Traveling wave solutions for the FPU chain: a constructive approach

TL;DR

This paper develops a computer-assisted method to construct traveling wave solutions for the FPU chain, accommodating large derivatives and sign changes, without requiring an attractive quadratic potential, focusing on supersonic solutions.

Contribution

It introduces a novel constructive approach that does not depend on the potential having an attractive quadratic term, broadening the class of solutions for the FPU chain.

Findings

Constructed traveling wave solutions with large derivatives and sign changes.

Method applies to non-attractive quadratic potentials.

Solutions are predominantly supersonic.

Abstract

Traveling waves for the FPU chain are constructed by solving the associated equation for the spatial profile $u$ of the wave. We consider solutions whose derivatives $u'$ need not be small, may change sign several times, but decrease at least exponentially. Our method of proof is computer-assisted. Unlike other methods, it does not require that the FPU potential has an attractive (positive) quadratic term. But we currently need to restrict the size of that term. In particular, our solutions in the attractive case are all supersonic.

Equations

$${\ddot{q}}_j={\varphi }^{\prime }(q_{j+1}-q_j)-{\varphi }^{\prime }(q_j-q_{j-1}),j\in \text{tends}Z,$$

$$\varphi (v)=\frac{1}{2}{\varphi }_2{v}^2+\frac{1}{m+1}{\varphi }_{m+1}{v}^{m+1}$$

$$u^{\prime\prime}(x)=\tau^{2}\phi^{\prime}\bigl{(}u(x+1)-u(x))-\tau^{2}\phi^{\prime}\bigl{(}u(x)-u(x-1)\bigr{)}\,,\qquad x\in{{\tends R}}\,.$$

$${\tau }^2{\varphi }^{\prime }(v)=\mu v+\nu v^m.$$

$$({\cal D}u)(x)=u\bigl{(}x+{\textstyle{1\over 2}}\bigr{)}-u\bigl{(}x-{\textstyle{1\over 2}}\bigr{)}\,.$$

$$v^{\prime \prime }={\tau }^2{\mathcal{D}}^2{\varphi }^{\prime }(v).$$

$$u_{\scriptscriptstyle L}(x)=\sum_{j=0}^{\infty}v\bigl{(}x-j-{\textstyle{1\over 2}}\bigr{)}\,,\qquad u_{\scriptscriptstyle R}(x)=-\sum_{j=0}^{\infty}v\bigl{(}x+j+{\textstyle{1\over 2}}\bigr{)}\,,$$

$$v={\tau }^2{A}^2{\varphi }^{\prime }(v),A=\mathcal{D}{D}^{-1}.$$

$$Av={\cal D}V=V\bigl{(}\hbox{\bf.}+{\textstyle{1\over 2}}\bigr{)}-V\bigl{(}\hbox{\bf.}-{\textstyle{1\over 2}}\bigr{)}=\int_{-1/2}^{1/2}v(\hbox{\bf.}+s)\,ds\,.$$

$$w=\sum_{j\in{Z}}{{P}}_{j}w\,,\qquad({{P}}_{j}w)(x)=\cases{w(x),&if $x\in I_{j}$;\cr 0,&otherwise;\cr}$$

$$w_j=\sum _{n\in N}{w}_{j,n}{\mathcal{P}}_n,{\mathcal{P}}_n(x)=P_n(2x),x\in I_0.$$

$$⟨{\mathcal{P}}_m,{\mathcal{P}}_n{⟩}_{\mathcal{H}}=\frac{2}{2n+1}{\delta }_{m,n},⟨g,f{⟩}_{\mathcal{H}}=2{\int }_{-1/2}^{1/2}\overline{g(x)}f(x)dx.$$

$$D^{-1}{\cal P}_{n}={1\over 4n+2}\bigl{(}{\cal P}_{n+1}-{\cal P}_{n-1}\bigr{)}\,,\qquad n\geq 1\,,$$

$$D^{-2}{\mathcal{P}}_n=C_n^{+}{\mathcal{P}}_{n+2}+C_n^{-}{\mathcal{P}}_{n-2}-(C_n^{+}+C_n^{-}){\mathcal{P}}_n,n\ge 2,$$

$$C_n^{+}=\frac{1/4}{(2n+1)(2n+3)},C_n^{-}=\frac{1/4}{(2n+1)(2n-1)}.$$

$$A_{\kappa}=\kappa^{-1}\bigl{[}T^{-\kappa/2}-T^{\kappa/2}\bigr{]}D^{-1}\,,\qquad A_{\kappa}^{2}=\kappa^{-2}\bigl{[}\hskip 0.7ptT^{-\kappa}+T^{\kappa}-2{\rm I}\hskip 0.7pt\bigr{]}D^{-2}\,.$$

$$w(x)=\cases{(2x)^{n},&if $-{1\over 2}<x<{1\over 2}$;\cr 0,&otherwise.}$$

$$\bigl{(}D^{-2}w\bigr{)}(x)=\cases{{1\over 4(n+1)(n+2)}(2x)^{n+2},&if $-{1\over 2}<x<{1\over 2}$;\cr{1\over 4(n+1)(n+2)}+{1\over 2(n+1)}\bigl{(}x-{1\over 2}\bigr{)},&if $x\geq{1\over 2}$;\cr(-1)^{n}{1\over 4(n+1)(n+2)}+(-1)^{n}{-1\over 2(n+1)}\bigl{(}x+{1\over 2}\bigr{)},&if $x\leq-{1\over 2}$.\cr}$$

$$\eqalign{A_{1}^{2}{\cal P}_{0}&=\textstyle T^{-1}\bigl{[}{1\over 12}{\cal P}_{2}+{1\over 4}{\cal P}_{1}+{1\over 6}{\cal P}_{0}\bigr{]}+\bigl{[}-{1\over 6}{\cal P}_{2}+{2\over 3}{\cal P}_{0}\bigr{]}\cr&\quad\textstyle+T\bigl{[}{1\over 12}{\cal P}_{2}-{1\over 4}{\cal P}_{1}+{1\over 6}{\cal P}_{0}\bigr{]}\,,\cr A_{1}^{2}{\cal P}_{1}&=\textstyle T^{-1}\bigl{[}{1\over 60}{\cal P}_{3}-{1\over 10}{\cal P}_{1}-{1\over 12}{\cal P}_{0}\bigr{]}+\bigl{[}-{1\over 30}{\cal P}_{3}+{1\over 5}{\cal P}_{1}\bigr{]}\cr&\quad\textstyle+T\bigl{[}{1\over 60}{\cal P}_{3}-{1\over 10}{\cal P}_{1}+{1\over 12}{\cal P}_{0}\bigr{]}\,.\cr}$$

