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.

## I. INTRODUCTION

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 lattice^{9–12} 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 technique^{23} 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 *l*^{2} 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.

## II. LATTICE MODEL AND FUNCTION SPACES

*i*th 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,

*V*to be defined by

*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., $W(X)=\u2211r=1k/2\u22121\kappa 2rX2r$.

*T*-periodic continuous functions $z(t):R\u2192\Omega $ that is endowed with the norm,

*T*-periodic continuously differentiable functions $z(t):R\u2192\Omega $ that is endowed with the norm,

*X*

_{1}and

*X*

_{0}by imposing some temporal symmetry conditions to $CT1(R;\Omega )$ and $CT0(R;\Omega )$, respectively, as follows:

*X*

_{1}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

*z*

_{i}(

*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:

*b*

_{1}and

*b*

_{0}as follows:

These spaces are obtained by imposing $CT1(R;R2)$ and $CT0(R;R2)$ the same types of temporal symmetries as to those of *X*_{1} and *X*_{0}, respectively. Each component of any ** z** ∈

*X*

_{1}(resp.

**∈**

*w**X*

_{0}) can be regarded as an element of

*b*

_{1}(resp.

*b*

_{0}).

## III. EXISTENCE OF DB SOLUTIONS

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.

*V*(

*X*) =

*V*(−

*X*). Let

*S*

_{O}and

*S*

_{E}be the linear mappings

*S*

_{O},

*S*

_{E}: Ω → Ω defined by

*z*

_{i}= (

*q*

_{i},

*p*

_{i}) and $z=(zi)i\u2208Z\u2208\Omega $. A

*T*-periodic solution

**(**

*z**t*) ∈

*X*

_{1}of Eq. (1) is said to have odd (resp. even) symmetry if it satisfies the condition

*S*

_{O}

**(**

*z**t*) =

**(**

*z**t*) [resp.

*S*

_{E}

**(**

*z**t*) =

**(**

*z**t*)] for $\u2200t\u2208R$. 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 $aO=(aiO)i\u2208Z$ and $aE=(aiE)i\u2208Z$ be two-sided real-valued bounded sequences, each of which satisfies the condition that there are $i1,i2\u2208Z$ and $aix\u22600$ only for *i*_{1} ≤ *i* ≤ *i*_{2}, otherwise $aix=0$, where the superscript x represents the sequence type *O* or *E*. We will choose *a*^{O} and *a*^{E} 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.

k = 4 | $a0O=0.3762$, $a\xb11O=\u22120.1968$, $a\xb12O=0.00867$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.32301$, $a1E=\u2212a\u22122E=\u22120.053551$, $aiE=0$ (otherwise) | |

(c, r) = (0.0015, 0.02) | |

k = 6 | $a0O=0.50566$, $a\xb11O=\u22120.25391$, $a\xb12O=0.00108$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.4166$, $a1E=\u2212a\u22122E=\u22120.015$, $aiE=0$ (otherwise) | |

(c, r) = (1.2 × 10^{−4}, 9 × 10^{−4}) | |

k = 8 | $a0O=0.55502$, $a\xb11O=\u22120.27764$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.44484$, $a1E=\u2212a\u22122E=\u22120.00365$, $aiE=0$ (otherwise) | |

(c, r) = (2 × 10^{−4}, 8 × 10^{−4}) | |

k = 10 | $a0O=0.58111$, $a\xb11O=\u22120.29057$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.45839$, $a1E=\u2212a\u22122E=\u22129.1\xd710\u22124$, $aiE=0$ (otherwise) | |

(c, r) = (3 × 10^{−5}, 2 × 10^{−4}) | |

k = 12 | $a0O=0.59730$, $a\xb11O=\u22120.29865$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.46649$, $aiE=0$ (otherwise) | |

(c, r) = (4 × 10^{−4}, 9 × 10^{−4}) | |

k ≥ 14 | $a0O=2\xd73\u2212(k\u22121)/(k\u22122)$, $a\xb11O=\u22123\u2212(k\u22121)/(k\u22122)$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=(1+2k\u22121)\u22121/(k\u22122)$, $aiE=0$ (otherwise) | |

