a − / (1 + ay )) Hi γ ( a − / (1 + ax ) e y − x − (cid:12)(cid:12)(cid:12)(cid:12) =(1 + ax ) − + γ (1 + ay ) − − γ (cid:12)(cid:12)(cid:12)(cid:12) e a ((1+ ax ) − (1+ ay ) )+( y − x ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:12)(cid:12)(cid:12)(cid:12) e a (1+ ax ) − ( y − x ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ca | x − y | The case x ≤ y is similar. (cid:3) The implicit function theorem
The operator T aγ in weighted function spaces. We rewrite the problemas a fixed point problem for the identity plus a compact map. Then the Fredholmalternative will allow us to apply the implicit function theorem. Things howeverare not as simple as they may appear from this description: The derivatives withrespect to a and µ are not bounded in this functional analytic setting. They haveto be handled by different arguments in the next section.The following result is the basic linear estimate for the operator T aγ . It is aweighted estimate with a weight tailored for the problem at hand. This is necessarilyinvolved.The asymptotics on the left is essentially given byHi γ ( a − / (1 + ax )) / Hi γ ( a − / ) and on the right by Gi γ ( a − / (1 + ax )) / Gi γ ( a − / ) . This decay is to a certain extent captured by the weights below.
Proposition 7.
Let k ≥ . There exists c > such that the following is true. Let < a ≤ / , | γ + | < and (70) w a ( x ) := e − a (1 + a − / | ax | ) − if x ≤ − a − exp( a [(1 + ax ) / − if − a − ≤ x ≤ x ) k (1 + ax ) − − γ − k if x ≥ and (71) w ai ( x ) = e − a (1 + a − / | ax | ) − if x ≤ − a − exp( a [(1 + ax ) − if − a − ≤ x ≤ x ) k (1 + ax ) − − γ − k if x ≥ and (72) w = (cid:26) e −| x | / if x < otherwise w i = (cid:26) e − | x | / if x < otherwiseThen (73) sup x, ≤ a ≤ , | + γ |≤ | T aγ f ( x ) | /w a ( x ) ≤ c sup x | f ( x ) | /w ai ( x ) . The complexity of the weight reflects the different asymptotic areas, and theproof consists in decomposing operator and domain in smaller pieces for which ele-mentary estimates become possible. The proposition is an immediate consequenceof
Lemma 8.
There exists c > independent of x , a and γ such that Z | K aγ ( x, y ) | w ai ( y ) dy ≤ cw a ( x ) for x ∈ R , ≤ a ≤ and | + γ | ≤ .Proof. We will restrict ourselves to k = 0, with marginal differences for positive k .We recall that(74) | Ai γ ( x ) | + | Gi γ ( x ) | + | Hi γ ( x ) | ≤ c (1 + | x | ) − for x ≤ | γ + | ≤ . Step 1: x ≥ . There are contributions from the integrals over ( −∞ , − a − ),( − a − , , x ) and ( x, ∞ ). We deal with them in reverse order, and we begin a − / Ai − γ ( a − / )(1 + ay )) Hi γ ( a − / (1 + ax )) - the kernel for y > x . Then a − / (1 + ax ) γ Hi γ ( a − / (1 + ax )) Z ∞ x Ai − γ ( a − / (1 + ay ))(1 + ay ) − − γ dy . Z ∞ x e a (cid:16) (1+ ax ) − (1+ ay ) (cid:17) (cid:18) ax ay (cid:19) γ [(1 + ax )(1 + ay )] − dy . Z ∞ x e (1+ ax ) / ( x − y ) (cid:18) ax ay (cid:19) + γ (1 + ax ) − dy . C holds uniformly in x ≥ | γ + | ≤ . Similarly a − / Ai γ ( a − / (1 + ax )) Z x Hi − γ ( a − / (1 + ay )) (cid:18) ax ay (cid:19) γ dy ≤ c. in the same range. ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 17
