On Non-degeneracy of Solutions to SU(3) Toda System

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1 On Non-degeneracy of Solutions to SU3 Toda System Juncheng Wei Chunyi Zhao Feng Zhou March Abstract We prove that the solution to the SU3 Toda system u + e u e v = 0 in R v e u + e v = 0 in R e u < e v < R R is nondegenerate i.e. the kernel of the linearized operator is exactly eight-dimensional. 1 Introduction Of concern is the nondegeneracy of solutions of the following two-dimensional SU3 Toda system u + e u e v = 0 in R v e u + e v = 0 in R e u < e v < R R Department of Mathematics Chinese University of Hong Kong Shatin Hong Kong. address: wei@math.cuhk.edu.hk Department of Mathematics Chinese University of Hong Kong Shatin Hong Kong. address: chunyi zhao@yahoo.com Department of Mathematics East China Normal University Shanghai P.R. China. address: fzhou@math.ecnu.edu.cn. 1 1

2 where = + is the usual Euclidean Laplacian in R. System 1 is x 1 x a natural generalization of the Liouville equation u + e u = 0 in R e u <. R The Liouville equation and the Toda system 1 arise in many physical models and geometrical problems. In Chern-Simons theories the Liouville equation is related to Abelian models while the Toda system is related to Non-Abelian models. We refer to the books by Dunne [] Yang [10] for physical backgrounds. The SU3 Chern-Simons model has been studied in many papers. We refer to Jost-Wang [4] Jost-Lin-Wang [3] Li-Li [6] Malchiodi-Ndiaye [8] Ohtsuka-Suzuki [9] and the references therein. Using algebraic geometry results Jost and Wang [5] classified all solutions to 1. More precisely when N = all solutions to 1 can be written as follows: uz = log 4 a 1 a + a 1 z + c + a z + bz + bc d a 1 + a z + b + z + cz + d 3 vz = log 16a 1 a a 1 + a z + b + z + cz + d a 1 a + a 1 z + c + a z + bz + bc d 4 where z = x 1 + ix C and a 1 > 0 a > 0 are real numbers and b = b 1 + ib C c = c 1 + ic C d = d 1 + id C. Note that in the above representation there are eight parameters a 1 a b 1 b c 1 c d 1 d R 8. In the study of the blow-up behaviors for solutions of SU3 Toda system a crucial question is the nondegeneracy of solutions. More precisely we need to study the elements in the kernel of the associated linearized operator i.e. the following linear system { ϕ 1 + e u ϕ 1 e v ϕ = 0 in R ϕ e u ϕ 1 + e v ϕ = 0 in R 5. Here u v are solutions to 1 given by 3-4. Certainly there are at least eight-dimensional kernels since any differentiation of u v with respect to the eight parameters satisfies 5. The following theorems shows that the converse is also true which shows that the solution u v is nondegenerate. Theorem 1.1. Let ϕ 1 ϕ satisfy 5. Assume that ϕ 1 C1 + x τ ϕ C1 + x τ forx R and some τ 0 1.

3 Then ϕ 1 ϕ belongs to the following linear space { a1 u a u b1 u b u c1 u span a1 v a v b1 v b v c1 v c u c v d1 u d1 v } d u. d v In the case of nondegeneracy of solutions of Liouville equation the problem becomes to a single linear equation ϕ + e u ϕ = 0 in R. 6 Using conformal transformations one can assume that u is radially symmetric. Then one can use the separation of variables to obtain the nondegeneracy result. See Lemma.3 of Chen-Lin [1]. Here we are dealing with a system. Firstly we can not find a conformal transformation to transform any solution u v to radially symmetric solution. Second even if u v is radially symmetric the new system is still too complicated to study. To overcome these difficulties we employ the invariants of the system 1. Using the invariants of 1 we also obtain invariants for 5 and thus prove Theorem 1.1. For convenience the language of complex variable is used in this paper. We refer z = x 1 ix to the usual conjugate of z = x 1 +ix C. In addition U z := z U = 1 U x 1 i U x U z := z U = 1 U x 1 + i U x. Notation C is a generic constant which may be different from line to line. We believe that our method may be used to deal with the general case SUN + 1. The major problem is how to obtain higher-order invariants as in Section. Invariants of 1 In this section we derive some invariants for 1. For more discussions we refer to Section 5.5 of the book by Leznov-Saveliev [7]. Let us first define the following transformation in whole C Uz z = u 3 + v 3 log 4 V z z = u 3 + v 3 log 4. 7 Note that = 4 z z. Then the Toda system 1 can be rewritten as U z z + e U V = 0 in C V z z + e V U = 0 in C R e U V < R e V U <. We prove now some preliminary lemmas. 8 3