$(c,r)=(3(1+2k\u22121)\u2212(k\u22121)/(k\u22122),1\xd710\u22123)$ |

k = 4 | $a0O=0.3762$, $a\xb11O=\u22120.1968$, $a\xb12O=0.00867$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.32301$, $a1E=\u2212a\u22122E=\u22120.053551$, $aiE=0$ (otherwise) | |

(c, r) = (0.0015, 0.02) | |

k = 6 | $a0O=0.50566$, $a\xb11O=\u22120.25391$, $a\xb12O=0.00108$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.4166$, $a1E=\u2212a\u22122E=\u22120.015$, $aiE=0$ (otherwise) | |

(c, r) = (1.2 × 10^{−4}, 9 × 10^{−4}) | |

k = 8 | $a0O=0.55502$, $a\xb11O=\u22120.27764$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.44484$, $a1E=\u2212a\u22122E=\u22120.00365$, $aiE=0$ (otherwise) | |

(c, r) = (2 × 10^{−4}, 8 × 10^{−4}) | |

k = 10 | $a0O=0.58111$, $a\xb11O=\u22120.29057$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.45839$, $a1E=\u2212a\u22122E=\u22129.1\xd710\u22124$, $aiE=0$ (otherwise) | |

(c, r) = (3 × 10^{−5}, 2 × 10^{−4}) | |

k = 12 | $a0O=0.59730$, $a\xb11O=\u22120.29865$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=0.46649$, $aiE=0$ (otherwise) | |

(c, r) = (4 × 10^{−4}, 9 × 10^{−4}) | |

k ≥ 14 | $a0O=2\xd73\u2212(k\u22121)/(k\u22122)$, $a\xb11O=\u22123\u2212(k\u22121)/(k\u22122)$, $aiO=0$ (otherwise) |

$a0E=\u2212a\u22121E=(1+2k\u22121)\u22121/(k\u22122)$, $aiE=0$ (otherwise) | |

$(c,r)=(3(1+2k\u22121)\u2212(k\u22121)/(k\u22122),1\xd710\u22123)$ |

*a*^{O}and/or

*a*^{E}. Given $m\u2208Z$ and a sequence $x=(xi)i\u2208Z$, let $Tm$ be the

*m*-site shift operator defined by

*a*^{O}and/or

*a*^{E}as follows:

*a*^{O}and

*a*^{E}has only a finite number of consecutive nonzero elements. When $J(Tmjaxj)={i1,i1+1,\u2026,i2}$, let $J\u0304(Tmjaxj)$ be the set defined by

*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 $\varphi \u0307(0;T)=0$.

*c*> 0 and 0 <

*r*< 1. This subset

*D*

_{c,r}(

*l*

_{1},

*l*

_{2}) is specified by the four parameters (

*l*

_{1},

*l*

_{2},

*c*,

*r*). Equation (18) shows that the interval of

*x*

_{i}rapidly (super-exponentially) decreases with increasing |

*i*| in

*D*

_{c,r}(

*l*

_{1},

*l*

_{2}). The existence theorem

^{26}is stated as follows:

*Suppose* *V*(*X*) *of* *Eq. (3)* *and (P1)–(P3)*. *Fix the value of* *k* *and choose* $aO=(aiO)i\u2208Z$*,* $aE=(aiE)i\u2208Z$*, and* (*c*, *r*) *as in Table I. Let* $sx\u0304,\theta \u0304,m\u0304$ *be an arbitrary superposition give*n *by Eq. (13) s*u*ch that* $J\u0304(Tmiaxi)\u2229J\u0304(Tmjaxj)=\varphi $ *for any* *i* ≠ *j**. Let* $l1=minJ\u0304(sx\u0304,\theta \u0304,m\u0304)$ *and* $l2=maxJ\u0304(sx\u0304,\theta \u0304,m\u0304)$*. Then, for any* *T* > 0*, there exists a unique* ** x** ∈