$$\eqalign{4A_{2}^{2}{\cal P}_{0}&=T^{-2}\Bigl{[}\textstyle{1\over 12}{\cal P}_{2}+{1\over 4}{\cal P}_{1}+{1\over 6}{\cal P}_{0}\Bigr{]}+T^{-1}\Bigl{[}\textstyle{1\over 2}{\cal P}_{1}+{\cal P}_{0}\Bigr{]}\cr&\quad+\Bigl{[}\textstyle-{1\over 6}{\cal P}_{2}+{5\over 3}{\cal P}_{0}\Bigr{]}+T\Bigl{[}\textstyle-{1\over 2}{\cal P}_{1}+{\cal P}_{0}\Bigr{]}+T^{2}\Bigl{[}\textstyle{1\over 12}{\cal P}_{2}-{1\over 4}{\cal P}_{1}+{1\over 6}{\cal P}_{0}\Bigr{]}\,,\cr 4A_{2}^{2}{\cal P}_{1}&=T^{-2}\Bigl{[}\textstyle{1\over 60}{\cal P}_{3}-{1\over 10}{\cal P}_{1}-{1\over 12}{\cal P}_{0}\Bigr{]}+T^{-1}\Bigl{[}\textstyle-{1\over 6}{\cal P}_{0}\Bigr{]}\cr&\quad+\Bigl{[}\textstyle-{1\over 30}{\cal P}_{3}+{1\over 5}{\cal P}_{1}\Bigr{]}+T\Bigl{[}\textstyle{1\over 6}{\cal P}_{0}\Bigr{]}+T^{2}\Bigl{[}\textstyle{1\over 60}{\cal P}_{3}-{1\over 10}{\cal P}_{1}+{1\over 12}{\cal P}_{0}\Bigr{]}\,.\cr}$$

$$(Bw{)}_{j,n}=\sum _{i,m}{B}_{j-i}^{n,m}{w}_{i,m},$$

$$W_{J,n}(w)=\sum _{j\ge J}{r}^{j-J}{w}_{j,n}$$

$$\Vert f{\Vert }_{\rho }{=}^{\mathrm{def}}\sum _{n\in N}|f_n|{\rho }^n,f=\sum _{n\in N}{f}_n{\mathcal{P}}_n.$$

$$\Vert w{\Vert }_{\rho ,r}{=}^{\mathrm{def}}\sum _{j\in Z}\Vert w_j{\Vert }_{\rho }{r}^{|j|}=\sum _{j\in Z}\sum _{n\in N}|w_{j,n}|r^{|j|}{\rho }^n.$$

$${\mathcal{P}}_k{\mathcal{P}}_l=\sum _m{C}_{k,l,m}{\mathcal{P}}_m,\sum _m{C}_{k,l,m}=1.$$

$$\eqalign{\|fg\|_{\rho}&=\sum_{m\in{N}}\,\Biggl{|}\,\sum_{k,l\in{N}}C_{k,l,m}\bigl{(}\rho^{k}f_{k}\bigr{)}\bigl{(}\rho^{l}g_{l}\bigr{)}\rho^{m-k-l}\Biggr{|}\cr&\leq\sum_{k,l\in{N}}\rho^{k}|f_{k}|\rho^{l}|g_{l}|\sum_{m\leq k+l}|C_{k,l,m}|=\|f\|_{\rho}\|g\|_{\rho}\,.\cr}$$

$$C_{k,l,m}=\frac{a(s-k)a(s-l)a(s-m)}{a(s)}\frac{2m+1}{2s+1},a(n)=2^{-n}\left(\genfrac{}{}{0ex}{}{2n}{n}\right),$$

$$G_q(w)=\nu A_{\kappa }^2{\Sigma }_q{w}^m+{\mu }^k{A}_{\kappa }^{2k}w,{\Sigma }_q=\sum _{n=0}^{k-1}{\mu }^n{A}_{\kappa }^{2n}.$$

$${\mathcal{N}}_q(h)=G_q(\bar{w}+\Lambda h)-\bar{w}+Mh,\Lambda =\mathrm{I}-M,$$

$$\Vert {\mathcal{N}}_q(0){\Vert }_{\rho ,r}\le \epsilon ,\Vert D{\mathcal{N}}_q(h){\Vert }_{\rho ,r}\le K,h\in B_{\delta },$$

Full text

Traveling wave solutions for the FPU chain:

a constructive approach

Gianni Arioli ††1

Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano.

${}^{\!\!\!,\!\!}$ ††2

Supported in part by the PRIN project “Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni”.

and Hans Koch ††3

Department of Mathematics, The University of Texas at Austin, Austin, TX 78712.

Abstract. Traveling waves for the FPU chain are constructed by solving the associated equation for the spatial profile $u$ of the wave. We consider solutions whose derivatives $u^{\prime}$ need not be small, may change sign several times, but decrease at least exponentially. Our method of proof is computer-assisted. Unlike other methods, it does not require that the FPU potential has an attractive (positive) quadratic term. But we currently need to restrict the size of that term. In particular, our solutions in the attractive case are all supersonic.

1. Introduction

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We consider a chain of interacting particles described by the equation

[TABLE]

where $\phi$ is a polynomial of degree at least $3$. The choice

[TABLE]

corresponds to the FPU model: the $\alpha$-model if $m=2$, or the $\beta$-model if $m=3$.

Our goal is to find traveling waves, meaning solutions of the form $q_{j}(t)=u(j-t/\tau)$. Substituting this ansatz into (1.1) yields the equation

[TABLE]

We focus on solutions $u$ that approach limit values at $\pm\infty$.

The first result on the existence of traveling waves for infinite chains of FPU type was obtained in [3], where solutions with prescribed energy are found as constrained minima of a suitable functional. This result has been extended to solutions with prescribed speed in [4], using the mountain pass theorem. A survey of variational results, covering both breathers and traveling waves, and including an extensive bibliography, is given in [12]. Some recent results based on variational techniques can be found e.g. in [17,19].

A perturbative approach, based on center manifold theory, has been developed in [6,7,13]. It yields small-amplitude solutions in cases where $\tau^{2}\phi_{2}\simeq 1$. A different type of perturbative approach exploits the fact that, for slowly varying functions, the equation (1.1) is close to integrable. This property has been used e.g. in [5,8,10,20] to construct and analyze solitary waves on FPU lattices.

These results all concern potentials that are attractive for small displacements, meaning $\phi_{2}>0$. In the approach considered here, the value of $\phi_{2}$ is allowed to be negative or zero as well, and we find several traveling wave solutions in this case. A similar approach might apply to other equations of this type. Advance-delay equations appear e.g. in models from biology [1], economics [15], and electrodynamics [9].

The method developed in this paper is constructive but not limited to small or near-integrable solutions. Starting with an approximate numerical solution, we use computer-assisted methods to prove that there exists a true solution nearby. Our current technique requires that $|\tau^{2}\phi_{2}|<1$, but we expect that this condition can be relaxed in future work.

