Discrete breathers are spatially localized periodic solutions in nonlinear lattices. The existence of odd and even symmetric single-pulse and multi-pulse discrete breathers has been proved in the one-dimensional Fermi–Pasta–Ulam–Tsingou lattices with even interaction potentials [Yoshimura and Doi, J. Differ. Equations 298, 560–608 (2021)]. We prove that those discrete breathers are exponentially localized in space.

A variety of physical systems are modeled as nonlinear space-discrete dynamical systems, i.e., nonlinear lattices. Examples of such physical systems are as diverse as solid crystals, polymer molecules, coupled nonlinear optical waveguide arrays, and Bose–Einstein condensates in optical lattice potentials. Takeno et al. found a time-periodic and spatially localized mode in the Fermi–Pasta–Ulam–Tsingou (FPUT) lattice based on approximate analytical calculation, and more importantly, they pointed out that spatially localized modes can exist ubiquitously in various nonlinear lattices.1,2 A few years later, a different type of localized mode was also found for the FPUT lattice.3 The localized modes are called discrete breathers (DBs) or intrinsic localized modes. The DB has attracted great interest, and considerable progress has been achieved in understanding the nature of DB so far (e.g., Refs. 4–8 and references therein).

The first existence proof of DB was given for the nonlinear Klein–Gordon lattice, based on the anti-continuous limit.9 This limit is a useful concept and has been applied to prove the existence of a variety of DBs, which are single-pulse and multi-pulse ones, for several lattice models such as the nonlinear Schrödinger lattice9–12 and the diatomic FPUT lattice.13 In addition, rigorous stability results for the multi-pulse DBs also have been given near the anti-continuous limit.14–20 

The one-dimensional FPUT lattice is one of the fundamental lattice models in physics, to which the anti-continuous limit approach is not applicable. Two types of single-pulse DBs having different spatial symmetries are known for this model: one is called the odd mode and has a site-center profile,1,2 and the other is called the even mode and has a bond-center profile.3 Normalized spatial profiles of the odd and even DBs are approximately given by (…, 0, −1/2, 1, −1/2, 0, …) and (…, 0, −1, 1, 0, …) in the regime of strong localization, respectively. In addition, multi-pulse DBs also have been found numerically.21 

For the FPUT model, the first proof of the odd and even DBs was given in a particular case of the homogeneous potential.22 In the more general case of non-homogeneous potentials, existence proofs of the odd and even DBs have been given by using a center manifold reduction technique23 and Schauder’s fixed point theorem.24 There has developed a different approach that constructs a DB solution in a homogeneous potential lattice by using Banach’s fixed point theorem and then continues it to non-homogeneous potential ones.25,26 One advantage of this approach is that it gives precise quantitative estimates for the core profiles of DBs and has justified the normalized profiles of the odd and even DBs predicted in Refs. 1–3, although the applicability of the approach is limited to non-homogeneous potential lattices close to a homogeneous potential one. More advantageously, the existence of infinitely many multi-pulse DBs has been proved by using the approach, each of which consists of a finite number of the odd-like and/or even-like DBs located separately on the lattice.26 

In this paper, we discuss the localization property of the odd, even, and multi-pulse DBs proved in Ref. 26. Their localization has been proved only in the l2 space. However, a common feature of DBs is the exponential localization of their profiles in space. We prove the exponential localization of those odd, even, and multi-pulse DBs by slightly modifying and using a technique developed in Ref. 9 to improve the estimation of the localization rate.

This paper is organized as follows: In Sec. II, we describe the FPUT lattice model and introduce some sequence and function spaces. In Sec. III, we describe a theorem for the existence of odd, even, and multi-pulse DBs, which was given in Ref. 26. In Sec. IV, we state the main theorem and describe its proof.