*D*

_{c,r}(

*l*

_{1},

*l*

_{2})

*such that*$\Gamma 0(t;T)=(ui\varphi (t;T),ui\varphi \u0307(t;T))i\u2208Z\u2208X1$

*with*$(ui)i\u2208Z=sx\u0304,\theta \u0304,m\u0304+x$

*is a*

*T*

*-periodic solution of FPUT lattice (1) with*

*μ*= 0

*. Moreover, there exists a neighborhood*$U\u2282R$

*of*

*μ*= 0

*and a family*

**Γ**(

*t*;

*T*,

*μ*)

*of*

*T*

*-periodic solutions of FPUT lattice (1) for*

*μ*∈

*U*

*such that it is a unique continuation of*

**Γ**

_{0}(

*t*;

*T*)

*in*

*X*

_{1}

*continuous with respect to*

*μ*

*. In particular, if*$sx\u0304,\theta \u0304,m\u0304=aO$

*(resp.*

*a*^{E}

*), then*

**Γ**(

*t*;

*T*,

*μ*)

*has odd (resp. even)*

*symmetry.*

## IV. EXPONENTIAL LOCALIZATION OF DB SOLUTIONS

We state the main theorem and then describe its proof after some preliminaries. The main theorem ensures that the DB solution **Γ**(*t*;*T*, *μ*) of Theorem III.1 is exponentially localized in space. Let **Γ**(*t*;*T*, *μ*) be simply denoted by ** z**(

*μ*) and its components as $z(\mu )=(zi(\mu ))i\u2208Z$, where

*z*

_{i}= (

*q*

_{i},

*p*

_{i}). The main theorem is stated as follows:

*Assume the conditions of Theorem III.1. There exist*

*K*> 0

*,*

*h*> 0

*,*

*ρ*∈ (0, 1)

*, and*

*μ*

_{0}> 0

*such that components*

*z*

_{i}(

*μ*)

*of the*

*T*

*-periodic solution*

**Γ**(

*t*;

*T*,

*μ*)

*satisfy*

*for all*

*μ*∈ [−

*μ*

_{0},

*μ*

_{0}] ⊂

*U*

*and all*$i\u2208Z$.

### A. Construction of the DB solution **Γ**(*t*;*T*, *μ*)

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

*Let*$F$

*be the operator defined on*$X1\xd7R$

*as follows:*

*where, denoting*$z=(qi,pi)i\u2208Z$

*and*$w=(si,ri)i\u2208Z$

*, we have*

*with*

*V*

*given by Eq. (3)*.

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

*Under assumptions (P1)–(P3),* $F$ *is an operator with values in* *X*_{0}*. Moreover, the operator* $F:X1\xd7R\u2192X0$ *is continuously differentiable with respect to* *z**and* *μ*.

*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*;*T*, *μ*). 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*) = *X*^{k}/*k*, i.e., *μ* = 0 in Eq. (3). In this particular lattice, it is possible to find a DB solution in the form ** q** =

*u**ϕ*(

*t*;

*T*), where $u=(ui)i\u2208Z\u2208l2(Z)$ is a time-independent constant vector describing the spatial profile of the solution. Given an approximate profile $sx\u0304,\theta \u0304,m\u0304$, the vector

**is determined by solving a set of infinite algebraic equations in a neighborhood of $sx\u0304,\theta \u0304,m\u0304$ with use of Banach’s fixed point theorem. The obtained solution is**

*u***Γ**

_{0}(

*t*;

*T*). That is, we have $F(z0,0)=0$, where

*z*_{0}≔

**Γ**

_{0}(

*t*;

*T*). In the second step, we consider the non-homogeneous potential FPUT lattice, i.e.,

*μ*≠ 0 in Eq. (3). The solution

**Γ**

_{0}(

*t*;

*T*) is uniquely continued to a non-homogeneous potential lattice for small

*μ*≠ 0 in

*X*

_{1}by applying the implicit function theorem to $F(z,\mu )=0$. Then,

**(**

*z**μ*) =

**Γ**(

*t*;

*T*,

*μ*) is obtained for

*μ*∈

*U*as the function such that $F(z(\mu ),\mu )=0$ and

**(0) =**

*z*

*z*_{0}.