Apart from this restriction, our method applies in principle to any polynomial potential $\phi$. But for simplicity, we restrict to the FPU model (1.2). Using as parameters $\mu=\tau^{2}\phi_{2}$ and $\nu=\tau^{2}\phi_{m+1}$, we have

[TABLE]

Without loss of generality, we may assume that $|\nu|=1$. In addition, we restrict to solutions $u$ that are either even or odd. In other words, $u(-x)=(-1)^{\sigma}u(x)$ for all $x$, with $\sigma\in\{0,1\}$. The number $\sigma$ will be referred to as the parity of $u$.

For any function $u$ on ${{\tends R}}$ define

[TABLE]

Notice that ${\cal D}u$ has parity $1-\sigma$, if $u$ has parity $\sigma$.

**Theorem 1.1. **Let $\nu=1$. Consider a fixed but arbitrary row in Table 1 and the data given in that row. Then for every value of $\mu$ in some open neighborhood of $\bar{\mu}$, the equation (1.3), with the potential given by (1.4), has a real analytic solution $u$ with parity $\sigma$. The function $v={\cal D}u$ satisfies a bound $|v(x)|\leq Cr^{-\kappa|x|}$ for some constant $C>0$. Its $\sup$-norm is given in column 5, where $p$ is some positive real number. This function $v$ has $E$ local extrema with values |v(t)|>\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 64}\hskip 1.0pt}. (The existence of others is not excluded.) The diagram in column 7 specifies the sequence and nature of these extrema, as described below.

Each diagram in Table 1 represent the graph of the function $v={\cal D}u$ associated with the solution $u$, where the endpoints correspond to $x=\pm\infty$. The vertices “

 ”, “

 ”, “

 ”, and “

 ” represent positive local maxima, negative local maxima, positive local minima, and negative local minima, respectively; and these extrema appear in the indicated order. More detailed (but purely numerical) graphs are shown in Figures 1–8.

The parameters $r$, $k$, and $\kappa$ that are listed in Table 1 are used during our construction of the solution, as will be explained later. Concerning our choice for the values of $\mu$, we note that similar solutions exist for many (if not all) other values in the interval $(-1,0)$ or $(-1,1)$, depending on whether $\mu$ is negative or nonnegative, respectively. Further details will be given at the end of this section.

If $u$ is a solution of the equation (1.3), then the function $v={\cal D}u$ satisfies

[TABLE]

We prove Theorem 1.1 by first solving this equation and verifying that the solution $v$ has the indicated properties. This involves estimates that are verified with the aid of a computer. Then we define two function $u_{\scriptscriptstyle L}$ and $u_{\scriptscriptstyle R}$ by setting

[TABLE]

for all $x\in{{\tends R}}$. It is not hard to see that both $u=u_{\scriptscriptstyle L}$ and $u=u_{\scriptscriptstyle R}$ satisfy the equation (1.3), and that the difference $u_{\scriptscriptstyle R}-u_{\scriptscriptstyle L}$ is constant. By construction, both satisfy ${\cal D}u=v$. The solution with the proper parity is u=\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}u_{\scriptscriptstyle L}+\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}u_{\scriptscriptstyle R}.

[TABLE]

Table 1. Parameter values and properties of solutions.

Our approach to solving the equation (1.6) is to turn it into a suitable fixed point problem. This is a common strategy in computer-assisted proofs. Alternatively, one could try a dynamical systems approach. Traveling waves can often be viewed as homoclinic (or heteroclinic) orbits of a dynamical system. This approach has been successful with systems of ordinary differential equations [14,21,22,26]. Integration methods have been applied also to dissipative systems in infinite dimensions [11,18,29] and to some delay equations [16,23,24,25]. But for an advance-delay equation like (1.3), this does not seem to be a workable approach. In this context, we should mention that the system (1.1) is Hamiltonian.

A fixed point equation $G(v)=v$ for the function $v={\cal D}u$ can be obtained by integrating both sides of (1.6) twice. The transformation $G$ improves regularity. But due to the non-compactness of the domain ${{\tends R}}$, we found it difficult to come up with an expansion for $v$ that allows for accurate approximations and is suitable for a rigorous computer-assisted analysis. The following turns out to work extremely well. A function $v$ in one of our spaces ${\cal B}_{\rho,r}$ is given by a sequence of “arcs” $v_{j}=v(\hbox{\bf.}-j)$, indexed by integers $j$ and defined on the interval [-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt},\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}]. Each arc $v_{j}$ is real analytic on this interval and represented by a rapidly converging Legendre series. So $v$ is real analytic outside the set {{\tends Z}}+\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}. Real analyticity on all of ${{\tends R}}$ is obtained if $v$ is a fixed point of $G$, due to the regularity-improving property of the transformation $G$.

Figure 1. Profile $u$ for the solutions 4 (left) and 22 (right).

1.1. Numerical observations

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As mentioned above, we solve the equation (1.6) by converting it to a suitable fixed point equation $G(v)=v$. Then, starting with an approximate numerical solution $\bar{v}$, we prove that $G$ has a true fixed point $v$ nearby.

To find approximate solutions we use a Newton-type iteration. Some approximate solutions were found by starting the iteration with a randomly generated function $v$. For others, we started with an initial guess, produced by combining one-bump or two-bump solutions into functions with multiple bumps. (In this discussion, a “bump” is a local maximum above \scriptstyle 1$$\scriptstyle/$$\scriptstyle 8  or local minimum below -\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 8}\hskip 1.0pt}.) Figures 4 and 6 suggest that solutions with an arbitrary large number of bumps can be obtained this way. To be more precise, multi-bump solutions were found only for negative value of $\mu$. Furthermore, the precision of our method deteriorates as $|\mu|$ approaches $1$. But for each of the numerical solutions that we found, it was possible to “continue” the solution to other values of $\mu$ of the same sign, without encountering any bifurcations. So the choice of $\mu$-values in Table 1 is somewhat random, except for the sign.

Let us call two functions $v$ and $w$ “independent” if they have disjoint supports, separated by an interval of length $1$. In this case ${\cal D}^{2}\phi^{\prime}(v+w)={\cal D}^{2}\phi^{\prime}(v)+{\cal D}^{2}\phi^{\prime}(w)$. So if an approximate solution $\bar{v}$ is small (in modulus) on an interval of length $1$ but has bumps on both sides of this interval, then it is a sum of two nearly independent approximate solutions. In such cases, our Newton iteration is pushing the two parts farther and farther apart. This suggests that there exists no true solution with nearly independent parts. (Near-independence is accompanied by eigenvalues close to $1$ for the linearized problem, so it was necessary to increase the numerical precision in such cases.)

For negative values of $\mu$, it appears that multi-bump solutions exist only for certain specific arrangements of the bumps. And for nonnegative values of $\mu$, no multi-bump solutions appear to exist. As $\mu$ approaches zero from below, the bumps approach near-independence in the sense described above; see also Figure 8 (left). In the case of solutions that have negative local maxima or positive local minima, these extreme values approach zero as $\mu\to 0$. This behavior is illustrated in Figure 8 (right).