Next we consider the contribution of the product of the functions Gi, usingGi γ ( a − / (1 + ax )) ∼ a (1+ γ ) (1 + ax ) − − γ for x > a − / Z x (cid:18) ax ay (cid:19) γ Gi γ ( a − / (1 + ax )) Gi − γ ( a − / (1 + ay )) dy . Z x a (1 + ay ) − dy . . The products a − / Ai − γ ( a − / (1 + ay )) Gi γ ( a − / (1 + ax ))and a − / Gi − γ ( a − / (1 + ay )) Ai γ ( a − / (1 + ax ))are much smaller.We observe that Ai γ ( a − / (1 + ax )) . Ai γ ( a − / ) w a ( x ) , Gi γ ( a − / (1 + ax ) . Gi γ ( a − / ) w a ( x )for x ≥ y ≤ x = 0 to get thesame bound for all nonnegative x .The estimates Z −∞ Hi − γ ( a − / (1 + ay )) w ai ( y ) dy . Hi − γ ( a − / ) , (75) e − a Z − a − −∞ (1 + a − / (1 + ay )) − dy . e − a and Z − a − (1 + a − / (1 + ay )) − − γ e a [(1+ a − / (1+ ay )) − dy . a (1+ γ ) ∼ Gi γ ( a − / )are straight forward. This completes the estimate for x > Step 2: x < . In view of the first substep above (with x = 0) the contributionfrom y > γ ( a − / (1 + ax )) . H γ ( a − / ) w a ( x ) . We consider the contribution from y ≤ x ∈ [ − a − , −∞ , − a − ), ( − a − , x ), and ( x, − a − ≤ x ≤ a − / Z − a − (1 + a − / (1 + ay )) γ − (1 + a − / (1 + ax )) γ +1 ×× e − a | (1+ ax ) − (1+ ay ) | e − a [(1+ ay ) − e a [(1 − ax ) − dy which is trivial once broken up into different cases: x = 0, y ≤ x , − a − ≤ x < y and − a − ≤ x ≤ − a . The contributions from the other terms in the Greensfunction are much smaller. Finally the contribution (to x ∈ [ − a − , y ≤− a − is controlled by (75). Step 3: The case x < − a − , contribution from y ≤ . Again we haveto consider the integrals over ( −∞ , x ), ( x, a − ) and ( a − , a − ,
0) has been evaluated above. The obvious estimates | Ai γ ( a − / (1 + ax )) | + | Gi γ ( a − / (1 + ax )) | + Hi γ ( a − / (1 + ax )) . w a ( x ) w a ( − a − ) complete that part.The kernel satisfies | K aγ ( x, y ) | . a − / (1 + a − / (1 + ax )) − (1 + a − / (1 + ay )) for x, y ≤ − a − . Now Z a − −∞ (1 + a − / (1 + ay )) − + dy . a − / completes the proof . (cid:3) We reformulate the bifurcation problem as a fixed point problem(76) u ( x ) = T aγ ( | u | p u − h u, Q x i Q x )where we search u in a neighborhood of Q .We introduce v = u/w a with w a from (70) and rewrite the problem as F ( v ) = 0with F ( v ) = v − ( w a ) − T aγ [ | vw a | p vw a − h vw a , Q x i Q x ] . Proposition 7 implies that the map is j times Frechet differentiable on the space ofbounded continuous functions, for every nonnegative integer j ≤ Q . We include the weightsinto the operator and consider˜ T ( a, p, γ, v ) u := ( w a ) − T a [( p + 1) | vw a | p w a u + h u, w a Q x i Q x ] . Let C b ( R ) denote the space of continuous functions equipped by the supremumsnorm, and the closed subspace of functions with limit 0 as x → ±∞ by C . Thespace of linear operators from the normed space X to the normed space Y is denotedby L ( X, Y ), which we equip with the operator norm.
Corollary 9.
