Stability of elliptic solutions to the sinh-Gordon equation
aa r X i v : . [ n li n . S I] A ug Stability of elliptic solutions to the sinh-Gordon equation
Wen-Rong Sun , and Bernard Deconinck
1. School of Mathematics and Physics,University of Science and Technology Beijing, Beijing 100083, China2. Department of Applied Mathematics,University of Washington, Seattle, WA 98195, USA
Abstract
Using the integrability of the sinh-Gordon equation, we demonstrate the spectralstability of its elliptic solutions. By constructing a Lyapunov functional using higher-order conserved quantities of the sinh-Gordon equation, we show that these ellipticsolutions are orbitally stable with respect to subharmonic perturbations of arbitraryperiod.
Key words. stability, sinh-Gordon equation, elliptic solution, subharmonic perturba-tions, integrability
AMS subject classications. . Introduction In real space-time coordinates, denoted (
X, T ), the sinh-Gordon equation reads [2, 22] u T T − u XX + sinh( u ) = 0 , (1)where partial derivatives are denoted by subscripts. Passing to light-cone coordinates ( x, t ),with x = 12 ( X + T ) , t = 12 ( X − T ) , (2)the sinh-Gordon equation becomes u xt = sinh u. (3)The sinh-Gordon equation (1) is rewritten in system form as u T = − p, (4a) p T + u XX − sinh( u ) = 0 . (4b)Here u is a real-valued function. The sinh-Gordon equation arises in the context of particularsurfaces of constant mean curvature. The geometrical interpretation of (1) was shown bystudying surfaces of constant Gaussian curvature in a three-dimensional pseudo-Riemannianmanifold of constant curvature [7]. It has appeared in differential geometry and variousapplications. For example, it can be used to describe generic properties of string dynamicsfor strings and multi-strings in constant curvature space [20]. Equation (3) is an integrablesystem and has a self-adjoint Lax pair [2]. The stability of periodic wave solutions wasstudied in [24] where it was shown that the periodic wave solutions are orbitally stable for( u, p ) ∈ H m,per ([0 , L ]) × L per ([0 , L ]), for specific choices of the traveling wave velocity. Here, H m,per ([0 , L ]) = n f ∈ H per ([0 , L ]); L R L f ( x ) dx = 0 o .In this paper, using the integrability of (1), we show the spectral and orbital stabilityof elliptic solutions of the sinh-Gordon equation with respect to arbitrary subharmonic per-turbations: perturbations that are periodic with period equal to an integer multiple of theperiod of the underlying solution. Using the same method as in [5, 6, 9, 10, 25], we provethe spectral stability of the elliptic solutions by providing an explicit analytic descriptionof the spectrum and the eigenfunctions associated with the linear stability problem of theelliptic solutions. Next, as in [5, 8, 9, 11, 25], we employ the Lyapunov method [3, 4], whichhas formed the crux of subsequent nonlinear stability techniques [16, 17, 29], to conclude theorbital stability with the help of classical results of Grillakis, Shatah and Strauss [14].
2. The elliptic solutions of the sinh-Gordon equation
We construct the real, bounded, periodic, traveling wave solutions to the sinh-Gordonequation. To obtain the traveling wave solutions, one rewrites the sinh-Gordon equation in2 frame moving with constant velocity c . With z = X − cT and τ = T , the sinh-Gordonequation becomes (cid:0) c − (cid:1) u zz − cu zτ + u ττ + sinh( u ) = 0 . (5)In what follows, we assume that c = ±
1. Stationary solutions are time-independent solutionsof (5). They satisfy the ordinary differential equation (cid:0) c − (cid:1) f ′′ ( z ) + sinh( f ( z )) = 0 , ′ := ddz . (6)Multiplying by f ′ ( z ) and integrating once,12 (cid:0) c − (cid:1) f ′ ( z ) + cosh( f ( z )) = E , (7)where E is a constant of integration referred to as the total energy.Equation (6) is rewritten as the first-order two-dimensional system f ′ ( z ) = g ( z ) , g ′ ( z ) = − f ( z )) c − , (8)with (0 ,