Solutions for $\mu<0$ appear not to change significantly as $\mu$ approaches the value $-1$. But the one-bump solution for $\mu>0$ widens, and its amplitude decreases, as $\mu$ approaches $1$. This behavior is illustrated in Figure 7. Presumably, the function $q$ associated with our one-bump solution $v$ approaches a small traveling wave of the type considered in [6,7] for $\mu\simeq 1$.

2. Arcs and Legendre series

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In this section we describe our decomposition of a function $w\in\mathop{\rm L}\nolimits^{2}({{\tends R}})$ into local “modes”. In order to motivate our choices, let us write (1.6) as the fixed point equation

[TABLE]

Here $D^{-1}$ is defined via integration from some arbitrary point $x_{0}\in{{\tends R}}$. The choice of $x_{0}$ does not matter, since for any antiderivative $V$ of $v$,

[TABLE]

Notice that $A$ is a convolution operator: If $\chi$ denotes the indicator function of the interval [-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt},\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}], then $Av=\chi\ast v$. Thus, the operator $A^{2}$ that appears in (2.1) is convolution with the function $\chi\ast\chi$, also known as the cardinal b-spline of order $2$, with separation $1$ between the knots. This suggests that we decompose a function $w$ on ${{\tends R}}$ as follows:

[TABLE]

where I_{j}=[\hskip 0.7ptj-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt},j+\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}] for every integer $j$.

Our goal is to compute $A^{2}v$ via antiderivatives, but to keep the computation as local as possible. To see how this can be achieved, consider a function $v$ on ${{\tends R}}$ that is supported on $I_{0}$. If $v$ is orthogonal to the constant function on $I_{0}$, then $v$ has an antiderivative $D^{-1}v$ on ${{\tends R}}$ that is again supported in $I_{0}$. More generally, if $v$ is orthogonal to all polynomials of degree less than $n$, then $v$ possesses antiderivatives $D^{-m}v$ of order $m\leq n$ that are all supported on $I_{0}$. This motivates the following choices.

Denote by $T$ translation by $1$; that is, $(Tw)(x)=w(x-1)$. We will represent a function $w\in\mathop{\rm L}\nolimits^{2}({{\tends R}})$ by the sequence of “arcs” $w_{j}=T^{-j}{{P}}_{j}w$ indexed by integers $j$. Each arc $w_{j}$ is supported in $I_{0}$, and when regarded as a function in ${\cal H}=\mathop{\rm L}\nolimits^{2}(I_{0})$, it admits a unique expansion

[TABLE]

Here $P_{n}$ denotes the Legendre polynomial of degree $n$. The sequence of polynomials $n\mapsto{\cal P}_{n}$ can be obtained from the sequence of monomials $n\mapsto(x\mapsto x^{n})$ via the Gram-Schmidt orthogonalization process, and then normalizing {\cal P}_{n}(\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt})=1. So the scaled Legendre polynomials ${\cal P}_{0},{\cal P}_{1},{\cal P}_{2},\ldots$ constitute a complete orthogonal set in ${\cal H}$. More specifically, we have

[TABLE]

Two other well-known facts are the following. The values of ${\cal P}_{n}$ on the interval $I_{0}$ are bounded in modulus by $1$. Furthermore, $(4n+2){\cal P}_{n}={\cal P}_{n+1}^{\prime}-{\cal P}_{n-1}^{\prime}$ for all positive integers $n$. Integrating both sides of this identity yields

[TABLE]

up to an additive constant. Notice that ${\cal P}_{n+1}$ and ${\cal P}_{n-1}$ agree at \pm\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}. Thus, the right hand side of (2.6) vanishes at \pm\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}. So we define $D^{-1}{\cal P}_{n}$ for $n\geq 1$ via integration from \scriptstyle 1$$\scriptstyle/$$\scriptstyle 2  or -\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}.

Applying the identity (2.6) twice yields

[TABLE]

where

[TABLE]

Notice that the function $D^{-2}{\cal P}_{n}$ and its first derivative vanish on the boundary of the interval $I_{0}$, for all $n\geq 2$. As mentioned above, this is one of our main reasons for having chosen scaled Legendre polynomials for our expansion of arcs.

3. The operator $A^{2}$ in more detail

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The goal here is to give an explicit description of how the operator $A^{2}$ acts on each term in the decomposition (2.4). To simplify notation, we define ${\cal P}_{n}(x)=0$ for |x|>\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}.

But first, let us consider a slight generalization of the approach described in the previous subsection. For a traveling wave $v$ that varies rapidly, it can be advantageous to partition ${{\tends R}}$ into subintervals of length $1/\kappa$, for some integer $\kappa>1$. Equivalently, we can reformulate our fixed point problem in terms of the scaled function $w=v(\hbox{\bf.}/\kappa)$. The resulting equation for $w$ is $w=\tau^{2}A_{\kappa}^{2}\phi^{\prime}(w)$, where

[TABLE]

Here $\kappa$ can be any positive integer. Notice that $A_{1}=A$.

By translation invariance and linearity, it suffices to compute $A_{\kappa}^{2}w$ for a function $w$ of the form

[TABLE]

We are interested mainly in the cases $n=0$ and $n=1$, where the identity (2.7) does not apply. Let $D^{-1}w$ be the antiderivative of $w$ that vanishes at the origin, and let $D^{-2}w$ be the antiderivative of $D^{-1}w$ that vanishes at the origin. Then

[TABLE]

Notice that the function $D^{-2}w$ is affine to the left and to the right of $I_{0}$. When we apply $T^{-\kappa}+T^{\kappa}-2{\rm I}$, these affine parts of $D^{-2}w$ cancels at a distance $\kappa$ or larger from $I_{0}$. Applying $T^{-\kappa}+T^{\kappa}-2{\rm I}$ to the part of $D^{-2}w$ that is supported in $I_{0}$ results in three copies: one in $I_{0}$, with a factor $-2$, and a translated copy in each of the intervals $I_{\pm\kappa}$.

In what follows, we restrict to $\kappa=1$ or $\kappa=2$. After a trivial but tedious computation, we end up with the following expressions. For $n<2$ and $\kappa=1$ we have

[TABLE]

For $n<2$ and $\kappa=2$ we obtain

[TABLE]

For $n\geq 2$ it suffices to combine the identities (3.1) and (2.7).

In our computations [30] we apply these formulas to finitely many terms in the expansion $w=\sum_{j,n}w_{j,n}T^{j}{\cal P}_{n}$. In order to estimate truncation errors, we use (among others) the following simple facts. Consider a convolution operator of the form

[TABLE]

where $i,j,m,n$ denote integers, with $m,n$ positive. Since we are interested in upper bounds only, assume for simplicity that the numbers $B^{n,m}_{k}$ and $w_{i,m}$ are all nonnegative. We consider bounds of the type

[TABLE]

with $r>1$, and weighted sums (over $n$) of such bounds.

Proposition 3.1. $W_{J,n}(Bw)=(BW(w))_{J,n}$ for all $J$ and $n$. If $w_{j,n}=0$ for all $j<L$, then $W_{J,n}(w)\leq r^{L-J}W_{L,n}$ for all $J$; and equality holds for $J\leq L$.