The map (cid:16) [0 , × h − − , −
12 + 18 i × [3 , q ) × C ( R ) (cid:17) → L ( C b , C b )( a, γ, p, v ) → (cid:16) u → ˜ T ( a, p, γ, v ) u (cid:17) is continuous.Proof. The map ( a, p ) → Q x /w ai ∈ L is clearly continuous as is ( v, u, a, p ) → | vw a | p w a u/w ai . This implies continuity with respect to v , uniform with respect to a and p . Henceit suffices to prove the continuity for the composition with the multiplication by acharacteristic function,( a, γ, p, v ) → (cid:16) u → ˜ T ( a, p, γ, v ) χ [ − R,R ] u (cid:17) . The proof of Proposition 7 implies that x → ˜ T ( a, p, γ, v ) χ [ − R,R ] u ( x )converges to zero as x → −∞ , uniformly for bounded v and u and a and p as inthe theorem. Continuity with respect to p is obvious. On the right hand side thesituation is slighty different: Since we apply the operator to a function with compactsupport, the only terms which does not decay as x → ∞ comes from Gi γ ( a − / (1 + ax )). This term is clearly continuous with respect to a and γ . Continuity with ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 19 respect to γ and a follows from the continuity of the Airy and Scorer functions,their asymptotics and the continuity of the Green’s function. (cid:3) The invertibility of the linearization in a neighborhood of the bifurcation pointis contained in the next proposition. We denote S av u = u − w − a T aγ (cid:2) ( p + 1) w p +1 a v p u − h u, w a Q x i Q x (cid:3) Proposition 10.
There exists δ > such that S av : C b → C b is invertible with aninverse whose norm is uniformly bounded for | γ + 12 | ≤ δ, ≤ a ≤ δ and sup | v − Q | w a ≤ δ. Proof.
The operator S av is bounded by Proposition 7 and the norm continuity at v = Q/w a is the content of Corollary 9. It thus suffices to consider invertibility at a = 0 and v = Q/w a . Clearly u → ( w ) − T [( p + 1) Q p ( w ) p +1 u ] − h u, w Q x i Q x ]is compact. We recall that the integral kernel of T is e −| x − y | .By the Fredholm alternative S Q is invertible if the null space is trivial. Consider(77) u = T γ [( p + 1) Q p u − h u, Q x i Q x ] . We claim that there is only the trivial bounded solution. Suppose that u satisfiesthe homogeneous equation (77). Since the kernel decays fast also u decays fast, andthe same holds for the derivatives. Hence u − u xx − ( p + 1) Q p u + h u, Q x i Q x = Lu − h u, Q x i Q x = 0 . We take the inner product with Q x . Then h u, Q x i = 0 since LQ x = 0. The nullspace of L is spanned by Q x and hence u = 0. This null space is trivial, by theFredholm alternative S Q is invertible, and this remains so in a small neighborhoodof the coefficients and Q/w a . (cid:3) We continue with an estimate which implies that Q is almost a solution to thefixed point problem. This is important since F fails to be differentiable with respectto a and γ . Lemma 11.
There exists
C > such that sup x | ( w a ) − ( Q − T aγ Q p +1 ) | ≤ Ca.
Proof.
We observe that Q − T γ (cid:2) Q p +1 + h Q, Q x i Q x (cid:3) = 0since Q satisfies the soliton equation. The assertion is equivalent to (cid:12)(cid:12) ( T γ − T aγ ) Q p +1 (cid:12)(cid:12) ≤ caw a ( x )which we address now. Since Q . e −| x | there exists c > a , γ and p sup | x |≥ c | ln a | | ( w ai ) − Q p +1 | ≤ a. so that with χ the characteristic function of the complement of [ − c | ln a | , c | ln( a ) | ]sup x ( w a ( x )) − | T aγ χQ p +1 ( x ) | . a. it suffices to verify (cid:12)(cid:12) ( T γ − T aγ ) χQ p +1 (cid:12)(cid:12) . aw a ( x ) . Checking the kernel we see that | T γ ˜ χQ p +1 ( x ) | + | T aγ ˜ χQ p +1 ( x ) | . aw a ( x )if | x | ≥ | ln( a ) | . Now | Q p | ≤ e − p | x | , Q s is integrable whenever s > Q . w ai .Thus the statement will follow from(78) sup | x | , | y |≤ c | ln a | e max {− x, } (cid:18) K aγ ( x, y ) − e −| x − y | (cid:19) e − | y | . a which is a consequence of Lemma 6. (cid:3) Proposition 12.