0) as a fixed point. The linearization about the origin has eigenvalues λ = ± r − c − . (9)Thus small-amplitude periodic solutions are expected for c − >
0. If c > E > f ( z )) = α − βv ( z ) + d, (10)where v ( z ) is a function to be determined, and α , β and d are parameters. Differentiatingand squaring (10), one obtains4 α β v (1 − βv ) (cid:18) dvdz (cid:19) = sinh ( f ( z )) (cid:18) dfdz (cid:19) . (11)Using (7), the above equation can be reduced to (cid:18) dvdz (cid:19) = [ α (1 − βv ) + 2 αd (1 − βv ) + ( d − − βv ) ][2 E − d − α − (2 E − d ) βv ]4 α β v ( c − . (12)Here sn( z, k ) denotes the Jacobi elliptic sine function with argument z and modulus k [21],which satisfies the first-order nonlinear equation (cid:18) dvdz (cid:19) = (cid:0) − v (cid:1) (cid:0) − k v (cid:1) . (13)3otivated by (13), we wish to eliminate the higher-order terms in the numerator and v from the denominator of (12). This is accomplished by equating d = 1 and α + 2 d = 0 or d = 1 and E = d + α . Case I: with the condition d = 1 and E = d + α , we cannot find elliptic solutionsfrom (12). In fact, with d = 1 and E = d + α , motivated by the form of (13), the expressionof (12) implies that β = 1 or β = 1 + α d . • For β = 1 + α d , Equation (12) becomes (cid:18) dvdz (cid:19) = (1 − βv )[2 αdβ (1 − v )]( − E + 2 d ) β α β ( c − . (14)Comparing (14) and (13), we obtain 0 < β < < k < d = 1 and E >
1, we know that α = E − d >
0. From β = 1 + α d , we know d = − < β <
1. From β = 1 − α >
0, we know α <
2. But when d = −
1, we know that α = E + 1 > • For β = 1 and d = 1, Equation (12) becomes (cid:18) dvdz (cid:19) = (1 − v ) ( α + 2 α ) (cid:2)(cid:0) − αα +2 α v (cid:1)(cid:3) ( − E + 2)4 α ( c − , (15)from which ( − E +2)( α +2 α )4 α ( c − <
0, and no elliptic solutions are obtained from (13). • For β = 1 and d = −
1, Equation (12) becomes (cid:18) dvdz (cid:19) = (1 − v ) ( α − α ) (cid:2)(cid:0) αα − α v (cid:1)(cid:3) ( − E − α ( c − . (16)Since d = − α = E − d >
2, which implies that − αα − α <
0. Therefore we cannot obtainelliptic solutions from (13).
Case II: we consider d = 1 and α + 2 d = 0.Equation (12) is reduced to (cid:18) dvdz (cid:19) = (1 − βv )[2 E − d − α − (2 E − d ) βv ]4 β ( c − , (17)so that v = sn( bz, k ) , k = r E − E + 1 , α = 2 , β = k , d = − , b = s E + 12( c − . (18)It follows that cosh( f ( z )) = 21 − k sn( bz, k ) − . (19)4he solution is periodic with period T ( k ) = b , where K ( k ) = Z π/ d y p − k sin ( y ) , (20)the complete elliptic integral of the first kind, see [21].
3. The linear stability problem
In this section, we examine the stability of the elliptic solutions obtained above. Con-sidering the perturbation of a stationary solution to (5), u ( z, τ ) = f ( z ) + ǫw ( z, τ ) + O (cid:0) ǫ (cid:1) , (21)where ǫ is a small parameter, we obtain the linear stability problem (cid:0) c − (cid:1) w zz − cw zτ + w ττ + cosh( f ( z )) w = 0 . (22)With w ( z, τ ) = w ( z, τ ) and w ( z, τ ) = cw z ( z, τ ) − w τ ( z, τ ), the linear problem is rewrittenas ∂∂τ w w ! = c∂ z − − ∂ z + cosh( f ( z )) c∂ z ! w w ! . (23)We note that (23) is autonomous in time. By separating variables, w ( z, τ ) w ( z, τ ) ! = e λτ W ( z ) W ( z ) ! , (24)the linear problem (23) is rewritten as λ W W ! = J L W W ! = c∂ z − − ∂ z + cosh( f ( z )) c∂ z ! W W ! , (25)where J = − ! , L = ∂ z − cosh( f ( z )) − c∂ z c∂ z − ! . (26)Note that L is formally self adjoint. We define σ J L = (cid:26) λ ∈ C : sup x ∈ R ( | W ( x ) | , | W ( x ) | ) < ∞ (cid:27) . (27)Spectral stability of an elliptic solution with respect to perturbations that are boundedon the whole line is established by demonstrating that the spectrum σ ( J L ) of the operator5 L does not intersect the right-half complex λ plane. Because the sinh-Gordon equation isa Hamiltonian partial differential equation [22], the spectrum is symmetric under reflectionwith respect to both the real and imaginary axes [30]. As a consequence, spectral stabilityrequires that σ ( J L ) is confined to the imaginary axis.