A proof of these properties is straightforward. Notice that $w\mapsto W(w)$ is a convolution, which explains most of why $WB=BW$.

4. Analyticity

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Here we describe the function spaces that are used in our analysis of the fixed point equation (2.1). Given $\rho\geq 1$, denote by ${\cal A}_{\rho}$ the space of all functions $f$ in ${\cal H}=\mathop{\rm L}\nolimits^{2}(I_{0})$ that have a finite norm

[TABLE]

The even and odd subspaces of ${\cal A}_{\rho}$ are denoted by ${\cal A}_{\rho}^{0}$ and ${\cal A}_{\rho}^{1}$, respectively. If $\rho>1$, then every function in ${\cal A}_{\rho}$ admits an analytic continuation to the complex open neighborhood ${\cal E}_{\rho}$ of $I_{0}$ whose boundary is the ellipse \bigl{|}4z+z^{-1}\bigr{|}=4\rho. This is a consequence of Bernstein’s lemma.

Consider now functions $w:{{\tends R}}\to{{\tends R}}$ that admit an expansion (2.3) with $w_{j}\in{\cal A}_{\rho}$ for all $j$. Given $r\geq 1$, define ${\cal B}_{\rho,r}$ to be the space of all such functions $w$ that have a finite norm

[TABLE]

The even and odd subspaces of ${\cal B}_{\rho,r}$ are denoted by ${\cal B}_{\rho,r}^{\,0}$ and ${\cal B}_{\rho,r}^{\,1}$, respectively. Since our fixed point equation (2.1) involves a product of functions, we will need the following.

**Proposition 4.1. **The spaces ${\cal A}_{\rho}$, ${\cal A}_{\rho}^{0}$, ${\cal B}_{\rho,r}$, and ${\cal B}_{\rho,r}^{\,0}$ are Banach algebras.

**Proof. **The product of two function in ${\cal A}_{\rho}$ can be estimated by using the linearization formula

[TABLE]

The existence of such an expansion follows by completeness, and the second identity is a consequence of the fact that {\cal P}_{n}(\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt})=1 for all $n$. Clearly $C_{k,l,m}=0$ for $m>k+l$. And by parity, we have $C_{k,l,m}=0$ unless $k+l+m$ is even. Below will see that the coefficients $C_{k,l,m}$ are all nonnegative. So if $f$ and $g$ belong to ${\cal A}_{\rho}$, then

[TABLE]

An analogous inequality $\|vw\|_{\rho,r}\leq\|v\|_{\rho,r}\|w\|_{\rho,r}$ for functions $v,w\in{\cal B}_{\rho,r}$ follows trivially.

The following expression for the coefficients $C_{k,l,m}$ was first found by Adams [2].

[TABLE]

where $s={1\over 2}(k+l+m)$. Our main reason for giving this formula here is that it is being used in our programs [30]. But it also shows that the coefficients $C_{k,l,m}$ are all nonnegative, as claimed above. QED

**Remark 1. ** A noteworthy consequence of (4.4) is the following. Let $f\in{\cal A}_{\rho}$. Then the spectral radius of the operator $g\mapsto fg$ acting on ${\cal A}_{\rho}$ is bounded by $\|f\|_{\rho}$. This implies e.g. that the analytic continuation of $f$ satisfies $|f(z)|\leq\|f\|_{\rho}$ for all $z\in{\cal E}_{\rho}$.

Notice that the translation operators $T$ and $T^{-1}$ are bounded on ${\cal B}_{\rho,r}$. So the following is essentially a consequence of the identity (2.7) and (2.8).

**Proposition 4.2. **The operator $A_{\kappa}$ defined by (3.1) is bounded on ${\cal B}_{\rho,r}$.

The following will be used to prove that a solution $v\in{\cal B}_{\rho,r}$ of the equation (1.6) is real analytic.

**Proposition 4.3. **Assume that $\rho>1$. Then there exists $\epsilon>0$ such that, if $w\in{\cal B}_{\rho,r}$ is of class $\mathop{\rm C}\nolimits^{\infty}$, then $w$ extends analytically to the strip $|\mathop{\rm Im}\nolimits z|<\epsilon$.

**Proof. **Let $w\in{\cal B}_{\rho,r}$. As described after (4.1), each arc $w_{j}$ extends analytically to a domain ${\cal E}_{\rho}$ that includes an open rectangle {\cal R}=(-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}-\epsilon,\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}+\epsilon)\times(-\epsilon,\epsilon). Thus ${{P}}_{j}v$ extends analytically to $T^{j}{\cal R}$, for each $j\in{{\tends Z}}$. Denote this extension by $E_{j}w$. Then the domains of both $E_{j}w$ and $E_{j+1}w$ include the rectangle {\cal R}_{j}=(j+\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}-\epsilon,j+\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}+\epsilon)\times(-\epsilon,\epsilon). Assume now that $w$ belongs to $\mathop{\rm C}\nolimits^{\infty}({{\tends R}})$. Then the derivatives $D^{n}E_{j}w$ and $D^{n}E_{j+1}w$ agree at the point j+\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}, for all $n\geq 0$. Thus, $E_{j}w$ and $E_{j+1}w$ agree on ${\cal R}_{j}$. This holds for all $j\in{{\tends Z}}$, so $w$ extends analytically to ${{\tends R}}\times(-\epsilon,\epsilon)$. QED

5. The fixed point equation

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Consider the parameter values $(m,\bar{\mu},\sigma,r,k,\kappa)$ associated with some fixed row in Table 1. In addition, we choose \rho=\hbox{\raise 2.2pt\hbox{\scriptstyle 17}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 16}\hskip 1.0pt}. Let $\mu$ be a real number of modulus $|\mu|<1$. Then the equation (2.1) for these parameters reads $v=\nu A^{2}v^{m}+\mu A^{2}v$. The corresponding equation for the function $w=v(\hbox{\bf.}/\kappa)$ is $w=\nu A_{\kappa}^{2}w^{m}+\mu A_{\kappa}^{2}w$, with $A_{\kappa}$ as defined by (3.1). This equation for $w$ is equivalent to the fixed point equation $G_{q}(w)=w$, where

[TABLE]

The subscript used here is $q=(m,\mu,k,\kappa)$. By Propositions 4.1 and 4.2, $G_{q}$ is differentiable as a map on ${\cal B}_{\rho,r}$. Notice that, if $w$ and $h$ belong to ${\cal B}_{\rho,r}$, and if $h$ is supported far away from the origin, then $DG_{q}(w)h$ is approximately equal to $\mu^{k}A_{\kappa}^{2k}h$. So the transformation $G_{q}$ contracts tails by roughly a factor $|\mu|^{k}$. This should make clear why we need larger values of $k$ when $|\mu|$ is close to $1$.

**Definition 5.1. **A compactly supported function $w:{{\tends R}}\to{{\tends R}}$ whose arcs $w_{j}$ are all polynomials will be called a spline.

