Recurrence and Resonance in the Cubic KleinGordon Equation
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Abstract
In a number of models for coupled oscillators and nonlinear wave equations primary resonances dominate the phasespace phenomena. A new feature is that in a Hamiltonian framework, the interaction of primary and higher order resonances is shown to be important and can be signaled by using recurrence properties. The interaction may involve embedded double resonance. We will demonstrate these phenomena for the cubic KleinGordon equation on a square with Dirichlet boundary conditions using normal form techniques. The results are qualitatively and quantitatively very different from the onedimensional spatial case.
Keywords
2dim nonlinear waves Hamiltonian Resonance zones Double resonanceMathematics Subject Classification
70H07 70H12 34E10 37J401 Introduction
Boundary value problems for nonlinear wave equation produce in a natural way problems with various resonances. We will consider these problems for a typical case, the cubic KleinGordon equation on a square. Galerkin projection and truncation will in this case lead to finitedimensional Hamiltonian systems.
After summarizing the approximation theory for nonlinear wave equations we will indicate the resonances in the case of the 2dimensional cubic KleinGordon equation in Sect. 2. The 1:1 resonances are a basic feature of this problem, they are analyzed in various combinations. Recurrence or lack of it will signal the presence or absence of resonance zones that may complicate the dynamics. Detuned resonances will produce embedded double resonance, see Sect. 2.7. Section 3 outlines the asymptotics of detuning.
1.1 Approximation of Wave Equations
We will employ two approximation steps for the cubic KleinGordon equation: Galerkin truncation of the system and averaging the resulting finitedimensional system; they produce an asymptotic estimate for the solution of the initialboundary value problem.
1.2 Normalizing Hamiltonian Systems
In many applications a combination of low and higher order resonances takes place. To avoid this one usually concentrates on the low order resonances neglecting the higher order ones. The purpose of this paper is a more complete theory by exploring the cases of combined low and higher order resonance. As we shall see, the tools will be averagingnormal form theory and the use of the Poincaré recurrence theorem to characterize the dynamics in resonance zones.
A powerful theorem on the stability of Hamiltonian systems in the sense of exponentiallylong time invariance of the actions was formulated and proved by Nekhoroshev [8]. This theorem presupposes steepness of the Hamiltonian and so the absence of first or second order resonances in the system. We cannot use the theorem in our case.
We will introduce nearidentity transformations producing normal forms; see [12] for theory and background literature. Prominent terms in the normal forms are produced by the resonances induced by the frequencies \(\omega \).
A two dof Hamiltonian system in first or second order resonance has an integrable normal form. The treatment of higher order resonance in [11] is quite general for two dof, for more than two dof the complexity of higher order resonance increases enormously. However, in the case of combined lower and higher order resonance we will, as in [16], consider for interactions the socalled resonance zones where the periodic solutions of the (primary) lower resonance are located.
1.3 Low and Higher Order Resonance
Assuming now more than two dof and that the frequency spectrum contains also first and/or second order frequency ratios. The corresponding low order frequency modes will dominate the phaseflow of higher order resonance except in primary resonance zones where the low order actions do not vary; in these zones the low order shortperiodic solutions are located.
In the case of many dof our strategy will be to locate the low order resonance zones (small neighborhoods of the resonance manifolds) and find out whether higher order resonance manifolds exist embedded in these zones; they will be called secondary resonance zones. This can be done analytically using normalization. The phenomenon will be called ‘embedded double resonance’, see [16]. For the general theory of double resonance see [3, 5] and more references there.
1.4 The Recurrence Theorem
Consider a dynamical system defined on a compact set in \({\mathbb{R}} ^{n}\) with the property that the flow induced by the system is measurepreserving. Poincaré uses the term volumepreserving for the phaseflow induced by a timeindependent Hamiltonian system without singularities on a compact domain, see [10], vol. 3, Chap. 26. Using the invariance of the volume of phaseelements under the flow, it is proved that most orbits return an infinite number of times arbitrarily close to their initial position; this is called recurrence. The recurrence time depends on the specific dynamical system considered, the initial condition chosen and the size of the neighborhood to be revisited. Consider for instance an initial point \(P_{0}\) and a ball with radius \(d>0\) centered around \(P_{0}\). The recurrence theorem states that after a finite time \(T_{d}\) an orbit starting in this ball will enter the ball again; there are exceptions for certain initial conditions but the exceptional initial conditions have measure zero.
It is easy to obtain an upper limit \(L\) for recurrence times, dependent on the Euclidean distance \(d(0)=d_{0}\) to the initial condition. Consider timeindependent Hamiltonian (6). Assume that \(H_{2}(p, q)\) is Morse at \((p, q)= (0, 0)\) and that the quadratic part is definite, so the origin is a stable equilibrium of the equations of motion. In [15] it is argued that:
Proposition 1
In the sequel we have often \(n {=} 3, d_{0} {=} 0.1\) producing \(L {=} 10^{5}\), if \(n {=} 4, d_{0} {=} 0.1, L {=} 10^{7}\). The actual Poincaré recurrence times are lower than \(L\) but passage of resonance zones can delay recurrence as the orbits will wind around the tori embedded in the resonance zones.
A preliminary test for embedded double resonance can be carried out using the recurrence theorem. Constructing numerical solutions of orbits passing the resonance zones, we expect fairly regular recurrent behavior if these zones contain no or very small resonance manifolds. Complicated and long time recurrent behavior points at motion around tori and other invariant manifolds during passage. In [16] the 1:1:4 resonance and the FermiPastaUlam \(\alpha \)chain were discussed as examples. Recurrence will be tested by computing the Euclidean distance \(d(t)\) to the initial conditions as a function of time.
2 The Cubic KleinGordon Equation
In the sequel we will consider problems derived from the cubic KleinGordon equation (1). The eigenfunctions of the linearized equation on a square are \(\phi _{kl}(x, y) = \sin kx \sin ly, k, l = 1, 2, \ldots \) .
2.1 Asymptotic Approximations