4 Lemma.1. We have in whole C that U zz + V zz U z V z + U z V z 0 U z z + V z z U z V z + U z V z 0. Proof. The proof is a straightforward calculation. We only prove the first identity because the second one can be dealt with similarly. Let fz z = U zz + V zz U z V z + U z V z. A straightforward computation and 8 show that in whole C U zz z = e U V U z V z V zz z = e V U V z U z U z z = U z e U V V z z = V z e V U U z V z z = e U V V z e V U U z. Thus it holds that f z 0 and thus f z z 0 in C. Since f is smooth and goes to 0 at by 3 7 we have that by Liouville s theorem f 0 in C. The first identity is then concluded. Simply exchanging z and z in the above proof leads to the second identity. The proof is then complete. Lemma.. We have U zzz 3U z U zz + U 3 z 0 V zzz 3V z V zz + V 3 z 0 9 U z z z 3U z U z z + U 3 z 0 V z z z 3V z V z z + V 3 z Proof. Since the proofs of 9 and 10 are similar we will only check the former. For convenience we denote that f 1 z z = U zzz 3U z U zz + U 3 z f z z = V zzz 3V z V zz + V 3 z. We claim that f 1 z 0 f z 0. In fact a direct calculation gives that U zzz z = e U V zz = [e U V U z V z ] z 4

5 = e U V 4U z + 4U z V z V z U zz + V zz 3U z U zz z = 3U z z U zz 3U z U zz z = 3e U V U zz + 3e U V U z U z V z and So we have = e U V 3U zz + 6U z 3U z V z U 3 z z = 3U z U z z = e U V 3U z. f 1 z = e U V U zz + V zz U z V z + U z V z. Then Lemma.1 implies that f 1 z 0. Similarly we also have f z 0. The claim is proved. Thus we know f 1z z 0 and so does f z z. Since f 1 0 and f 0 as z again by Liouville s theorem we get 9. This concludes the proof. 3 Proof of the main theorem In what follows we discuss the kernel of the corresponding linearized operator of 8 which is equivalent to 5. All functions and equations discussed here are defined in the whole plane C. Let φ ψ be functions satisfy φ z z + e U V φ ψ = 0 ψ z z + e V U ψ φ = We prove the following proposition which gives the proof of Theorem 1.1. Proposition 3.1. Let φ ψ satisfy 11. Assume that φ C1 + z τ ψ C1 + z τ for some τ Then φ ψ belongs to the following linear space { a1 U a U b1 U b U c1 U span a1 V a V b1 V b V c1 V c U c V d1 U d1 V } d U. d V Remark 3.. Under the assumption 1 we know that all the derivatives of φ and ψ approach to 0 as z goes to. Indeed if we define that for x R φx = 1 log x y e Uy V y [φy ψy]dy 8π R 5

6 then φ C log1 + x and φ φ = 0. Therefore φ = φ + C. The potential theory implies that φ s derivatives vanish at infinity. So do the derivatives of ψ. Lemma 3.3. Under the assumption of Proposition 3.1 it holds that φ zz + ψ zz U z φ z V z ψ z + U z ψ z + V z φ z 0 φ z z + ψ z z U z φ z V z ψ z + U z ψ z + V z φ z 0. Proof. The proof is similar as that of Lemma.1 using Remark 3.. Lemma 3.4. Under the assumption of Proposition 3.1 we have φ zzz 3φ zz U z 3φ z U zz + 3U z φ z 0 φ z z z 3φ z z U z 3φ z U z z + 3U z φ z 0 ψ zzz 3ψ zz V z 3ψ z V zz + 3V z ψ z 0 ψ z z z 3ψ z z V z 3ψ z V z z + 3V z ψ z 0. Proof. We only check the first one since the others are similar. By direct computation we get using 8 φ zzz z = [e U V φ ψ] zz = [e U V U z V z φ ψ] z [e U V φ z ψ z ] z = e U V U z V z φ ψ e U V U zz V zz φ ψ e U V U z V z φ z ψ z e U V φ zz ψ zz = e U V 8U z φ + 4U z ψ + 8U z V z φ 4U z V z ψ V z φ + V z ψ 4U zz φ + U zz ψ + V zz φ V zz ψ 8U z φ z + 4U z ψ z + 4V z φ z V z ψ z φ zz + ψ zz 3φ zz U z z = 3[e U V φ ψ] z U z + 3e U V φ zz = e U V 1U z φ 6U z ψ 6U z V z φ + 3U z V z ψ + 6U z φ z 3U z ψ z + 3φ zz 3φ z U zz z = e U V 6U zz φ 3U zz ψ + 6U z φ z 3V z φ z 3U z φ z z = e U V 6U z φ z 6U z φ + 3U z ψ. 6