### B. Differential equation for the continued DB solution

**. It has been shown that $DF(z,\mu )$ is invertible at (**

*z*

*z*_{0}, 0), where

*z*_{0}=

**Γ**

_{0}(

*t*;

*T*), in the above-mentioned second step.

^{26}The inverse $DF\u22121(z,\mu )$ exists in a neighborhood of (

*z*_{0}, 0), and it is a bounded linear operator. The implicit function theorem tells that the solution

**(**

*z**μ*) =

**Γ**(

*t*;

*T*,

*μ*) satisfies the following differential equation defined in

*X*

_{1}:

*μ*. We denote $F\mu (z,\mu )=\u2212\Delta (z)$, where the operator Δ:

*X*

_{1}→

*X*

_{0}is given in terms of its components as follows:

*i*th component of $DF(z,\mu )\delta z$ and

*δz*

_{i}= (

*δq*

_{i},

*δp*

_{i}). The linear operators

*L*

_{−,i},

*L*

_{0,i}, and

*L*

_{+,i}are given by

*∂*

_{−}

*q*

_{i}=

*q*

_{i}−

*q*

_{i−1}and

*∂*

_{+}

*q*

_{i}=

*q*

_{i+1}−

*q*

_{i}. Equation (24) indicates that the linear operator $DF(z,\mu )$ has the block tridiagonal form, where

*L*

_{0,i},

*L*

_{+,i}, and

*L*

_{−,i}give blocks on the diagonal, the first super-diagonal, and the first sub-diagonal, respectively,

*L*

_{0},

*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

*X*

_{1}to

*X*

_{0}. We have $DF(z,\mu )=L0+L++L\u2212$ 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 $\Vert \u22c5\Vert L(A,B)$. We will abbreviate the subscript $L(A,B)$ when it is clear from the context. There exist finite values of $\Vert L0\Vert L(X1,X0)$, $\Vert L\xb1\Vert L(X1,X0)$, and $\Vert DF\u22121(z,\mu )\Vert L(X0,X1)$ for *μ* close to zero.

### C. Inverse of block tridiagonal operator

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 *L*_{0,i} given by Eq. (25) and consider general block tridiagonal operators from *X*_{1} to *X*_{0}.

*Suppose that a bounded linear operator*

*L*:

*X*

_{1}→

*X*

_{0}

*has the block tridiagonal form*

*L*=

*L*

_{0}+

*L*

_{+}+

*L*

_{−}

*, where*

*L*

_{0}

*,*

*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*:

*X*

_{0}→

*X*

_{1}

*. Then, there exist constants*

*C*> 0

*and*

*λ*∈ (0, 1)

*such that the norms of blocks of*

*M*

*satisfy*

*where*

*C*

*and*

*λ*

*depend only on*

*c*

_{M}

*and*

*c*

_{L}

*such that*$\Vert M\Vert L(X0,X1)\u2264cM$

*and*$\Vert L\xb1\Vert L(X1,X0)\u2264cL$.

*λ*

_{m}=

*c*

_{M}

*c*

_{L}. There exist $K\u2208N$ and

*κ*> 1 such that

*g*(

*κ*) =

*κ*−

*κ*

^{−K}. Choose

*K*large enough so that $2\lambda mK+2<1+K$ 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*).

*w*_{0}∈

*X*

_{0}be a vector supported on a single site only, and we can assume the site is

*i*= 0 without loss of generality. Let

*L*

**=**

*Z*

*w*_{0}. For an integer

*N*≥ 1, let

*Z*^{≥N}be the projection of

**onto sites**

*Z**i*≥

*N*. Then,

*L*

*Z*^{≥N}has components zero except for two sites

*N*− 1 and

*N*, and it is given by

*Z*_{i}is the projection of

**onto a site**

*Z**i*. Choosing

*Q*≥

*K*+ 1, let

*Z*_{i}‖

_{1}= |

*Z*

_{i}|

_{1}. In addition, using the fact $|Wi\u2032|0=0$ for $i\u2209I$ and Eq. (34), we can evaluate ‖