1:3:3 resonances with
\(kl = 11, 15, 51; \omega _{11}^{2}= 3\), \(\omega _{15}^{2}= 27\)
\(kl = 22, 48, 84; \omega _{22}^{2}= 9\), \(\omega _{48}^{2}= 81\)
\(kl = 13, 31, 77; \omega _{13}^{2}= 11\), \(\omega _{77}^{2}= 99\)

1:1:3:3 resonance
\(kl = 12, 21, 27, 72; \omega _{12}^{2}= 6\), \(\omega _{27}^{2}= 54\)

1:1:1 resonance
\(kl = 55, 17, 71; \omega ^{2}= 51\)

1:1:1:1 resonances
\(kl = 47, 74, 18, 81; \omega ^{2}= 66\)
\(kl = 67, 76, 29, 92; \omega ^{2}= 86\)

Detuned 1:2:2 resonance
\(kl = 33, 57, 75; \omega _{33}^{2}= 19, \omega _{57}^{2}=\omega _{75} ^{2}= 75\)
\(kl = 14, 41, 66; \omega _{14}^{2}=\omega _{41}^{2}= 18, \omega _{66} ^{2}= 73\)

Detuned 1:1:2:2 resonance
\(kl = 24, 42, 19, 91; \omega _{24}^{2}= 21, \omega _{19}^{2}= 83\)

Detuned 1:1:3:3 resonance
\(kl = 13, 31, 49, 94; \omega _{13}^{2}= 11, \omega _{49}^{2}= 98\)