We consider the one-dimensional infinite FPUT lattice described by the equations of motion,
q̇i=pi,ṗi=V(qi+1qi)V(qiqi1),iZ,
(1)
where qi,piR are the position and momentum of ith particle of unit-mass, respectively, and V is a potential function of the nearest neighbor interaction. Equation (1) forms an infinite system of ordinary differential equations, which has the Hamiltonian,
H=i=12pi2+i=V(qi+1qi).
(2)
Equation (1) is derived by q̇i=H/pi and ṗi=H/qi.
Let XR and μR be a parameter. We assume the interaction potential V to be defined by
V(X)=μW(X)+1kXk,
(3)
where
(P1)k4 is an even integer;(P2)W(X):RRis aC3function ofX;(P3)W(X)=W(X)forXRandW(0)=0.
Condition (P3) ensures that V(X) is an even function of X. A typical non-homogeneous potential often used in the literature is a polynomial potential. Equation (3) gives this case when W is an even polynomial of order less than k, i.e., W(X)=r=1k/21κ2rX2r.
Let l2(Z) be the Hilbert space of square-summable two-sided real-valued sequences endowed with the norm xl2=x,x, which is derived from the inner product x,y=iZxiyi, where x=(xi)iZ and y=(yi)iZ. Let q=(qi)iZl2(Z) and p=(pi)iZl2(Z). We choose the phase space of Eq. (1) as Ω=l2(Z)×l2(Z) with the norm,
zΩ=ql22+pl22,
(4)
where z=(zi)iZΩ and zi=(qi,pi)R2. The phase space Ω is also the Hilbert space with the inner product z,zΩ=iZqiqi+pipi.
Let CT0(R;Ω) be the space of T-periodic continuous functions z(t):RΩ that is endowed with the norm,
z0=supt[0,T]z(t)Ω.
(5)
Let CT1(R;Ω) be the space of T-periodic continuously differentiable functions z(t):RΩ that is endowed with the norm,
z1=supt[0,T]z(t)Ω+supt[0,T]ż(t)Ω.
(6)
Both CT0(R;Ω) and CT1(R;Ω) are Banach spaces.
We define two Banach spaces X1 and X0 by imposing some temporal symmetry conditions to CT1(R;Ω) and CT0(R;Ω), respectively, as follows:
X1=(qi(t),pi(t))iZCT1(R;Ω);qi(t)=qi(t),pi(t)=pi(t),qi(t+T/2)=qi(t),pi(t+T/2)=pi(t),iZ,
(7)
and
X0=(si(t),ri(t))iZCT0(R;Ω);si(t)=si(t),ri(t)=ri(t),si(t+T/2)=si(t),ri(t+T/2)=ri(t),iZ.
(8)
Existence of the odd and even symmetric single-pulse DB solutions and multi-pulse DB solutions has been proved in the space X1 for the FPUT lattice given by Eq. (1).26 In this study, we address the exponential localization of these DB solutions. To discuss their localization property, we need to measure the amplitude of each component zi(t) of a DB solution z(t). Let CT0(R;R2) and CT1(R;R2) be the spaces of T-periodic continuous functions and T-periodic continuously differentiable functions with values in R2, which are endowed with the following norms, respectively:
|z|0=supt[0,T]|z(t)|,
(9)
and
|z|1=supt[0,T]|z(t)|+supt[0,T]|ż(t)|,
(10)
where z(t)=(q(t),p(t))R2 and |·| represents the Euclidean norm of R2. Both CT0(R;R2) and CT1(R;R2) are Banach spaces.
Define two Banach spaces b1 and b0 as follows:
b1=(q(t),p(t))CT1(R;R2);q(t)=q(t),p(t)=p(t),q(t+T/2)=q(t),p(t+T/2)=p(t),
(11)
and
b0=(s(t),r(t))CT0(R;R2);s(t)=s(t),r(t)=r(t),s(t+T/2)=s(t),r(t+T/2)=r(t).
(12)

These spaces are obtained by imposing CT1(R;R2) and CT0(R;R2) the same types of temporal symmetries as to those of X1 and X0, respectively. Each component of any zX1 (resp. wX0) can be regarded as an element of b1 (resp. b0).

In this section, we describe a theorem for the existence of DB solutions, which has been proved in Ref. 26, after introducing some preliminary notions to precisely state the theorem.