4. The Lax pair restricted to the elliptic solution
Equation (3) admits the following Lax pair [2]: ψ x = − iζ u x u x iζ ! ψ, ψ t = i cosh u ζ − i sinh u ζi sinh u ζ − i cosh u ζ ! ψ. (28)From (28), one has the following spectral problem: i∂ x − i u xi u x − i∂ x ! ψ = ζ ψ. (29)Since the spectral problem is self adjoint, the Lax spectrum σ L is a subset of the real line: σ L := { ζ ∈ C : sup x ∈ R ( | ψ | , | ψ | ) < ∞} ⊂ R .Through (2), (1) admits the following Lax pair: ψ X = − i ζ + i cosh u ζ u X + u T − i sinh u ζu X + u T + i sinh u ζ i ζ − i cosh u ζ ! ψ,ψ T = − i ζ − i cosh u ζ u X + u T + i sinh u ζu X + u T − i sinh u ζ i ζ + i cosh u ζ ! ψ. We transform the Lax pair by moving into a traveling reference frame, letting z = X − cT , τ = T and u ( z, τ ) = f ( z ). The Lax pair restricted to the stationary solution is ψ z = − i ζ + i cosh f ( z )8 ζ (1 − c ) f ′ ( z )4 − i sinh f ( z )8 ζ (1 − c ) f ′ ( z )4 + i sinh f ( z )8 ζ i ζ − i cosh f ( z )8 ζ ! ψ, (30) ψ τ = A BC − A ! ψ = − i (1+ c )2 ζ + ( c − i cosh f ( z )8 ζ (1 − c )( c +1) f ′ ( z )4 + (1 − c ) i sinh f ( z )8 ζ (1 − c )( c +1) f ′ ( z )4 + ( c − i sinh f ( z )8 ζ ( c + 1) i ζ + (1 − c ) i cosh f ( z )8 ζ ! ψ. Since A , B and C are independent of τ , we separate variables to look for solutions of theform ψ ( z, τ ) = e Ω τ ϕ ( z ) , (31)where Ω is independent of τ . We substitute (31) into the τ -part of the Lax pair and obtain A − Ω BC − A − Ω ! ϕ = 0 . (32)6his implies that the existence of nontrivial solutions requiresΩ = A + BC = − ζ ( c + 1) − ζ ( c − E + ( c − ζ , (33)where we have used the explicit form of the stationary solution f ′ ( z ) derived earlier. Equa-tion (33) determines Ω in terms of ζ . Since ζ is real, Ω is real or imaginary. Further, Ω isan even function of ζ . From the discriminant of (33), we know that with E > c > = − ( ζ − ζ )( ζ − ζ )( ζ + ζ )( ζ + ζ )4 ζ , (34)where ζ = q ( c − E−√E − c +1) and ζ = q ( c − E + √E − c +1) , see Figure 1. - Ζ - W Figure 1
The graph of Ω as a function of real ζ . The eigenvectors corresponding to the eigenvalue Ω are ϕ ( z ) = γ ( z ) − B ( z ) A ( z ) − Ω ! , (35)where γ ( z ) is a scalar function. It is determined by substitution of (35) into the z -part ofthe Lax pair, resulting in a first-order scalar differential equation for γ ( z ), so that γ ( z ) = exp Z i ζ − i cosh f ( z )8 ζ − B h (1 − c ) f ′ ( z )4 + i sinh f ( z )8 ζ i + A ′ A − Ω d z = 1 A − Ω exp Z i ζ − i cosh f ( z )8 ζ − B h (1 − c ) f ′ ( z )4 + i sinh f ( z )8 ζ i A − Ω d z . (36)7xcluding the branch points, where Ω = 0, each ζ results in two values of Ω. Thus (35)represents two linearly independent solutions. When ζ = ζ or ζ = ζ , only one solution isobtained. The second one may be obtained using the reduction of order method.Since the vector part of the eigenvector ϕ ( z ) is bounded in z , we need to check for which ζ the scalar function γ is bounded for all z , including as | z | → ∞ . A necessary and sufficientcondition for this is that [5, 6] * Re − B h (1 − c ) f ′ ( z )4 + i sinh f ( z )8 ζ i + A ′ A − Ω + = 0 . (37)Here h·i = T R T − T · dz denotes the average over a period. The explicit form of the abovedepends on Ω. Since ζ is real, Ω is real or imaginary from (33). Recently, Upsal andDeconinck [28] demonstrated that purely real Lax spectrum implies spectral stability. Weshow this explicitly below.a) When Ω is imaginary, the integrand of (37) isRe − B h (1 − c ) f ′ ( z )4 + i sinh f ( z )8 ζ i A − Ω + Re (cid:18) − A ′ A − Ω (cid:19) . (38)The second term is a total derivative, with zero average over a period. For the first term,Re − B h (1 − c ) f ′ ( z )4 + i sinh f ( z )8 ζ i A − Ω = f ′ ( z ) (cid:20) sinh f ( z )(1 − c ) + sinh f ( z )(1 − c ) ζ ( − iA + i Ω) (cid:21) , (39)which is a total derivative, resulting in zero average over a period. Therefore, all ζ valuesfor which Ω is imaginary are in the Lax spectrum.b) If Ω is real, the second term is still a total derivative, thus giving zero average over aperiod. We consider the first term:Re − B h (1 − c ) f ′ ( z )4 + i sinh f ( z )8 ζ i A − Ω = Ω (1+ c )(1 − c ) f ′ ( z ) + ( c − ( f ( z )))64 ζ Ω + A + f ′ ( z ) F ( f ( z )) . (40)The second term is a total derivative, thus giving zero average over a period. The first termresults in a zero average only when Ω is zero. Thus all values of ζ for which Ω is real (exceptfor the four branch points, where Ω = 0) are not part of the Lax spectrum.We conclude that the Lax spectrum consists of all ζ values for which Ω ≤ σ L = ( −∞ , − ζ ] ∪ [ − ζ , ∪ (0 , ζ ] ∪ [ ζ , ∞ ) . (41)Moreover, Ω takes on all negative values for ( −∞ , − ζ ], implying that Ω covers the imagi-nary axis. The same is true for the other intervals. ThusΩ ∈ ( i R ) , ζ ∈ σ L , (42)8here the exponent denotes multiplicity.
5. The squared eigenfunction connection
It is well known that there exists a connection between the eigenfunctions of the Laxpair of an integrable equation and the eigenfunctions of the linear stability problem for thisintegrable equation [2, 5, 6, 9, 10, 25–27].
Theorem 1
The difference of squares, w ( z, τ ) = ψ ( z, τ ) − ψ ( z, τ ) , (43) satisfies the linear stability problem (22). Here ψ = ( ψ , ψ ) T is any solution of the Laxpair (30). To establish the connection between the σ J L spectrum and the σ L spectrum we examinethe right- and left-hand sides of (24). Substituting (43) and (31) into (24), e τ ϕ − ϕ −
2Ω ( ϕ − ϕ ) + 2 c ( ϕ ϕ z − ϕ ϕ z ) ! = e λτ W ( z ) W ( z ) ! , (44)so that λ = 2Ω( ζ ) , (45)with W ( z ) W ( z ) ! = ϕ − ϕ −
2Ω ( ϕ − ϕ ) + 2 c ( ϕ ϕ z − ϕ ϕ z ) ! . (46) We note that all but three solutions of (25) may be written in this form. Specifically, allsolutions of (25) bounded on the whole real line are obtained through the squared eigenfunctionconnection.
This is proven as in [5, 6, 9, 25]: For a fixed λ , (25) is a second-order linear ordinarydifferential equation, thus it has two linearly independent solutions. We know that (46)provides solutions of this ordinary differential equation. We count how many solutions areobtained in this way for a given λ . Note that Ω is an even function of ζ . Excluding the twovalues of λ for which the discriminant of (35) as a function of ζ is zero, (33) gives rise to fourvalues of ζ ∈ C . We revisit these values in (b), below. For all other values of λ = 2Ω, anyfixed Ω and ζ defines a unique solution (up to a multiplicative constant) of the Lax pair. Asin [5, 6, 9, 25], there are two parts for this.(a) For any λ not equal to the two values mentioned earlier, we obtain four solutionsthrough the squared eigenfunction connection. Since Ω is an even function of ζ , the Laxparameters come in {− ζ , ζ } pairs. As in [9], only one element of these pairs gives rise toan independent solution of the stability problem, eliminating two of these four solutions.On the other hand, as in [6], if there is an exponential contribution from γ , the remaining9wo solutions are linearly independent. The only possibility for the exponential factor from γ not to contribute is if Ω = 0. In that case, only one linearly independent solution isobtained through the squared eigenfunction connection, corresponding to ( f z , cf zz ) T . Theother solution is obtained through the reduction of order method, and introduces algebraicgrowth.(b) Let us consider the two excluded values of λ . For the two λ values where Ω reaches itsmaximum value, only one solution is obtained through the squared eigenfunction connection,which is unbounded. The second one may be constructed using reduction of order, andintroduces algebraic growth.As a consequence of the discussion above, the double covering in the Ω representation(42) drops to a single covering. In summary, we have the following theorem. Theorem 2
The periodic traveling wave solutions of the sinh-Gordon equation are spec-trally stable. The spectrum of their associated linear stability problem is explicitly given by σ ( J L ) = i R , or, accounting for multiple coverings, σ ( J L ) = ( i R ) . (47)An application of the SCS basis lemma in [18] establishes that the eigenfunctions form abasis for L per ([ − N T , N T ]), for any integer N , when the potential f is periodic in z with pe-riod T . This allows us to conclude linear stability with respect to subharmonic perturbations.