7 So it holds that φ zzz 3φ zz U z 3φ z U zz + 3U z φ z z = e U V [U zz + V zz U z V z + U z V z φ ψ] + e U V φ zz + ψ zz U z φ z V z ψ z + U z ψ z + V z φ z. Then Lemma.1 Lemma 3.3 and Remark 3. yield that The proof is completed. φ zzz 3φ zz U z 3φ z U zz + 3U z φ z 0. Proof of Proposition 3.1. Let φ 1 = e U φ. Since we have easily that φ zzz 3φ zz U z 3φ z U zz + 3U z φ z = e U [ φ 1zzz + U zzz 3U z U zz + U 3 z φ 1 ] using Lemma. and Lemma 3.4 we have φ 1zzz 0. holds that φ 1 z z z 0. This implies that Similarly it also φ 1 = kl=0 Since φ 1 is real it must hold that α kl z k z l with all α kl C. 13 α 00 α 11 α R and α 01 = ᾱ 10 α 0 = ᾱ 0 α 1 = ᾱ 1. On the other hand denote that ψ 1 = e V ψ. Similarly we can also obtain that ψ 1 = β kl z k z l with all β kl C 14 where β kl satisfy kl=0 β 00 β 11 β R and β 01 = β 10 β 0 = β 0 β 1 = β 1. Rewriting 11 in the term of φ 1 and ψ 1 we have φ 1z z + U z φ 1z + U z φ 1 z + e U V + U z U z φ 1 e U ψ 1 = 0 15 ψ 1z z + V z ψ 1z + V z ψ 1 z + e V U + V z V z ψ 1 e V φ 1 = Substituting into 15 and using M athematica we find that β 00 = α 11a 1 + α 00 a α 10 a b α 01 a b + α 11 a b + α 00 c α 10 cd α 01 c d + α 11 d /3 3 a 1 a 7

8 β 11 = 3 α a 1 + α 00 + a α b α 0 d α 0 d + α d 3 a 1 a β = α a + α 11 α 1 c α 1 c + α c /3 3 a 1 a 3 α1 a 1 α 0 a β 01 = b + α 1 a b + α 00 c α 10 d α 0 c d + α 1 d 3 a 1 a β 0 = α 0a α 1 ba α 01 c + α 0 c + α 11 d α 1 cd /3 3 a 1 a 3 α ba + α 01 α 0 c α 1 d + α cd β 1 = 3 a 1 a β 10 = β 01 β 0 = β 0 β 1 = β 1. Finally we insert all the above quantities into 16 again and thus obtain another relation α = α 11a 1 + α 1ca 1 + α 1 ca 1 α 00a + α 10a b + α 01a b α 11 a b a 1 a + c a 1 + a b c a b cd a bc d + a d + α 0a bc α 0a b c + α 1 a b c + α 1 a b c + α 0 a d + α 0a d a 1 a + c a 1 + a b c a b cd a bc d + a d + α 1 a bd α 1 a b d a 1 a + c a 1 + a b c a b cd a bc d + a d from which we know that φ 1 and ψ 1 actually depend on 8 real parameters rather than formally 9. Therefore the dimension of the space {φ ψ} is exactly 8. Since it is known that a1 U a1 V a U a V b1 U b1 V b U b V c1 U c1 V c U c V d1 U d1 V d U d V are linearly independent and satisfy 11 we then complete the proof of Proposition 3.1. Finally let ϕ 1 = φ ψ ϕ = ψ φ where φ ψ satisfy 11. It is easy to check that ϕ 1 ϕ satisfy 5. Thus Theorem 1.1 is equivalent to Proposition 3.1. Theorem 1.1 is concluded. Acknowledgments: The research of the first and second author is partially supported by a research Grant from GRF of Hong Kong and a Focused Research Scheme of CUHK. The third author is supported in part by NSFC 8

9 No and Shanghai project 09XD Part of this work was done while the third author was visiting CUHK he thanks the department of Mathematics for its warm hospitality. We thank Prof. Changshou Lin for useful discussions. References [1] C.C. Chen and C.S. Lin Sharp estimates for solutions of multi-bubbles in compact Riemann surface Comm. Pure Appl. Math no [] G. Dunne Self-dual Chern-Simons theories. Lecture Notes in Physics No. 36 Springer Berlin [3] J. Jost C.S. Lin and G.F. Wang Analytic aspects of Toda system. II. Bubbling behavior and existence of solutions. Comm. Pure Appl. Math no [4] J. Jost and G.F. Wang Analytic aspects of the Toda system. I. A Moser-Trudinger inequality Comm. Pure Appl. Math no [5] J. Jost and G.F. Wang Classification of solutions of a Toda system in R. Int. Math. Res. Not. 00 no [6] J.Y. Li and Y.X. Li Solutions for Toda systems on Riemann surfaces Ann. Sc. Norm. Super. Pisa Cl. Sci no [7] A.N. Leznov and M.V. Saveliev Group-theoretical methods for integration of nonlinear dynamical systems Progress in Physics Vol. 15. Birkhauser 199. [8] A. Malchiodi and C.B. Ndiaye Some existence results for the Toda system on closed surfaces Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 Math. Appl no [9] H. Ohtsuka and T. Suzuki Blow-up analysis for SU3 Toda system J. Diff. Eqns no [10] Y. Yang Solitons in field theory and nonlinear analysis. Springer Monographs in Mathematics. Springer New York