Let q > and ≤ p ≤ q . Then there exists ε and C > so thatthere is a unique fixed point u to u = T aγ ( | u | p u − h u, Q x i Q x ) with (79) sup x ( w a ( x )) − | u ( x ) − Q ( x ) | + |h u, Q x i| . a for max (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)
12 + γ (cid:12)(cid:12)(cid:12)(cid:12) , a (cid:27) ≤ ε. The map ( a, γ, p ) → ( w a ) − u ∈ C b ( R ) is continuous.Proof. We write u = w a v − Q . Then we search a fixed point to v =( w a ) − (cid:0) T aγ ( | w a v + Q | p ( w a v + Q ) − h v, Q x i Q x ) − Q (cid:1) =( w a ) − (cid:0) T aγ ( | w a v + Q | p ( w a v + Q ) − Q p +1 − h v, Q x i Q x ) (cid:1) + ( w a ) − ( T aγ Q p +1 − Q ) . The second term on the right hand side is bounded by a constant times a by Lemma11. The derivative at v = 0 is invertible by Lemma 10 with a uniformly boundedinverse. The existence of a unique fixed point with the desired properties followsnow by the implicit function theorem. (cid:3) Asymptotics and differentiability
In the last section we have constructed a unique fixed point u to(80) u = T aγ (cid:16) | u | p u − h u, Q x i Q x (cid:17) with k ( u − Q ) /w a k sup << . Moreover it satisfies(81) | u − Q | ≤ caw a with a constant which is uniform in a , p and γ . It follows immediately from theintegral representation and the decay that u x is square integrable. Moreover u/w a depends continuously on a , p and γ considered as a map to C b ( R ). It remains toshow that this map is smooth for every x , to give bounds for the derivatives, andto prove the uniqueness statement. Here we turn to differentiability and boundsfor the derivatives.As a first step and a warm up we consider the simpler term first. This term willnot enter the asymptotics of the fixed point, but we need it to prove differentiabilitywith respect to a . ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 21
The asymptotics of v aγ := T aγ Q x . We define(82) c = c ( a, p, γ ) = π Z Ai − γ ( a − / (1 + ay )) Q ( y ) dy and(83) d = d ( a, p, γ ) = π Z Gi − γ ( a − / (1 + ay )) Q ( y ) dy. Proposition 13.
The following estimates hold for κ < e − κx (cid:12)(cid:12)(cid:12) ∂ jx ∂ ka ∂ lp ( T aγ Q x + c Hi γ ( a − / (1 + ax ))) (cid:12)(cid:12)(cid:12) ≤ c ( j, k, l ) √ − κ if x ≤ and if x ≥ e κx (cid:12)(cid:12)(cid:12) ∂ jx ∂ ka ∂ lp ( T aγ Q x + d Gi γ ( a − / (1 + ax ))) (cid:12)(cid:12)(cid:12) ≤ c ( j, k, l ) √ − κ . Moreover there are the asymptotic series c H γ ( a − / ) = ∞ X k =0 α k ( γ, p ) a k and d Gi γ ( a − / ) = ∞ X k =2 β k ( γ, p ) a k with nontrivial leading term β resp α . The coefficients are smooth functions of γ and p , with bounds depending only on k .Proof. We can define a solution to the linear equation(86) a ( γv + xv x ) − v xxx + v x = Q xx by an integral kernel K L , supported in y < x , which is given by (compare withTheorem 5) a / K L ( x, y ) π = Hi − − γ ( a − / (1 + ay )) Ai γ ( a − / (1 + ax ))+ Ai − − γ ( a − / (1 + ay )) Hi γ ( a − / (1 + ax ))+ sin( γπ ) (cid:16) Gi − − γ ( a − / (1 + ay )) Gi γ ( a − / (1 + ax ))+ Ai − − γ ( a − / (1 + ay )) Ai γ ( a − / (1 + ax )) (cid:17) + cos( γπ ) (cid:16) Ai − − γ ( a − / (1 + ay )) Gi γ ( a − / (1 + ax )) − Gi − − γ ( a − / (1 + ay )) Ai γ ( a − / (1 + ax )) (cid:17) . (87)Two solutions to (86) differ by a solution to the homogeneous problem. The formula Z x −∞ K L ( x, y ) Q xx ( y ) dy − c Hi γ ( a − / (1 + ax ))defines a solution to (86) hence it differs from v aγ by a solution to the homogeneousequation. Both functions and their derivatives are bounded by a multiple of w a for x ≤