Let $\tau=1-\sigma$. Our choice of the exponent $m$ in (1.4) guarantees that $G_{q}(w)$ has parity $\tau$ whenever $w$ has parity $\tau$. Given a function $\bar{w}\in{\cal B}_{\rho,r}^{\,\tau}$, and a bounded linear operator $M$ on ${\cal B}_{\rho,r}^{\,\tau}$, define

[TABLE]

for every function $h\in{\cal B}_{\rho,r}^{\,\tau}$. Clearly, if $h$ is a fixed point of ${\cal N}_{q}$ then $\bar{w}+\Lambda h$ is a fixed point of $G_{q}$. For practical reasons, we choose $\bar{w}$ to be a spline that is an approximate fixed point of $G_{\bar{q}}$ with $\bar{q}=(m,\bar{\mu},k,\kappa)$. And for $M$ we choose a finite rank operator such that $\Lambda$ is an approximate inverse of ${\rm I}-DG_{\bar{q}}(\bar{w})$. If $\mu$ is sufficiently close (but not necessarily equal) to $\bar{\mu}$, then we can expect ${\cal N}_{q}$ to be a contraction near the origin.

**Lemma 5.2. **Consider a fixed but arbitrary row in Table 1 and the parameter values given in that row. Let \rho=\hbox{\raise 2.2pt\hbox{\scriptstyle 17}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 16}\hskip 1.0pt} and $\tau=1-\sigma$. Then there exists a spline $\bar{w}$, a bounded linear operator $M$ on ${\cal B}_{\rho,r}^{\,\tau}$, and positive constants $\varepsilon,K,\delta$ satisfying $\varepsilon+K\delta<\delta$, such that for every value of $\mu$ in some open neighborhood of $\bar{\mu}$, the transformation ${\cal N}_{q}$ defined by (5.2) satisfies

[TABLE]

where $B_{\delta}$ denotes the closed ball of radius $\delta$ in ${\cal B}_{\rho,r}^{\,\tau}$, centered at the origin. Furthermore, if $w=\bar{w}+\Lambda h$ with $h\in B_{\delta}$, then the function $v=w(\kappa\hskip 0.7pt\hbox{\bf.})$ has the properties listed in (the given row of) Table 1 concerning the $\sup$-norm and the local extrema.

Notice that $\|w-\bar{w}\|_{\rho,r}\leq\delta^{\prime}$, where $\delta^{\prime}=\|\Lambda\|\delta$. Our proof of Lemma 5.2 yields $\delta^{\prime}<2^{-32}$ for all solutions. This bound can be made as small as desired by running our programs (which also determine $\bar{w}$) at higher numerical precision.

Based on this lemma, we can now give a

**Proof of Theorem 1.1. ** By the contraction mapping principle, the given bounds imply that ${\cal N}_{q}$ has a unique fixed point $h$ in $B_{\delta}$. The corresponding function $w=\bar{w}+\Lambda h$ is a fixed point of $G_{q}$. Given that $A_{\kappa}^{2}$ includes two antiderivatives, the identity $w=G_{q}(w)$ implies that $w$ is of class $\mathop{\rm C}\nolimits^{\infty}$. So by Proposition 4.3, $w$ extends to an analytic function on some strip $|\mathop{\rm Im}\nolimits z|<\epsilon$. This extension still decreases exponentially: by Remark 1 we have a uniform bound $|w(x+iy)|\leq Cr^{-|x|}$ that holds for all $x,y\in{{\tends R}}$ with $|y|<\epsilon$. These properties of $w$ imply that the function $v=w(\kappa\hskip 0.7pt\hbox{\bf.})$ extends analytically to the strip $|\mathop{\rm Im}\nolimits z|<\epsilon/\kappa$, decreases exponentially, and satisfies the the equation (1.6).

Consider now the function $u=u_{\scriptscriptstyle L}$ defined by the equation (1.7). The above-mentioned properties of $v$ imply that $u$ is real analytic. Furthermore, ${\cal D}u=v$. The equation (1.6) implies that the function $g$ defined by $g=u^{\prime\prime}-\tau^{2}{\cal D}\phi^{\prime}(v)$ satisfies ${\cal D}g=0$. Thus $g$ is periodic with period $1$. The function ${\cal D}\phi^{\prime}(v)$ vanishes at $\pm\infty$, so $u^{\prime\prime}$ approaches the periodic function $g$ at $\pm\infty$. But $u^{\prime\prime}=u_{\scriptscriptstyle L}^{\prime\prime}$ vanishes at $-\infty$. Thus $g=0$, which implies that $u^{\prime\prime}=\tau^{2}{\cal D}\phi^{\prime}({\cal D}u)$. In other words, $u$ satisfies the equation (1.1).

The same arguments apply to the function $u=u_{\scriptscriptstyle R}$. The difference $f=u_{\scriptscriptstyle L}-u_{\scriptscriptstyle R}$ is periodic, since ${\cal D}f=0$. But $f^{\prime\prime}$ vanishes at infinity, so $f$ is constant. Now define u=\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}u_{\scriptscriptstyle L}+\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}u_{\scriptscriptstyle R}. Then $u$ is real analytic, satisfies the equation (1.1), and has parity $\sigma$.

The claims in Theorem 1.1 concerning the $\sup$-norm and the local extrema of the function $v={\cal D}u$ follow from the last statement in Lemma 5.2. QED

6. Computer estimates

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What remains to be done is to verify Lemma 5.2. This is carried out with the aid of a computer. To be more specific, consider the parameter values $(m,\bar{\mu},\sigma,r,k,\kappa)$ from a fixed but arbitrary row in Table 1. As a first step, we determine an approximate fixed point $\bar{w}$ of $G_{q}$ and an approximate inverse of ${\rm I}-DG_{\bar{q}}(\bar{w})$ of the form $\Lambda={\rm I}-M$, with $M$ of finite rank. The remaining steps are rigorous: We compute an upper bound $\varepsilon$ on the norm of ${\cal N}_{q}(0)$, and an upper bound $K$ on the operator norm of $D{\cal N}_{q}(h)$ that holds for all $h$ of norm $4\varepsilon$ or less. This is done simultaneously for all values of $\mu$ in some open interval centered at $\bar{\mu}$. After verifying that K<\hbox{\raise 2.2pt\hbox{\scriptstyle 7}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 8}\hskip 1.0pt}, we choose a positive $\delta<8\varepsilon$ in such a way that $\varepsilon+K\delta<\delta$. The last statement in Lemma 5.2 is verified by estimating $v(x_{i})$ at a finite number of points $x_{i}\in{{\tends R}}$.

The rigorous part is still numerical, but instead of truncating series and ignoring rounding errors, it produces guaranteed enclosures at every step along the computation. This part of the proof is written in the programming language Ada [31]. The following is meant to be a rough guide for the reader who wishes to check the correctness of our programs. The complete details can be found in [30].