We describe the odd and even symmetries in the case of even interaction potentials, i.e., V(X) = V(−X). Let SO and SE be the linear mappings SO, SE: Ω → Ω defined by
SO:(SOz)i=zi,iZ,SE:(SEz)i=z(i+1),iZ,
where zi = (qi, pi) and z=(zi)iZΩ. A T-periodic solution z(t) ∈ X1 of Eq. (1) is said to have odd (resp. even) symmetry if it satisfies the condition SOz(t) = z(t) [resp. SEz(t) = z(t)] for tR. The odd and even symmetric solutions have a spatial profile centered at i = 0 site and that centered between i = −1 and 0 sites, respectively.

The existence theorem uses approximations for the spatial profiles of DB solutions. Let aO=(aiO)iZ and aE=(aiE)iZ be two-sided real-valued bounded sequences, each of which satisfies the condition that there are i1,i2Z and aix0 only for i1ii2, otherwise aix=0, where the superscript x represents the sequence type O or E. We will choose aO and aE as in Table I to approximate the odd and even single-pulse DB profiles in a lattice with μ = 0 and an even integer k ≥ 4.

TABLE I.

Approximation of odd and even single-pulse DB profiles.

k = 4 a0O=0.3762, a±1O=0.1968, a±2O=0.00867, aiO=0 (otherwise) 
a0E=a1E=0.32301, a1E=a2E=0.053551, aiE=0 (otherwise) 
(c, r) = (0.0015, 0.02) 
k = 6 a0O=0.50566, a±1O=0.25391, a±2O=0.00108, aiO=0 (otherwise) 
a0E=a1E=0.4166, a1E=a2E=0.015, aiE=0 (otherwise) 
(c, r) = (1.2 × 10−4, 9 × 10−4
k = 8 a0O=0.55502, a±1O=0.27764, aiO=0 (otherwise) 
a0E=a1E=0.44484, a1E=a2E=0.00365, aiE=0 (otherwise) 
(c, r) = (2 × 10−4, 8 × 10−4
k = 10 a0O=0.58111, a±1O=0.29057, aiO=0 (otherwise) 
a0E=a1E=0.45839, a1E=a2E=9.1×104, aiE=0 (otherwise) 
(c, r) = (3 × 10−5, 2 × 10−4
k = 12 a0O=0.59730, a±1O=0.29865, aiO=0 (otherwise) 
a0E=a1E=0.46649, aiE=0 (otherwise) 
(c, r) = (4 × 10−4, 9 × 10−4
k ≥ 14 a0O=2×3(k1)/(k2), a±1O=3(k1)/(k2), aiO=0 (otherwise) 
a0E=a1E=(1+2k1)1/(k2), aiE=0 (otherwise) 
(c,r)=(3(1+2k1)(k1)/(k2),1×103) 
k = 4 a0O=0.3762, a±1O=0.1968, a±2O=0.00867, aiO=0 (otherwise) 
a0E=a1E=0.32301, a1E=a2E=0.053551, aiE=0 (otherwise) 
(c, r) = (0.0015, 0.02) 
k = 6 a0O=0.50566, a±1O=0.25391, a±2O=0.00108, aiO=0 (otherwise) 
a0E=a1E=0.4166, a1E=a2E=0.015, aiE=0 (otherwise) 
(c, r) = (1.2 × 10−4, 9 × 10−4
k = 8 a0O=0.55502, a±1O=0.27764, aiO=0 (otherwise) 
a0E=a1E=0.44484, a1E=a2E=0.00365, aiE=0 (otherwise) 
(c, r) = (2 × 10−4, 8 × 10−4
k = 10 a0O=0.58111, a±1O=0.29057, aiO=0 (otherwise) 
a0E=a1E=0.45839, a1E=a2E=9.1×104, aiE=0 (otherwise) 
(c, r) = (3 × 10−5, 2 × 10−4
k = 12 a0O=0.59730, a±1O=0.29865, aiO=0 (otherwise) 
a0E=a1E=0.46649, aiE=0 (otherwise) 
(c, r) = (4 × 10−4, 9 × 10−4
k ≥ 14 a0O=2×3(k1)/(k2), a±1O=3(k1)/(k2), aiO=0 (otherwise) 
a0E=a1E=(1+2k1)1/(k2), aiE=0 (otherwise) 
(c,r)=(3(1+2k1)(k1)/(k2),1×103) 
Approximations for the profiles of multi-pulse DBs are constructed by combining aO and/or aE. Given mZ and a sequence x=(xi)iZ, let Tm be the m-site shift operator defined by
(Tmx)i=xim,iZ.
Let nN, x̄=(x1,,xn){O,E}n, θ̄=(θ1,,θn){1,1}n, and m̄=(m1,,mn)Zn. Given (x̄,θ̄,m̄), define the superposition of shifted aO and/or aE as follows:
sx̄,θ̄,m̄=j=1nθjTmjaxj.
(13)
For each Tmjaxj in the sum, we define its support by
J(Tmjaxj)=iZ;(Tmjaxj)i0.
(14)
This support is a finite set since each of aO and aE has only a finite number of consecutive nonzero elements. When J(Tmjaxj)={i1,i1+1,,i2}, let J̄(Tmjaxj) be the set defined by
J̄(Tmjaxj)=J(Tmjaxj){i11,i2+1},
(15)
which is an extended support by adding two adjacent sites of J(Tmjaxj). Using these notations, we define the extended support of sx̄,θ̄,m̄ by
J̄(sx̄,θ̄,m̄)=j=1nJ̄(Tmjaxj).
(16)
Consider the scalar differential equation,
ϕ̈+ϕk1=0,
(17)
where k ≥ 4 is an even integer. Given any T > 0, let ϕ(t;T) be the T-periodic solution of Eq. (17) with initial conditions ϕ(0;T) > 0 and ϕ̇(0;T)=0.
Let l1,l2Z,l1l2 and Dc,r(l1,l2)l2(Z) be a closed convex subset defined by
Dc,r(l1,l2)=x;|xi|cforl1il2,|xi|cr(k1)l1iforil11,|xi|cr(k1)il2foril2+1,
(18)
where c > 0 and 0 < r < 1. This subset Dc,r(l1, l2) is specified by the four parameters (l1, l2, c, r). Equation (18) shows that the interval of xi rapidly (super-exponentially) decreases with increasing |i| in Dc,r(l1, l2). The existence theorem26 is stated as follows:

Theorem III 1.

Suppose V(X) of Eq. (3) and (P1)–(P3). Fix the value of k and choose aO=(aiO)iZ, aE=(aiE)iZ, and (c, r) as in Table I. Let sx̄,θ̄,m̄ be an arbitrary superposition given by Eq. (13) such that J̄(Tmiaxi)J̄(Tmjaxj)=ϕ for any ij. Let l1=minJ̄(sx̄,θ̄,m̄) and l2=maxJ̄(sx̄,θ̄,m̄). Then, for any T > 0, there exists a unique xDc,r(l1, l2) such that Γ0(t;T)=(uiϕ(t;T),uiϕ̇(t;T))iZX1 with (ui)iZ=sx̄,θ̄,m̄+x is a T-periodic solution of FPUT lattice (1) with μ = 0. Moreover, there exists a neighborhood UR of μ = 0 and a family Γ(t;T, μ) of T-periodic solutions of FPUT lattice (1) for μU such that it is a unique continuation of Γ0(t;T) in X1 continuous with respect to μ. In particular, if sx̄,θ̄,m̄=aO (resp. aE), then Γ(t;Tμ) has odd (resp. even) symmetry.

We state the main theorem and then describe its proof after some preliminaries. The main theorem ensures that the DB solution Γ(t;Tμ) of Theorem III.1 is exponentially localized in space. Let Γ(t;Tμ) be simply denoted by z(μ) and its components as z(μ)=(zi(μ))iZ, where zi = (qi, pi). The main theorem is stated as follows:

Theorem IV 1.
Assume the conditions of Theorem III.1. There exist K > 0, h > 0, ρ ∈ (0, 1), and μ0 > 0 such that components zi(μ) of the T-periodic solution Γ(t;Tμ) satisfy
|zi(μ)|1Kexph|μ|ρ|i|,
(19)
for all μ ∈ [−μ0μ0] ⊂ U and all iZ.