0, and hence their difference is a multiple of Hi γ . But the coefficients of theleading term are the same because of the choice of c , and hence both are the same.We have ∂ jx h v − c Hi γ ( a − / (1 + ax ) i = Z x −∞ ∂ jx K L ( x, y ) Q xx ( y ) dy if j = 0 , ,
2. For j ≥ Q and its derivatives. The kernel obviously reproduces exponential decay up topolynomial factors. The leading contribution for κ → x close to 1.We argue similarly for x ≥
0, but this time with the standard kernel K aγ . Theleading part comes from a − / Gi γ ( a − / (1+ ax ) Gi − γ ( a − / (1+ ax )). The productsof Ai γ Hi − γ and Ai − γ Hi γ and reproduce exponential decay if κ < √ − µ if µ is close to 1. The other components of the kernel are supportedin y ≥ x . They reproduce the exponential decay of Q xx . We verify the asymptoticformulas for c and d and we begin with d . Let m k = Z x k Q ( x ) dx. be the moments of Q . They are smooth functions of p , and independent of γ and a . A Taylor expansion of Gi − γ gives the asymptotic series Z Gi − γ ( a − / (1 + ay )) Q ( y ) dy ∼ ∞ X k =0 k ! m k a k/ Gi k − γ ( a − / ) ∼ a − γ ( m + ( m + m ) a + . . . ) ∼ (Gi γ ( a − / )) − ( ∞ X k =2 β k a k )with β = 0.Differentiability of the coefficients with respect to p is obvious. Differentiabilitywith respect to γ follows from the differentiability of Gi − γ with respect to γ andthe corresponding bounds. The difference to a partial sum is easily controlled bythe estimates for Gi γ and its derivatives.For the expansion of c we write c = a − / Z Ai − γ ( a − / (1 + ay )) e y ( e − y Q ( y )) dy = − a − / Z Z y h Ai − γ ( a − / (1 + as )) e s i ds ddy ( e − y Q ( y )) dy. The function ddy ( e − y Q ( y )) is a Schwartz function with exponential decay. Thezeroth moment is −
1, but R x Ai − γ vanishes at x = 0, and the leading contributioncomes from the next term, c = a − / Ai − γ ( a − / ) Z y ddy ( e − y Q ( y )) dy + (cid:18)
14 + γ (cid:19) ( a / Ai − γ ( a − / )) Z y ddy e − y Q ( y ) dy . . . =(Hi γ ( a − / )) − ( ∞ X j =0 α j a j )with α = 0. Any derivative on Ai − γ ( a − / (1 + ax )) e x gains us a factor a . Thisis a consequence of the multiplication by e x . We used the expansion of Hi γ in thisexpansion. The derivatives with respect to p fall only on Q , and hence they areeasy to estimate. The Scorer functions are differentiable with respect to γ . Thisimplies the statement on the differentiablity of the coefficients. (cid:3) As a consequence we have |h u, Q x i ( v aγ − c Hi γ ( a − / (1 + ax ))) | ≤ cae κx ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 23 hence |h u, Q x i| v aγ | ≤ ca Hi γ ( a − (1 + ax ))Hi γ ( a − ) + e κx √ − κ ! for x ≤ x > |h u, Q x i v a | ≤ ca (cid:18) a Gi γ ( a − / (1 + ax ))Gi − γ ( a − / ) + e − κx √ − κ (cid:19) In the sequel we will only rely on those two estimates, and not on the full statementof Proposition 13.7.2.