In the present context, a “bound” on a map $f:{\cal X}\to{\cal Y}$ is a function $F$ that assigns to a set $X\subset{\cal X}$ of a given type (Xtype) a set $Y\subset{\cal Y}$ of a given type (Ytype), in such a way that $y=f(x)$ belongs to $Y$ for all $x\in X$. In Ada, such a bound $F$ can be implemented by defining a procedure F(X: in Xtype; Y: out Ytype).

For balls in a real Banach algebra ${\cal X}$ with unit ${\bf 1}$, we use a data type Ball. A Ball consists of a pair B=(B.C,B.R), where B.C is a representable number (Rep) and B.R a nonnegative representable number (Radius). The corresponding ball in ${\cal X}$ is the set ${\tt B}_{\scriptscriptstyle{\cal X}}=\{x\in{\cal X}:\|x-({\tt B.C}){\bf 1}\|\leq{\tt B.R}\}$. Our bounds on some standard functions involving the type Ball are defined in the packages Std_Balls. Other basic functions are covered in the packages Vectors and Matrices. Bounds of this type have been used in many computer-assisted proofs; so we focus here on the more problem-specific aspects of our programs.

6.1. Analytic arcs

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Consider the space ${\cal A}_{\rho}$ for a given Radius $\rho\geq 1$. Our enclosures for functions in ${\cal A}_{\rho}$ are associated with a data type Legend, based on scalars of type Ball, with ${\cal X}={R}$. Given a fixed odd integer $D>1$, a Legend is in essence a pair G=(G.C,G.E), where G.C is an array(0 .. D) of Ball and G.E is an array(0 .. D+2) of Radius. The corresponding set ${\tt G}_{\!\scriptscriptstyle{\cal A}}\subset{\cal A}_{\rho}$ consists of all functions $g$ that admit a representation

[TABLE]

with $c_{n}\in{\tt G.C}(n)_{\scriptscriptstyle{R}}$ and $E_{m}\in{\cal A}_{\rho,m}$, satisfying $\|E_{m}\|_{\rho}\leq{\tt G.E}(m)$. Here ${\cal A}_{\rho,m}$ denotes the subspace of ${\cal A}_{\rho}$ consisting of all functions $E\in{\cal A}_{\rho}$ that are orthogonal to all polynomials of degree less than $m$ and have the same parity as ${\cal P}_{m}$.

The type Legend is defined in the package Legends. This package also implements basic bounds on functions to/from the space ${\cal A}_{\rho}$. This includes a bound Prod on the product $(f,g)\mapsto fg$, based on the identities (4.3) and (4.5). These bounds are quite straightforward, so we refer to [30] for details.

In Legends.Chain we use (2.7) to define a bound DDInvHigh on the operator $D^{-2}$ on ${\cal A}_{\rho}$, restricted to function with $f_{0}=f_{1}=0$. For compactly supported functions $w\in{\cal A}_{\rho}^{{Z}}$ we use enclosures of a type LVector. This type is defined as an unconstrained array (Integer range <>) of Legend. Using DDInvHigh, as well as the identities (3.1), (3.4), and (3.5), we define a bound AA on the linear operator $A_{\kappa}^{2}$ for such functions.

6.2. Piecewise real analytic functions

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Let $L$ be a fixed integer larger than $1$. Then a function $w\in{\cal B}_{\rho,r}^{\,\tau}$ has a unique decomposition

[TABLE]

with $G\in{\cal B}_{\rho,r}^{\,\tau}$ supported outside (\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}-j,j-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}). The sum in this equation will be referred to as the “center” of $w$, and $G$ will be referred to as the “tail” of $w$.

Let $r\geq 1$ be a fixed Radius. An enclosure for a tail in ${\cal B}_{\rho,r}^{\,\tau}$ is defined by a Legend G with the property that {\tt G.C\hbox{(n)}.C} is zero for all $n$. The corresponding set ${\tt G}_{\scriptscriptstyle{\cal B}}\subset{\cal B}_{\rho,r}^{\,\tau}$ consists of all functions $G$ that admit a representation

[TABLE]

with coefficients $c_{j,m}\in{{\tends R}}$ and functions $E_{j,m}\in{\cal A}_{\rho,m}$ satisfying the bounds

[TABLE]

The coefficients $c_{j,m}$ and functions $E_{j,m}$ for $j\leq-L$ are determined by the requirement that $G$ has parity $\tau$.

For more general subsets of ${\cal B}_{\rho,r}^{\,\tau}$ we use a data type LChain, which consists of a triple W=(W.R,W.Par,W.C), where ${\tt W.R}=r$, ${\tt W.Par}=\tau$, and where W.C is an LVector(0 .. L). The component W.C(0) must have parity $\tau$. The corresponding set ${\tt W}_{\scriptscriptstyle{\cal B}}$ is defined as the set of all functions (6.2), where w_{j}\in{\tt W.C\hbox{(j)}}_{\scriptscriptstyle{\cal A}} for $0\leq j<L$, and G\in{\tt W.C\hbox{(L)}}_{\scriptscriptstyle{\cal B}}. The arcs $w_{j}$ for $-L<j<0$ are determined by the requirement that $w$ has parity $\tau$.

These types are defined in the package Legends.Chain which takes ${\tt JCMax}=L-1$ and ${\tt Scale}=\kappa$ as arguments. This package also implements basic bounds on functions to/from the spaces ${\cal B}_{\rho,r}^{\,\tau}$. Most are straightforward combinations of bounds defined in Legends, such as the bound Norm on $w\mapsto\|w\|_{\rho,r}$ or the bound Prod on $(v,w)\mapsto vw$. The representation (6.3) for the tail $G$ has been chosen in such a way that the tail component W.C(L) of an LChain W can often be treated the same way as the other components W.C(J).

6.3. Transformations and their derivatives

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Consider first the operators $A_{\kappa}^{2k}$ and $A_{\kappa}^{2}\Sigma_{q}$ that appear in the definition (5.1) of the transformation $G_{q}$. Our bounds on these two operators are given by the two procedures AAPower and SumAAPowers. They are more elaborate than the bounds discussed so far, due to the fact that $A_{\kappa}^{2}$ is nonlocal. In particular, $A_{\kappa}^{2}$ couples the center and tail of a function $w$.