The DB solutions of Eq. (1) are formulated as zeros of the following μ-dependent operator F, i.e., z such that F(z,μ)=0.

Definition IV 1.
Let F be the operator defined on X1×R as follows:
F(z,μ)=w,(z,μ)X1×R,
where, denoting z=(qi,pi)iZ and w=(si,ri)iZ, we have
si=q̇ipi,
(20)
ri=ṗiV(qi+1qi)+V(qiqi1),
(21)
with V given by Eq. (3).

For this operator, we have the following lemma (cf. Lemma 7.1 in Ref. 26).

Lemma IV 1.

Under assumptions (P1)–(P3), F is an operator with values in X0. Moreover, the operator F:X1×RX0 is continuously differentiable with respect to z and μ.

Remark IV 1.

Continuous differentiability of F with respect to μ was not shown in Ref. 26, but it is obvious since F is linearly dependent on μ.

We outline the construction procedure of the DB solution Γ(t;Tμ). The construction consists of two steps. In the first step, we consider the homogeneous potential FPUT lattice which is described by Eq. (1) with the potential V(X) = Xk/k, i.e., μ = 0 in Eq. (3). In this particular lattice, it is possible to find a DB solution in the form q = uϕ(t;T), where u=(ui)iZl2(Z) is a time-independent constant vector describing the spatial profile of the solution. Given an approximate profile sx̄,θ̄,m̄, the vector u is determined by solving a set of infinite algebraic equations in a neighborhood of sx̄,θ̄,m̄ with use of Banach’s fixed point theorem. The obtained solution is Γ0(t;T). That is, we have F(z0,0)=0, where z0Γ0(t;T). In the second step, we consider the non-homogeneous potential FPUT lattice, i.e., μ ≠ 0 in Eq. (3). The solution Γ0(t;T) is uniquely continued to a non-homogeneous potential lattice for small μ ≠ 0 in X1 by applying the implicit function theorem to F(z,μ)=0. Then, z(μ) = Γ(t;Tμ) is obtained for μU as the function such that F(z(μ),μ)=0 and z(0) = z0.

Let DF(z,μ) be the Fréchet derivative of F(z,μ) with respect to z. It has been shown that DF(z,μ) is invertible at (z0, 0), where z0 = Γ0(t;T), in the above-mentioned second step.26 The inverse DF1(z,μ) exists in a neighborhood of (z0, 0), and it is a bounded linear operator. The implicit function theorem tells that the solution z(μ) = Γ(t;Tμ) satisfies the following differential equation defined in X1:
dzdμ=DF1(z,μ)Fμ(z,μ),
(22)
where Fμ(z,μ) is the derivative of F with respect to μ. We denote Fμ(z,μ)=Δ(z), where the operator Δ: X1X0 is given in terms of its components as follows:
Δ(z)=0,W(qi+1qi)W(qiqi1)iZ.
(23)
The Fréchet derivative DF(z,μ) acts on δz=(δzi)iZX1 as follows:
DF(z,μ)δzi=L,iδzi1+L0,iδzi+L+,iδzi+1,
(24)
where the left-hand side is ith component of DF(z,μ)δz and δzi = (δqi, δpi). The linear operators L−,i, L0,i, and L+,i are given by
L±,i=00V(±qi)0,L0,i=d/dt1V(qi)+V(+qi)d/dt,
(25)
where qi = qiqi−1 and +qi = qi+1qi. Equation (24) indicates that the linear operator DF(z,μ) has the block tridiagonal form, where L0,i, L+,i, and L−,i give blocks on the diagonal, the first super-diagonal, and the first sub-diagonal, respectively,
DF(z,μ)=L,i1L0,i1L+,i1L,iL0,iL+,iL,i+1L0,i+1L+,i+1.
(26)
Let L0, L+, and L be the operators consisting of blocks on the diagonal, the first super-diagonal, and the first sub-diagonal, respectively. It can be shown from Eq. (25) that each of them is a bounded linear operator from X1 to X0. We have DF(z,μ)=L0+L++L in these notations.