6. Orbital stability
In this section, we study the nonlinear stability of the elliptic solutions. We are concernedwith the stability of T -periodic solutions of equation with respect to subharmonic perturba-tions of period N T , for any fixed positive integer N . We require u and its derivatives of upto order three to be square-integrable.To prove orbital stability, we prove formal stability first: we construct a Lyapunov func-tional for the elliptic solutions using the conserved quantities of the sinh-Gordon equation.We introduce the Hamiltonian structure of the sinh-Gordon equation and its hierarchy. Weuse the system form of the sinh-Gordon equation: u τ = − p + cu z , (48) p τ = cp z − u zz + sinh( u ) . (49)The Hamiltonian structure is [2, 22] ∂u/∂τ∂p/∂τ ! = J δH/δuδH/δp ! , (50)with H = − Z N T − N T (cid:20) p + 12 ( u z ) + cosh u − cpu z (cid:21) dz, (51)10nd J = − ! .The Hamiltonian is one of an infinite number of conserved quantities of the sinh-Gordonequation. We label these quantities { H j } ∞ j =0 . We need the first three conserved quantities: H = − Z N T − N T pu z dz,H = − Z N T − N T (cid:20) p + 12 ( u z ) + cosh u (cid:21) dz,H = Z N T − N T (cid:2) − u zz p z − u z ) p − u z p + 48 p z sinh u (cid:3) dz. In fact, all the functionals H i are mutually in involution under the Poisson bracket. Theconservation and the involution properties for H , H , and H are straightforward to verify.Each H i defines an evolution equation with respect to a time variable τ i by ∂∂τ i up ! = J H ′ i ( p, u ) , (52)where H ′ i ( u, p ) denotes the variational gradient of H i . The collection of (52) with i = 0 , , . . . is the sinh-Gordon hierarchy. Its first three members are ∂∂τ up ! = − u z − p z ! ,∂∂τ up ! = − p sinh u − u zz ! ,∂∂τ up ! = − u z − u z p − u z cosh u + 64 u zzz − p z cosh u − p p z − p z u z − pu zz u z + 64 p zzz ! . Since the flows in the sinh-Gordon hierarchy commute, we may take any linear combinationof the above Hamiltonians to define a new Hamiltonian system. We define the n -th sinh-Gordon equation with evolution variable t n as ∂∂t n up ! = J ˆ H ′ n ( u, p ) , (53)ˆ H n = H n + n − X i =0 c n,i H i , n > , (54)where the coefficients c n,i are constants. For example, ˆ H = H − cH = H is the Hamiltonianof the sinh-Gordon equation (48) and (49), as shown in (51).11very member of the sinh-Gordon hierarchy has a Lax pair. These Lax pairs share thesame first component − T , while the second component ψ τ j = T j ψ is different: ψ τ = T ψ = − i cosh u ζ − i ζ ( u z − p ) − i sinh u ζ ( u z − p ) + i sinh u ζ − i cosh u ζ + i ζ ! ψ,ψ τ = T ψ = − i cosh u ζ − i ζ ( u z − p ) + i sinh u ζ ( u z − p ) − i sinh u ζ i cosh u ζ + i ζ ! ψ,ψ τ = T ψ = A B C − A ! ψ, where A = 32 iζ − i cosh u ζ + 8 iζ (cid:18) − pu z + 12 p + 12 u z (cid:19) + 2 ζ (cid:18) − ip z csch u + 2 ip z cosh u coth u − ipu z cosh u − ip cosh u − iu z cosh u − iu zz csch u + 2 iu zz cosh u coth u − i cosh u + i (cid:1) ,B = i sinh u ζ + 116 iζ (256 iu z − ip ) + p + u z ζ + 8 iζ (2 p z − u zz + sinh u )+ i sinh uζ ( − p z coth u + 2 pu z + p + u z − u zz coth u + 2 cosh u ) + i