**′‖**

*W*_{0}as follows:

*M*on Eq. (32), and the last one from Eq. (35). Let

*g*

_{i}be defined by

*λ*

_{m}, we have

*G*≔ max{

*g*

_{n};

*Q*−

*K*− 1 ≤

*n*≤

*Q*− 1 }. Equation (38) combined with our choice of

*κ*yields

*g*

_{Q}≤

*G*for an arbitrary

*Q*≥

*K*+ 1. By induction, we can obtain

*Z*

_{i}|

_{1}≤ ‖

**‖**

*Z*_{1}, we can evaluate

*G*

_{0}as follows:

*w*

_{0}is the nonzero component of

*w*_{0}for the site

*i*= 0 and we used ‖

*w*_{0}‖

_{0}= |

*w*

_{0}|

_{0}. From Eqs. (37), (40), and (41), we have

*λ*= 1/

*κ*and

*C*=

*κ*

^{K}

*c*

_{M}. 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

*j*= 0. If we make the same argument using a vector

*w*_{j}which is supported on a site

*j*instead of

*w*_{0}, we can prove Eq. (43) for any

*j*≠ 0.

### D. Proof of Theorem IV.1

*B*

_{r}= {

**; ‖**

*z***−**

*z*

*z*_{0}‖

_{1}≤

*r*} ⊂

*X*

_{1}and $D=Br\xd7[\u2212\mu \u0304,\mu \u0304]$, where

*z*_{0}=

**Γ**

_{0}(

*t*;

*T*). Given

*r*> 0 and $\mu \u0304>0$, there exists a constant

*α*

_{1}such that

*z*_{i},

*μ*

_{i}) ∈

*D*,

*i*= 1, 2. This follows from the fact that

*V*″(

*X*,

*μ*), which appears in Eq. (25), is

*C*

^{1}and then Lipschitz continuous with respect to both

*X*and

*μ*over the region $[\u2212r\u0304,r\u0304]\xd7[\u2212\mu \u0304,\mu \u0304]$, where $r\u0304=2(r+\Vert z0\Vert 1)$. As for the map Δ(

**) given by Eq. (23), ‖Δ(**

*z***)‖**

*z*_{0}≤

*c*

_{Δ}holds with some constant

*c*

_{Δ}for all

**∈**

*z**B*

_{r}. In addition, there exists a constant

*α*

_{2}such that

*z*_{i}∈

*B*

_{r},

*i*= 1, 2. These properties of Δ(

**) follow from the fact that**

*z**W*′(

*X*) ∈

*C*

^{2}is Lipschitz continuous for $X\u2208[\u2212r\u0304,r\u0304]$. Denote $M(z,\mu )\u2254DF\u22121(z,\mu )$. Existence of the inverse $M(z,\mu )\u2208L(X0,X1)$ has been proved when (

**,**

*z**μ*) = (

*z*_{0}, 0).

^{26}Choose sufficiently small

*r*and $\mu \u0304$. Then, the inverse

*M*(

**,**

*z**μ*) exists and ‖

*M*(

**,**

*z**μ*)‖ ≤

*c*

_{M}holds with some constant

*c*

_{M}for all (

**,**

*z**μ*) ∈

*D*, and $[\u2212\mu \u0304,\mu \u0304]\u2282U$. We have

*z*_{i},

*μ*

_{i}) ∈

*D*,

*i*= 1, 2.