Bounds for the fixpoints and derivatives with respect to x . After thiswarm-up we turn to the nonlinear term. Let c = a − / Z Ai − γ ( a − / (1 + ay )) u p +1 ( y ) dy and d = a − / Z Gi − γ ( a − / (1 + ay )) u p +1 ( y ) dy. These integrals exist since | u | . w a ( x ) . Using the bound | u − Q | ≤ caw a ( x )of the previous section we see that | u p +1 − Q p +1 | ≤ caw ai ( x )and, as in the previous subsection c = (1 + O ( a )) a − / Ai − γ ( a − / ) Z Q p +1 ( y ) dy. Thus(89) c ∼ (Hi γ ( a − / ) − ) . and similarly(90) d ∼ a − γ )3 . The function u p +1 decays sufficiently fast to repeat the argument of the lastsection. Thus w L ( x ) := u ( x ) − v a − c Hi γ ( a − / (1 + ax )) = Z x −∞ K aL ( x, y ) ∂ y u p +1 ( y ) dy and we also have the obvious integral representation for w R ( x ) := u ( x ) − v a − d Gi γ . Thus we obtain the very rough estimate, using p ≥ γ close to − / | w L ( x ) | ≤ c p +1 Z x −∞ | ∂ y K L ( x, y ) | ( w a ( x )) p +1 dx ≤ c (cid:18) Hi γ ( a − / (1 + ax ))Hi γ ( a − / ) (cid:19) . We put this information in the expansion. The oscillatory part of the kernel givesa small contribution when applied to | x | − ( γ +1)( p +1) resp. (Hi γ ( a − / (1 + ax ))) p +1 hence, with ω = − (1 + γ )( p + 1) − p − | w L ( x ) | . a − / | a − / (1 + ax ) | ω (Hi γ ( a − / )) − p − if x ≤ − a − | a / + (1 + ax ) | − / (cid:16) Hi γ ( a − / (1+ ax ))Hi γ ( a − / ) (cid:17) p +1 if − a − ≤ x ≤ . . Similarly we repeat the arguments from the last section on the right hand side.In a first step | w R ( x ) | ≤ c (cid:18) e − x + a Gi γ ( a − / (1 + ax ))Gi γ ( a − / ) (cid:19) . We plug this into the integral operator. The exponential part with Ai and Hi repro-duces the decay ( Q + ca (1+ ax ) − − γ ) p +1 . The second potentially large contributionis bounded by a Z ∞ x (1 + ay ) − γ ( Q + ca (1 + ax ) − − γ ) − ( p +1) dy (1 + ax ) − − γ . ae − x + a p +1 (1 + ax ) − ( p +1) γ − ( p +2) The exponential part again reproduces the decay and we arrive at | w R ( x ) | ≤ ce − x + a p +1 (1 + ax ) − (1+ γ )(1+ p ) − for x > Lemma 14.
There exists ε > so that for ≤ a ≤ ε, (cid:12)(cid:12)(cid:12)(cid:12)
12 + γ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε, ≤ p ≤ q and the fixed point u the following is true. Let w L = u − v a − c Hi γ ( a − / (1 + ax )) − Q. Then | ∂ kx w L | ≤ c a − / | a − / (1 + ax ) | − ( p +1) γ − p − (Hi γ ( a − / )) p +1 if x ≤ − a − ( a / + (1 + ax )) k − (cid:16) Hi γ ( a − / (1+ ax ))Hi γ ( a − / ) (cid:17) p +1 if − a − ≤ x ≤ . and with w R = u − v a − ad Gi γ ( a − / (1 + ax )) − Q the estimate | ∂ kx w R | ≤ c k a h e − x + a p + k (1 + ax )) − ( p +1) γ − p − − k i holds. The sum ( c + c ) und d are bounded and bounded from below by a positiveconstant, independent of p , γ and a . Finally (92) c − Hi γ ( a − / (1 + ax ))Hi γ ( a − ) ≤ u ≤ c Hi γ ( a − / (1 + ax ))Hi γ ( a − ) if x ≤ and (93) c − ( e − x + a (1 + ax ) − − γ ) ≤ u ≤ c ( e − x/ + a (1 + ax ) − − γ ) if x ≥ .Proof. Only the last two statements need to be shown. Since | u − Q | ≤ aw a the statement is obvious for | x | ≤ | ln a | / x ≤ −| ln a | + R and a sufficiently small the term Hi γ ( a − / (1+ ax ))Hi γ ( a − / ) becomesdominant and ensures positivity for those x . The same argument applies on theright hand side. (cid:3) In particular u is positive and bounded from below by the same type of boundsas from above. ELF SIMILAR SOLUTIONS TO SUPER-CRITICAL GKDV 25
Derivatives with respect to γ and a . The result of this subsection con-cludes the proof.
Proposition 15.