Let L(A,B) denote the space of bounded linear operators from A to B, where A and B are Banach spaces. Denote the operator norm in L(A,B) with L(A,B). We will abbreviate the subscript L(A,B) when it is clear from the context. There exist finite values of L0L(X1,X0), L±L(X1,X0), and DF1(z,μ)L(X0,X1) for μ close to zero.

MacKay and Aubry proved a lemma that if a block tridiagonal operator is invertible, then elements of the inverse matrix decay exponentially with distance between sites.9 The function spaces and norms used here are different from those in their study. Thus, we use their lemma by slightly modifying it to incorporate the differences. In this subsection, we do not assume the specific forms of blocks L±,i and L0,i given by Eq. (25) and consider general block tridiagonal operators from X1 to X0.

Lemma IV 2.
Suppose that a bounded linear operator L: X1X0 has the block tridiagonal form L = L0 + L+ + L, where L0, L+, and L are bounded linear operators consisting of blocks on the diagonal, the first super-diagonal, and the first sub-diagonal, respectively, and that L has a bounded inverse operator M: X0X1. Then, there exist constants C > 0 and λ ∈ (0, 1) such that the norms of blocks of M satisfy
|Mij|L(b0,b1)Cλ|ij|,
(27)
where C and λ depend only on cM and cL such that ML(X0,X1)cM and L±L(X1,X0)cL.

Proof.
Let λm = cMcL. There exist KN and κ > 1 such that
κ21λmK+2κκK.
(28)
Let f(κ)=(κ21)λmK+2 and g(κ) = κκK. Choose K large enough so that 2λmK+2<1+K will hold. Then, f(1) = g(1) and f′(1) < g′(1). Thus, there exists κ > 1 close to unity such that f(κ) ≤ g(κ).
Let w0X0 be a vector supported on a single site only, and we can assume the site is i = 0 without loss of generality. Let
Z=Mw0X1.
(29)
We denote its components as Z=(Zi)iZ. By definition, we have LZ = w0. For an integer N ≥ 1, let ZN be the projection of Z onto sites iN. Then, LZN has components zero except for two sites N − 1 and N, and it is given by
LZN=L+ZNLZN1,
(30)
where Zi is the projection of Z onto a site i. Choosing QK + 1, let
Z=N=QKQκNZNX1,
(31)
and let
W=LZX0.
(32)
Using Eqs. (30)(32), we have
W=κQKL+ZQK+i=QKQ1κi+1L+Zi+1κiLZi1κQLZQ1.
(33)
Denote the components as W=(Wi)iZ. Equation (33) implies that Wi is zero except for iI{QK1,,Q}. As for the norms |Wi|0 for iI, we have
|WQK1|0κQKL+|ZQK|1,
|Wi|0κi+1L+|Zi+1|1+κiL|Zi1|1fori=QK,,Q1,
(34)
|WQ|0κQL|ZQ1|1,
where we used ‖Zi1 = |Zi|1. In addition, using the fact |Wi|0=0 for iI and Eq. (34), we can evaluate ‖W′‖0 as follows:
W0=supt[0,T]iI|Wi(t)|21/2K+2maxiI|Wi|0.
(35)
Denote Z=(Zi)iZ. For the component ZQ, we have
|ZQ|1Z1MW0MK+2maxiI|Wi|0,
(36)
where the first inequality follows from definitions of the two norms, the second one from the action of M on Eq. (32), and the last one from Eq. (35). Let gi be defined by
gi=κi|Zi|1.
(37)
Then, |ZQ|1=(κK++1)gQ. If we use Eqs. (34) and (36) and note the definition of λm, we have
κK++1gQλmK+2maxgQK,κgi1+gi+1,κgQ1;QKiQ1.
(38)
Let G≔ max{gnQK − 1 ≤ nQ − 1 }. Equation (38) combined with our choice of κ yields
gQ(κ+1)1maxG,(κ+1)G,κG+gQ,κG.
(39)
This implies gQG for an arbitrary QK + 1. By induction, we can obtain
giG0max{gn;0nK}fori0.
(40)
Using Eq. (37) and |Zi|1 ≤ ‖Z1, we can evaluate G0 as follows:
G0κKZ1κKMw00κKcM|w0|0,
(41)
where w0 is the nonzero component of w0 for the site i = 0 and we used ‖w00 = |w0|0. From Eqs. (37), (40), and (41), we have
|Zi|1Cλ|i||w0|0fori0,
(42)
with λ = 1/κ and C = κKcM. The same estimate can be proved for i < 0 in a similar manner. Since |Mi0|L(b0,b1)=supw0|Zi|1/|w0|0, we arrive at
|Mij|L(b0,b1)Cλ|ij|,
(43)
for j = 0. If we make the same argument using a vector wj which is supported on a site j instead of w0, we can prove Eq. (43) for any j ≠ 0.