128 ( − ip cosh u − ip + 2048 ip zz − ipu z + 768 ip u z + 256 iu z − iu zzz + 1536 iu z cosh u ) ,C = − i sinh u ζ + 116 iζ (256 iu z − ip ) + p + u z ζ − iζ (2 p z − u zz + sinh u ) − i sinh uζ ( − p z coth u + 2 pu z + p + u z − u zz coth u + 2 cosh u ) + i
128 ( − ip cosh u − ip + 2048 ip zz − ipu z + 768 ip u z + 256 iu z − iu zzz + 1536 iu z cosh u ) . We construct the Lax pair for the n -th sinh-Gordon equation by taking the same linearcombination of the lower-order flows as for the nonlinear hierarchy: ψ t n = ˆ T n ψ = ˆ A n ˆ B n ˆ C n − ˆ A n ! ψ, (55)ˆ T n = T n + n − X i =0 c n,i T i , ˆ T = T . (56)We study the stationary solutions of the sinh-Gordon hierarchy. Since the flows commute,a stationary solution of the n th equation remains a stationary solution after evolving underany of the other flows, for a suitable choice of the coefficients c n,i .For example, the traveling wave solutions ( f, cf z ) are the stationary solutions of the firstequation in the sinh-Gordon hierarchy with c , = − c . They are also stationary solutions of12he second equation in the sinh-Gordon hierarchy provided16 ( − c − E ( c − − c , c − c , = 0 . (57)This gives one condition for the two coefficients c , and c , . In order to proceed as inRefs. [5, 15], we consider stability in the space of subharmonic functions of period N T , for1 ≤ N ∈ N , i.e., V ,N = (cid:26) W : W ∈ H per ([ − N T , N T (cid:27) . (58)To prove the orbital stability of the solution ( u, p ) in this space, we construct a Lyapunovfunction [15, 23], i.e., a constant of the motion E ( u, p ) for which ( u, p ) is an unconstrainedminimizer: d E ( u, p ) dτ = 0 , E ′ ( u, p ) = 0 , h v, L ( u, p ) v i > , ∀ v ∈ V , v = 0 , (59)where E ′ ( u, p ) denotes the variational gradient of E and L is the Hessian of E . The existenceof such a function implies formal stability. We know that ( f z , cf zz ) T is in the kernel of H ′′ = L . This is obtained from the action of the infinitesimal generator ∂ z , acting on( f ( z ) , p ( z )) T , where p ( z ) = cf z . Following Grillakis, Shatah, and Strauss [14, 15], underextra conditions (see the orbital stability theorem in [5, 8, 9, 25]) this allows one to concludeorbital stability. Since the sinh-Gordon equation is an integrable Hamiltonian system, all theconserved quantities of the equation satisfy the first two conditions. It suffices to constructone that satisfies the third requirement.To prove orbital stability, we check the Krein signature K [14], associated with ˆ H : K = h W, L W i = Z N T − N T W ∗ L W dz, (60)where L = L . Using the squared eigenfunction connection, we have W ∗ L W = 2Ω W ∗ J − W = 2Ω ( W W ∗ − W W ∗ )= 8Ω ( | ϕ | + | ϕ | − ϕ ϕ ∗ − ϕ ϕ ∗ ) + 2Ω(2 cϕ ϕ ∗ ϕ ∗ z + 2 cϕ ϕ ∗ ϕ ∗ z − cϕ ϕ ∗ ϕ ∗ z − cϕ ϕ ∗ ϕ ∗ z − cϕ ∗ ϕ ϕ z − cϕ ∗ ϕ ϕ z + 2 cϕ ∗ ϕ ϕ z + 2 cϕ ∗ ϕ ϕ z ) , (61)with ϕ = − γ ( z ) B ( z ) and ϕ = γ ( z )( A ( z ) − Ω). Using (36), we have | γ | = 1 | A − Ω | . (62)13ith Ω = A + | B | , we have | ϕ | = − ( A + Ω) , | ϕ | = − ( A − Ω) , ϕ ϕ ∗ = − B ∗ , ϕ ϕ ∗ = − B ,ϕ ϕ ∗ ϕ ∗ z − ϕ ∗ ϕ ϕ z = A + Ω A − Ω ( BB ∗ z − B ∗ B z ) − (Ω + A ) (Θ ∗ − Θ) ,ϕ ϕ ∗ ϕ ∗ z − ϕ ∗ ϕ ϕ z = − ( A − Ω) (Θ ∗ − Θ) ,ϕ ∗ ϕ ϕ z − ϕ ϕ ∗ ϕ ∗ z = B ∗ (Θ ∗ − Θ) + B ∗ B ∗ z − B ∗ A − Ω A z ,ϕ ∗ ϕ ϕ z − ϕ ϕ ∗ ϕ ∗ z = B (Θ ∗ − Θ) − BB z + B A − Ω A z , where Θ = i ζ − i cosh f ( z )8 ζ − B h (1 − c ) f ′ ( z )4 + i sinh f ( z )8 ζ i + A z A − Ω . (63)It follows that the Krein signature K can be expressed as K = h W, L W i = − Ω Z N Kb − N Kb (cid:18) − ζ ( c + 1) − ζ cosh( f ( z )) + c − ζ (cid:19) dz = − N Ω b (cid:20)(cid:18) − cζ − ζ (1 + c ) + 8 (cid:19) K ( k ) − E ( k )1 − k (cid:21) (64)= − N Ω P ( ζ ) bζ , (65)where E ( k ) is the complete elliptic integral of the second kind [21]: E ( k ) = Z π/ q − k sin y d y. (66)We have the following properties:1. P ( ζ ) is an even function and the discriminant of P ( ζ ) is positive. Since − c − c ) < P ( ζ ) = 0 has two real roots ± ζ c ( ζ c > K ( ζ ) = 0, when Ω( ζ ) = 0 or ζ = ± ζ c . Since d P ( ζ ) ζ dζ = − ( ζ ( c +1)+ c − ) K (cid:16)q E− E +1 (cid:17) ζ , we know that for c >
1, when ζ > P ( ζ ) ζ decreases along ζ and when ζ < P ( ζ ) ζ increases along ζ . For c < −
1, when ζ > P ( ζ ) ζ increases along ζ and when ζ < P ( ζ ) ζ decreases along ζ .2. Since ζ < ζ c < ζ , ± ζ c is not in σ L (see Appendix), K = 0 is obtained only on the ker-nel of L , i.e., when Ω = 0. For c < − ζ c = r − √ c K ( k ) + E ( k )( E +1) − E ( k )( E +1) K ( k ) − E ( k )( E +1)+ K ( k )( c +1) K ( k ) .For c > ζ c = r √ c K ( k ) + E ( k )( E +1) − E ( k )( E +1) K ( k ) − E ( k )( E +1)+ K ( k )( c +1) K ( k ) . We note that ζ c , ζ and ζ are all greater than zero.3. When c >
1, since P ( ζ ) ζ > | ζ | < ζ , K > | ζ | < ζ and since P ( ζ ) ζ < | ζ | > ζ , K < | ζ | > ζ . When c < −
1, since P ( ζ ) ζ < | ζ | < ζ , K < | ζ | < ζ P ( ζ ) ζ > | ζ | > ζ , K > | ζ | > ζ . Thus no conclusion about orbitalstability can be drawn from K .It follows that ˆ H is not a Lyapunov function. Thus we need to consider different con-served quantities. Linearizing the n -th sinh-Gordon equation about the equilibrium solution f , one obtains w t n = J L n w, (67)where L n is the Hessian of ˆ H n evaluated at the stationary solution.Using the squared-eigenfunction connection with separation of variables gives2Ω n W ( z ) = J L n W ( z ) , (68)where Ω n is defined through ψ ( z, t n ) = e Ω n t n ϕ ( z ) . (69)Due to the commuting property of the different flows in the sinh-Gordon hierarchy, theLax hierarchy shares the common set of eigenfunctions ϕ ( z ) from before (still assuming thesolution is stationary with respect to the first, and hence all higher flows). Substituting (69)into the Lax pair of the n -th sinh-Gordon equation determines a relationship between Ω n and ζ Ω n ( ζ ) = ˆ A n + ˆ B n ˆ C n . (70)To find a Lyapunov functional, we check K : K = Z N T − N T W ∗ L W dz = 2Ω Z N T − N T W ∗ J − W dz = Ω Ω Z N T − N T W ∗ L W dz. (71)Therefore, we have K ( ζ ) = Ω ( ζ ) K ( ζ )Ω( ζ ) . (72)Here, we use that ( f, cf z ) are the stationary solutions of the second flow. To calculate K ,we also need ˆ T = T + c , T + c , T , (73)where, from before, 16 ( − c − E ( c − − c , c − c , = 0 . (74)The second