Consider the task of implementing a bound on $A_{1}^{2}$. Given an LChain W, consider a fixed but arbitrary function $w\in{\tt W}_{\scriptscriptstyle{\cal B}}$ of the form (6.2). The goal is to find an enclosure ${\tt U}_{\scriptscriptstyle{\cal B}}$ for the function $u=A_{1}^{2}w$ that only depends on W. By linearity, we can consider centers and tails separately. Assume first that W has a zero tail W.C(L). Then, by using the above-mentioned bound AA for compactly supported chains, we obtain a LVector-type enclosure ${\tt P(0\hskip 1.0pt{..}\hskip 1.0ptL)}_{{\cal A}^{Z}}$ for $u$. Setting U.C(0 .. L-1) := P(0 .. L-1) and converting P(L) to a tail U.C(L), we obtain the desired enclosure ${\tt U}_{\scriptscriptstyle{\cal B}}$. This part, generalized to $A_{\kappa}^{2k}$, is implemented by the procedure LA_AAPower. Next, consider the case where W.C(J) is zero for all J$<$L. In order to determine the center part of U, the tail W(L) can be considered to be a bound on $w_{\scriptscriptstyle L}$ only, since $A_{1}^{2}$ is a convolution with a kernel supported in $[-1,1]$. So the center components U.C(0 .. L-1) of W can be obtained again via the procedure AA. This part, generalized to $A_{\kappa}^{2k}$, is implemented by HL_AAPower. An enclosure ${\tt U.C(L)}_{\scriptscriptstyle{\cal B}}$ for the tail of $u$ is constructed in the procedure HH_AAPower. In this part we use Proposition 3.1. The enclosure U for $u=A_{\kappa}^{2k}w$ returned by AAPower is the Sum of the enclosures returned by LA_AAPower, HL_AAPower and HH_AAPower. Our bound SumAAPowers on $A_{\kappa}^{2}\Sigma_{q}$ is very similar. For more details we refer to the program code [30].

Our bounds GMap and DGmap on the map $G_{q}$ and its derivative, respectively, are defined in the package Legends.Chain.Fix. They are obtained simply by combining lower-level bounds like Prod, AAPower, and SumAAPowers. The construction (5.3) of a quasi-Newton map ${\cal N}$ from a given map $G$ is sufficiently general and useful that it has been implemented in a generic package Linear.Contr. The same package has been used before in [27,28]. Our instantiation of Linear.Contr defines bounds Contr and Contr on the transformation ${\cal N}_{q}$ and its derivative, respectively. The type LMode that is used to instantiate Linear will be described below.

6.4. Operator norms

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Consider the task of estimating the norm of a linear operator on ${\cal B}_{\rho,r}^{\,\tau}$. Let $\hat{\tau}=1-2\tau$. Denote by ${\cal S}$ the set of all pairs $s=(j,n)$ of integers $j,n\geq 0$ with the property that $n\equiv\tau\,({\rm mod~{}}2)$ whenever $j=0$. If we define $\varrho_{(j,n)}=r^{j}\rho^{n}$ and

[TABLE]

then a function $w\in{\cal B}_{\rho,r}^{\,\tau}$ and its norm $\|w\|=\|w\|_{\rho,r}$ can be written as

[TABLE]

A useful feature of such weighted $\ell^{1}$ spaces is the following. Let ${\cal L}$ be a continuous linear operator on ${\cal B}_{\rho,r}^{\,\tau}$. Then the operator norm of ${\cal L}$ is simply \|{\cal L}\|=\sup_{s\in{\cal S}}\,\bigl{\|}{\cal L}h^{s}\bigr{\|}. In order to estimate this norm, we first choose a suitable partition $\{S_{1},S_{2},\ldots,S_{M}\}$ of ${\cal S}$. Then

[TABLE]

The sets $S_{m}$ that we use in our partitions of ${\cal S}$ are specified by data of type LMode. A partition is represented by an array (1 .. M) of LMode. Such partitions are created by the procedure Make in Legends.Chain. To simplify notation, let us identify a LMode S with the corresponding subset $S\subset{\cal S}$. A procedure Assign(S: in LMode; H: out LChain) defines a set ${\tt H}_{\scriptscriptstyle{\cal B}}\subset{\cal B}_{\rho,r}^{\,\tau}$ that contains all functions $h^{s}$ with $s\in S$.

The way this is being used is as follows. Let LinOp be a bound on the operator ${\cal L}$. Then a Ball-type enclosure ${\tt B}_{\scriptscriptstyle{R}}$ for the constant $b_{m}$ is obtained by calling Assign(S,H) with $S=S_{m}$, followed by LinOp(H,G) and then Norm(G,B). For the operator ${\cal L}=D{\cal N}_{q}(w)$, this is carried out by the procedure DContrNorm in Legends.Chain.Fix. This procedure is little more than an instantiation of the procedure Op_Norm from the generic package Linear, with LinOp being in essence DContr.

For the complete details we refer to the source code of our programs [30]. For the set of representable numbers (Rep) we choose standard extended floating-point numbers [33] that support controlled rounding, and for bounds on non-elementary Rep-operations we use the open source MPFR library [34]. Our programs were run successfully on a standard desktop machine, using a public version of the gcc/gnat compiler [32].

7. Appendix

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The figures below show graphs of the functions $v={\cal D}u$ associated with our solutions $u$ of the equation (1.3). Recall that $v$ is a solution of the equation

[TABLE]

Notice that odd solutions ($\sigma=0$) of this equation require $m$ to be odd.

All solutions depicted here are for the $\beta$-model ($m=3$). The solutions 1 and 2 for the $\alpha$-model ($m=2$) are similar to the $\beta$-model solutions 4 and 8.

7.1. Some solutions with $\sigma=1$

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Figure 2. $v={\cal D}u$ for the solutions 6, 8, 11, 14, and 17; for $m=3$ and \mu=-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 4}\hskip 1.0pt}.

Figure 3. $v={\cal D}u$ for the solutions 12, 15, 13, and 18; for $m=3$ and \mu=-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 4}\hskip 1.0pt}.

Figure 4. Comparison of graphs from Figures 2 and 3.

7.2. Some solutions with $\sigma=0$

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Figure 5. $v={\cal D}u$ for the solutions 19, 20, 21, and 22; for $m=3$ and \mu=-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt}.

Figure 6. Comparison of graphs from Figure 5.

7.3. Limit behavior

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Figure 7. Behavior of $v={\cal D}u$ as $\mu{\scriptstyle\nearrow\hskip 0.7pt}1$; \mu=-\hbox{\raise 2.2pt\hbox{\scriptstyle 63}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 64}\hskip 1.0pt},0,\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt},\hbox{\raise 2.2pt\hbox{\scriptstyle 3}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 4}\hskip 1.0pt},\hbox{\raise 2.2pt\hbox{\scriptstyle 7}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 8}\hskip 1.0pt},\hbox{\raise 2.2pt\hbox{\scriptstyle 63}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 64}\hskip 1.0pt}.

Figure 8. Behavior of $v={\cal D}u$ as $\mu{\scriptstyle\nearrow\hskip 0.7pt}0$;

\mu=-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt},-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 8}\hskip 1.0pt},-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 64}\hskip 1.0pt},-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 1024}\hskip 1.0pt} (left); \mu=-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 2}\hskip 1.0pt},-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 4}\hskip 1.0pt},-\hbox{\raise 2.2pt\hbox{\scriptstyle 1}\hskip-1.2pt{\scriptstyle/}\hskip-0.9pt\lower 2.2pt\hbox{\scriptstyle 256}\hskip 1.0pt} (right).

References \pdfdestnum xyz \pdfoutlinegoto num count 0 References

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