Let Br = {z; ‖zz01r} ⊂ X1 and D=Br×[μ̄,μ̄], where z0 = Γ0(tT). Given r > 0 and μ̄>0, there exists a constant α1 such that
DF(z2,μ2)DF(z1,μ1)α1z2z11+|μ2μ1|,
(44)
for any (ziμi) ∈ D, i = 1, 2. This follows from the fact that V″(Xμ), which appears in Eq. (25), is C1 and then Lipschitz continuous with respect to both X and μ over the region [r̄,r̄]×[μ̄,μ̄], where r̄=2(r+z01). As for the map Δ(z) given by Eq. (23), ‖Δ(z)‖0cΔ holds with some constant cΔ for all zBr. In addition, there exists a constant α2 such that
Δ(z2)Δ(z1)0α2z2z11,
(45)
for any ziBr, i = 1, 2. These properties of Δ(z) follow from the fact that W′(X) ∈ C2 is Lipschitz continuous for X[r̄,r̄]. Denote M(z,μ)DF1(z,μ). Existence of the inverse M(z,μ)L(X0,X1) has been proved when (zμ) = (z0, 0).26 Choose sufficiently small r and μ̄. Then, the inverse M(zμ) exists and ‖M(zμ)‖ ≤ cM holds with some constant cM for all (zμ) ∈ D, and [μ̄,μ̄]U. We have
M(z2,μ2)M(z1,μ1)M(z2,μ2)DF(z1,μ1)DF(z2,μ2)M(z1,μ1)α1cM2z2z11+|μ2μ1|,
(46)
for any (ziμi) ∈ D, i = 1, 2.
Consider Eq. (22) in X1, which Γ(t;Tμ) satisfies. Denoting f(zμ) = M(zμ)Δ(z), we rewrite Eq. (22) into
dzdμ=f(z,μ).
(47)
The map f(zμ) is Lipschitz continuous with respect to both z and μ in D. In fact, we have
f(z2,μ2)f(z1,μ1)1M(z2,μ2)M(z1,μ1)Δ(z2)0+M(z1,μ1)Δ(z2)Δ(z1)0α1cMcΔ+α2cMz2z11+α1cM2cΔ|μ2μ1|,
(48)
for any (ziμi) ∈ D, i = 1, 2. By the Picard–Lindelöf theorem, there exists a unique solution z(μ) of Eq. (47) such that z(0) = z0 over the interval μ ∈ [−μ0μ0], where μ0=min{μ̄,r/cf} and cf = sup(z,μ)∈Df(zμ)‖1. This solution z(μ) of Eq. (47) coincides with Γ(t;Tμ), since they satisfy Eq. (47) and the same initial condition. Equation (47) with the initial condition z(0) = z0 is equivalent to an integral equation, and it can be solved by the iteration method,
z(k+1)(μ)=z0+0μf(z(k)(s),s)ds,
(49)
where {z(k)(μ)}k=0 is the approximation sequence in X1 with z(0)(μ) = z0 and it satisfies z(k)(μ) ∈ Br × [−μ0μ0] for all k. The solution z(μ) is given by z(μ) = limk→∞z(k)(μ).
Lemma IV.2 applies to M(zμ) for all (zμ) ∈ Br × [−μ0μ0]. Its blocks satisfy the inequality
|Mij(z,μ)|L(b0,b1)Cλ|ij|,
(50)
with some constants C > 0 and λ ∈ (0, 1), which are independent of (zμ). Let denote z(k)(μ)=(zi(k)(μ))iZ. We claim that for any choice of ρ ∈ (λ, 1), there exists K > 0 and each component zi(k)(μ) satisfies
|zi(k)(μ)|1Kexph|μ|ρ|i|,
(51)
for all μ ∈ [−μ0μ0] and all iZ, where h is given by
h=βC(1+λ)221+λ+ρ1λρ+1ρλ,
(52)
with
β=sup|X|r̄W(X).
(53)