sinh-Gordon equation can be expressed as ∂∂t up ! = J ( H ′ + c , H ′ + c , H ′ ) = 0 . (75)15 direct calculation givesΩ = [ c (64 ζ + 16 ζ E + ζ c , + 4) + c (4 − ζ ) + ζ (16 E − c , )] c ( c − ζ Ω . (76)We can choose c , = − c ζ c + 4 c ζ c E + c − cζ c + c + 4 ζ c E )( c − ζ c , (77)to ensure K has definite sign. With this choice of c , and c , determined by (74), ˆ H is aLyapunov functional for the dynamics (with respect to any of the time variables in the sinh-Gordon hierarchy) of the stationary solutions. Therefore, whenever solutions are spectrallystable with respect to subharmonic perturbations of period N , they are formally stable in V ,N .Since the infinitesimal generators of the symmetries correspond to the values of ζ forwhich Ω( ζ ) = 0, the kernel of the functional ˆ H ′′ ( u, p ) consists of the infinitesimal generatorsof the symmetries of the solution ( u, p ). On the other hand, since ± ζ c is not in σ L , K ( ζ ) = 0is obtained only when Ω = 0 for ζ ∈ σ L . We have proved the following theorem. Theorem 3 (Orbital stability)
The elliptic solutions (19) of the sinh-Gordon equationare orbitally stable with respect to subharmonic perturbations in V ,N , N ≥ Acknowledgments
WS has been supported by the National Natural Science Foundationof China under Grant No.61705006, and by the Fundamental Research Funds of the CentralUniversities (No.230201606500048).
Appendix
Lemma.
For c > , P ( ζ ) ζ > and P ( ζ ) ζ < , while for c < − , P ( ζ ) ζ < and P ( ζ ) ζ > . Proof. • For c > P ( ζ ) ζ = 8 c √E − K r E − E + 1 ! + 8( E + 1) K r E − E + 1 ! − E + 1) E r E − E + 1 ! . (78)Since E ( k ) < K ( k ), c > E >
1, we have P ( ζ ) ζ > P ( ζ ) ζ = 8 c (cid:16) −√E − (cid:17) K r E − E + 1 ! + 8( E + 1) K r E − E + 1 ! − E + 1) E r E − E + 1 ! . (79)Let P ( ζ ) ζ = F ( c ). We note that F ′ ( c ) = 8 (cid:0) −√E − (cid:1) K (cid:16)q E− E +1 (cid:17) <
0. We have F ( c ) < F (1) = 8 (cid:16) −√E − (cid:17) K r E − E + 1 ! + 8( E + 1) K r E − E + 1 ! − E + 1) E r E − E + 1 ! . (80)16sing E ( k ) K ( k ) > k ′ = √ − k , see [1, 19.9.8], we have8 (cid:16) −√E − (cid:17) + 8( E + 1) − E + 1) E (cid:16)q E− E +1 (cid:17) K (cid:16)q E− E +1 (cid:17) < (cid:16) −√E − (cid:17) + 8( E + 1) − √ √E + 1 . (81)Let Q ( E ) = 8 (cid:0) −√E − (cid:1) + 8( E + 1) − √ √E + 1. We note Q ′ ( E ) = − E√E − + 8 − √ √E +1 < − √ √E +1 <
0. So we have Q ( E ) < Q (1) = 0 for E >
1. Therefore, we have P ( ζ ) ζ = F ( c )
0. We have G ( c ) < G ( −
1) = 8 (cid:16) −√E − (cid:17) K r E − E + 1 ! + 8( E + 1) K r E − E + 1 ! − E + 1) E r E − E + 1 ! . (83)Again, using E ( k ) K ( k ) > k ′ = √ − k , we have8 (cid:16) −√E − (cid:17) + 8( E + 1) − E + 1) E (cid:16)q E− E +1 (cid:17) K (cid:16)q E− E +1 (cid:17) < (cid:16) −√E − (cid:17) + 8( E + 1) − √ √E + 1 . (84)We know Q ( E ) < Q (1) = 0 for E >
1. Therefore, P ( ζ ) ζ = G ( c ) < G ( − < K (cid:16)q E− E +1 (cid:17) Q ( E ) < P ( ζ ) ζ = 8 c (cid:16) −√E − (cid:17) K r E − E + 1 ! + 8( E + 1) K r E − E + 1 ! − E + 1) E r E − E + 1 ! . (85)Since E ( k ) < K ( k ), c < − E >
1, we have P ( ζ ) ζ >
0. This finishes the proof of theLemma.