*X*

_{1}, which

**Γ**(

*t*;

*T*,

*μ*) satisfies. Denoting

*f*(

**,**

*z**μ*) =

*M*(

**,**

*z**μ*)Δ(

**), we rewrite Eq. (22) into**

*z**f*(

**,**

*z**μ*) is Lipschitz continuous with respect to both

**and**

*z**μ*in

*D*. In fact, we have

*z*_{i},

*μ*

_{i}) ∈

*D*,

*i*= 1, 2. By the Picard–Lindelöf theorem, there exists a unique solution

**(**

*z**μ*) of Eq. (47) such that

**(0) =**

*z*

*z*_{0}over the interval

*μ*∈ [−

*μ*

_{0},

*μ*

_{0}], where $\mu 0=min{\mu \u0304,r/cf}$ and

*c*

_{f}= sup

_{(z,μ)∈D}‖

*f*(

**,**

*z**μ*)‖

_{1}. This solution

**(**

*z**μ*) of Eq. (47) coincides with

**Γ**(

*t*;

*T*,

*μ*), since they satisfy Eq. (47) and the same initial condition. Equation (47) with the initial condition

**(0) =**

*z*

*z*_{0}is equivalent to an integral equation, and it can be solved by the iteration method,

*X*

_{1}with

*z*^{(0)}(

*μ*) =

*z*_{0}and it satisfies

*z*^{(k)}(

*μ*) ∈

*B*

_{r}× [−

*μ*

_{0},

*μ*

_{0}] for all

*k*. The solution

**(**

*z**μ*) is given by

**(**

*z**μ*) = lim

_{k→∞}

*z*^{(k)}(

*μ*).

*M*(

**,**

*z**μ*) for all (

**,**

*z**μ*) ∈

*B*

_{r}× [−

*μ*

_{0},

*μ*

_{0}]. Its blocks satisfy the inequality

*C*> 0 and

*λ*∈ (0, 1), which are independent of (

**,**

*z**μ*). Let denote $z(k)(\mu )=(zi(k)(\mu ))i\u2208Z$. We claim that for any choice of

*ρ*∈ (

*λ*, 1), there exists

*K*> 0 and each component $zi(k)(\mu )$ satisfies

*μ*∈ [−

*μ*

_{0},

*μ*

_{0}] and all $i\u2208Z$, where

*h*is given by

The proof is as follows: The initial approximation is given by $z(0)(\mu )=(ui\varphi (t;T),ui\varphi \u0307(t;T))i\u2208Z$ by Theorem III.1, and |*u*_{i}| decreases super-exponentially as *i* → ±*∞* due to the definition of *D*_{c,r}(*l*_{1}, *l*_{2}) given by Eq. (18). This implies that $|zi(0)(\mu )|1$ also decreases super-exponentially. Thus, it is possible to choose *K* large enough so that $|zi(0)(\mu )|1\u2264K\rho |i|$ will hold for all $i\u2208Z$, and then Eq. (51) holds for all *μ* ∈ [−*μ*_{0}, *μ*_{0}] when *k* = 0. We prove the claim by induction.

*μ*∈ [0,

*μ*

_{0}]. If we take the norm of each component of Eq. (49), we have

*z*

_{0,i}and Δ

_{j}represent

*i*th component of

*z*_{0}and

*j*th component of Δ, respectively. Let

_{t∈[0,T]}|

*q*

_{j}(

*t*)| ≤ |

*z*

_{j}|

_{0}. Using Eqs. (50) and (56), we have

*z*

_{j}|

_{0}≤ |

*z*

_{j}|

_{1}and defined

*J*

_{d}by

*z*^{(k)}(

*μ*) lies in the set defined by Eq. (51) for all

*μ*∈ [−

*μ*

_{0},

*μ*

_{0}]. When

*i*> 0, from Eq. (57), we have the following inequality for

*s*∈ [0,

*μ*]:

*i*< 0. In the case of

*i*= 0, we have

*z*

_{0,i}|

_{1}≤

*Kρ*

^{|i|}in Eq. (54), we have

*μ*∈ [−

*μ*

_{0}, 0]. Thus,

*z*^{(k)}(

*μ*) lies in the set defined by Eq. (51) for all

*μ*∈ [−

*μ*

_{0},

*μ*

_{0}] and all integers

*k*≥ 0. Equation (19) has been proved by taking the limit

*k*→ ∞ in Eq. (51). □

## ACKNOWLEDGMENT

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).

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

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

## DATA AVAILABILITY

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

## REFERENCES

*Progress in Nanophotonics 3*