The proof is as follows: The initial approximation is given by z(0)(μ)=(uiϕ(t;T),uiϕ̇(t;T))iZ by Theorem III.1, and |ui| decreases super-exponentially as i → ± due to the definition of Dc,r(l1, l2) given by Eq. (18). This implies that |zi(0)(μ)|1 also decreases super-exponentially. Thus, it is possible to choose K large enough so that |zi(0)(μ)|1Kρ|i| will hold for all iZ, and then Eq. (51) holds for all μ ∈ [−μ0μ0] when k = 0. We prove the claim by induction.

Assume μ ∈ [0, μ0]. If we take the norm of each component of Eq. (49), we have
|zi(k+1)(μ)|1|z0,i|1+0μj=|Mij(z(k)(s),s)|L(b0,b1)|Δj(z(k)(s))|0ds,iZ,
(54)
where z0,i and Δj represent ith component of z0 and jth component of Δ, respectively. Let
Fi(z,s)=j=|Mij(z,s)|L(b0,b1)|Δj(z)|0.
(55)
From Eq. (23), we have
|Δj(z)|0=supt[0,T]W(qj+1qj)W(qjqj1)supt[0,T]β|qj+1qj|+β|qjqj1|β|zj+1|0+2|zj|0+|zj1|0,
(56)
where we used supt∈[0,T]|qj(t)| ≤ |zj|0. Using Eqs. (50) and (56), we have
Fi(z,s)βCj=λ|ij||zj+1|0+2|zj|0+|zj1|0βCj=J|ij||zj|1,
(57)
where we used |zj|0 ≤ |zj|1 and defined Jd by
Jd=2(1+λ),ifd=0,(1+λ)2λd1,ifd1.
(58)
Suppose that z(k)(μ) lies in the set defined by Eq. (51) for all μ ∈ [−μ0μ0]. When i > 0, from Eq. (57), we have the following inequality for s ∈ [0, μ]:
Fi(z(k)(s),s)Kexp(hs)βC(1+λ)2×2ρi1+λ+ρiλl=1iλρl+λiρ+ρi+1l=0(λρ)lhKexp(hs)ρ|i|.
(59)
The same result holds for i < 0. In the case of i = 0, we have
F0(z(k)(s),s)Kexp(hs)βC(1+λ)221+λ+2ρl=0(λρ)lhKexp(hs).
(60)
If we use Eqs. (59) and (60) and |z0,i|1|i| in Eq. (54), we have
|zi(k+1)(μ)|1Kexp(h|μ|)ρ|i|,iZ.
(61)
The same result holds for μ ∈ [−μ0, 0]. Thus, z(k)(μ) lies in the set defined by Eq. (51) for all μ ∈ [−μ0μ0] and all integers k ≥ 0. Equation (19) has been proved by taking the limit k → ∞ in Eq. (51). □

This work was supported by a Grant-in-Aid for Scientific Research (C), Grant No. 22K03451 from the Japan Society for the Promotion of Science (JSPS).

The author has no conflicts to disclose.

Kazuyuki Yoshimura: Conceptualization (lead); Formal analysis (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available within the article.

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