The fixed point u is infinitely often differentiable with respect to x , a , γ and p up to a = 0 . Moreover the estimate | ∂ kx ∂ la ∂ mp ∂ nγ u | . ( a − / + | ax | ) − − γ − k − n | ln(2 + | ax | ) | n / Hi γ ( a − / ) if x < − a − Hi γ ( a − / (1 + ax )) / Hi γ ( a − / ) if − a − ≤ x ≤ e − x + a k (1 + ax ) − − γ − k − n | ln(2 + ax ) | n if x ≥ holds with a constant depending only on k, l, m and n . The bounds are exactly the bounds for the derivatives of a (1 + ax ) − − γ plus a Schwartz function, resp. for x ≤ γ ( a − / (1 + ax )) / Hi γ ( a − / )Proposition 15 completes the proof of Theorem 1.Despite considering a nonsmooth nonlinearity the fixed point will be smooth.This is compatible with the nonregularity of the power function since the fixedpoint u is positive. Proof.
The differentiation with respect to p is simpler than the differentiation withrespect to a and γ , and we ignore it. We differentiate a ((1 + γ ) u + xu x ) − u xxx + u x + ∂ x ( | u | p u + h u, Q x i Q x ) = 0with respect to γ formally and denote the derivative again by ˙ u . It satisfies a ((1 + γ ) ˙ u + x ˙ u x ) − ˙ u xxx + ˙ u x + ∂ x (( p + 1) | u | p ˙ u + h ˙ u, Q x i Q x ) = − au By Proposition 10 the linear operator is invertible, and we want estimate ˙ u in termsof u . However, we do not have the bound | u | ≤ w ai for | x | ≤ − a − since there w ai is not bounded by w a .We choose a smooth monotone function η + , supported in [ − , ∞ ) and identicallyone in [1 , ∞ ). Let η ( x ) = 1 − η + ( x ). We denote˙Hi γ = ∂∂γ Hi γ and ˙Gi γ = ∂∂γ Gi γ Then a ((1 + γ ) ˙Hi γ + x ˙Hi ′ γ ) − ˙Hi ′′′ γ + ˙Hi ′ γ = − a Hi γ Let ˙ v = ˙ u − ( c + c ) η − ˙Hi γ − a ( d + d ) η + ˙Gi γ Then a (1 + γ ) ˙ v + x ˙ v x ) − ˙ v xxx + ˙ v x + ∂ x (( p + 1) | u | p ˙ v + h ˙ v, Q x i Q x )= φ − a ( u − ( c + c ) η − Hi γ − a ( d + d ) η + Gi γ ) − ∂ x (( p + 1) | u | p h ( c + c ) η − ˙Hi γ + a ( d + d ) η + ˙Gi γ i + h ( c + c ) η − ˙Hi γ + a ( d + d ) η + ˙Gi γ , Q x i Q x . for some smooth function φ supported in [ − , differences. Continuity with respect to all parameters is obvious. This argumentcan be iterated.Similarly we deal with derivatives with respect to a . The partial derivatives ∂ na Gi γ ( a − / (1 + ax )) behave similarly as ∂ na (1 + ax ) − − γ = c γ,n x n (1 + ax ) − n − γ . Again Proposition 10 implies differentiability with respect to a , for a >
0, but thistime we have to use weights with k > a , p or x , using crucially the estimate (92) and (93). (cid:3) Expansion of the selfsimilar solution.
The argument above gives infor-mation on the asymptotics of the self similar solutions which we state below.
Proposition 16.
Let u = u ( a ) be the selfsimilar solution orthogonal to Q x . Thenexists a unique expansion (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ( x ) − a (1 + ax ) − /p N X j =0 d j ( a )(1 + ax ) − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ac N (1 + ax ) − N − for x > , where d j are bounded uniformly in a , and c N is independent of a and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ( x ) − (Hi γ ( a − / )) − | ax | − /p N X j =0 d j ( a )(1 + ax ) − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ac N (1 + | ax | ) − N − for x < − a − . A numerical simulation
The selfsimilar solutions have been computed numerically by N. Strunk in hisdiploma thesis. The first curve shows 1 /p as a function of a . There is a smallartefact near a = 0 .
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