A higher order 3:3:7:7 resonance with \(kl= 14, 49\)
We will start with a sketch of the basic 1:1 resonances; after this we study the dynamics and weak interactions with other resonances. Our focus will be on the interesting case where recurrence highlights more complicated dynamics involving higher order resonance.
2.2 The Basic 1:1 Resonances
An additional feature of the Galerkin projection for the cubic KleinGordon equation is that there exist other phaselocked solutions if \(\cos 2 \chi _{1}= 1/4\). Substituting this value into system (17) we find heteroclinic invariant manifolds connecting the unstable normal modes. Both the periodic solutions in the resonance zones and the heteroclinic solutions will be a returning feature in what follows.
2.3 Combining Independent 1:1 Resonances
Suppose we have a Galerkin truncation with a finite number \(M\) of basic 1:1 resonances \(k_{i}, l_{i}, k_{i} \neq l_{i}\) for certain indices \(k_{i}, l_{i}\). We exclude the special resonance cases as presented in Sect. 2.1: (1:1:1:1), (1:1:2:2) and (1:1:3:3).
With this assumption the \(M\) basic 1:1 resonances will be independent of each other. The resulting approximation will be a superposition of the individual basic resonances.
Note that for values of \(\omega \) such that \(1/ \omega ^{2} \leq \varepsilon \) the contribution of such a 1:1 resonance will be of order \(\varepsilon ^{2}\).
2.4 The First Three Modes
As the frequency ratio \(\omega _{11}: \omega _{12}\) is close to 7:10, one could look for the 7:10 higher order resonance in the resonance zones. As we find from Eq. (27) that the combination angle \((10 \psi _{1}7 \psi _{2})\) is timelike, this higher order resonance does not arise.
2.5 Coupled 1:1 Resonances with Different Frequencies
Considering a truncation to 4 modes generated by \(k_{1}l_{1}, k_{2}l _{2}\) (\(k_{1} \neq l_{1}, k_{2} \neq l_{2}\)) we may find interaction if a resonance exists between the frequencies \(\omega _{k_{1}l_{1}}, \omega _{k_{2}l_{2}}\). As a typical example we consider the case of the 1:1:3:3 resonance with \(kl = 12  21  27  72\); \(\omega _{12}^{2}= 6\), \(\omega _{27}^{2}= 54\). Coupled 1:1 resonances were discussed in [9] without the additional 1:3 resonance.
If \(r_{3}(0)= r_{4}(0)=0\) we find 4 periodic solutions in a submanifold from the conditions \(r_{1}^{2}(0)= r_{2}^{2}(0)=E_{1}, \sin 2 \chi _{1} =0\). If \(r_{3}(0) r_{4}(0)\neq 0\), the approximate solutions in the resonance zone \(r_{1}^{2}(0)= r_{2}^{2}(0)=E_{1}, \sin 2 \chi _{1} =0\) are in general not periodic anymore as \(u_{12}, u_{21}\) are periodic with period depending on \(E_{1}\) and \(E_{2}\), but \(u_{27}, u_{72}\) are in general not periodic. An analogous reasoning applies when considering \(r_{1}(0)= r_{2}(0)=0\) and the resonance zone \(r_{3}^{2}(0)= r_{4} ^{2}(0)=E_{2}, \sin 2 \chi _{2} =0\). A different case arises when the resonance zones intersect, we have a double resonance. In this case we find quasiperiodic solutions with two periods depending on \(E_{1}, E_{2}\). We can characterize the intersection by putting \(u_{12}^{2}=u_{21}^{2}, u_{27}^{2}=u_{72}^{2}\) in system (29) and applying averagingnormalization to the resulting equations. Although there is anglecoupling between the modes, the 1:3 resonance is not present in the intersection.
2.6 Coupled 1:1 Resonances, the 1:1:1:1 Resonance
 6 (2 dof) 4dimensional invariant manifolds \(M_{12}, M_{13}, M_{14}, M _{23}, M_{24}, M_{34}\) with for instance \(M_{12}\) given by \(u_{3}(t)=u _{4}(t)=0, t \geq 0\). The phaseflow in these invariant manifolds has been described in Sect. 2.2, the normal modes are unstable. Two periodic solutions in general position in \(M_{12}\) are given by:$$ u_{1}(t)=u_{2}(t),\quad \quad \ddot{u}_{1} + u_{1}= \varepsilon \frac{21}{16 \omega ^{2}}u^{3}_{1}. $$(33)
 4 (3 dof) 6dimensional invariant manifolds \(M_{123}, M_{124}, M_{234}, M_{134}\). Apart from the periodic solutions in 4dimensional submanifolds we have two periodic solutions in general position given bywith for instance in \(M_{123}, u=u_{1}=u_{2}=u_{3}\).$$ \ddot{u}+ u= \varepsilon \frac{33}{16 \omega ^{2}}u^{3}, $$(34)
2.7 Embedded Double Resonance in a Detuned Case
Normalization in \(M_{12}\)
Normalization in a General Position Resonance Zone
Exact solutions as in Sect. 2.2.
3 Appendix on the Analysis of Detuning
Our results on normalization and error estimates are based on [12] with slight modifications. We assume existence and uniqueness of solutions of initial value problems, sufficient smoothness and \(T\)periodicity of vector fields. To study the dynamics of secondary resonance in the detuned case, we have to normalize to higher order, in Hamiltonian terms to \(H_{6}\). Detuning produces in a number of cases interesting bifurcation phenomena, so we outline the procedure here in a rather general context.
Applying higher order normalization to a detuned resonance like system (38) we find at \(O(\varepsilon ^{2})\) terms of the form \(r_{1}^{3}r_{2}^{2} \sin (4 \psi _{1}2 \psi _{2})\) and similar terms involving \(r_{3}\). These terms introduce the 4:2 resonances with corresponding periodic solutions discussed in the preceding section.
4 Conclusions
 1.
In mathematical physics PDEs have often been analyzed in the case of one space dimension. We have shown that when allowing more space dimensions, the results may change remarkably. We discuss a typical case of mathematical physics, the cubic KleinGordon equation.
 2.
The validity of asymptotic approximations holds on intervals of time proportional to \(1/ \varepsilon \) and \(\omega _{kl} \). Our analysis yields some conclusions using the truncation procedure of series (3). First, if \(\omega _{kl} \) is large enough the tail of the series will take an extremely long time to become effective. Secondly, the modes in 1:1 resonance are ubiquitous but combination of independent 1:1 systems produces no new phenomena. Thirdly, interesting phenomena like embedded double resonance arise from detuning with new resonant interactions. There will exist an infinite number of such detuned systems, but they arise for large values of \(\omega _{kl} \).
For the complicated dynamics described in the preceding sections to be observed one has to choose the corresponding initial conditions producing the resonant modes. Exciting for instance only one mode, there will be nontrivial evolution if this mode is unstable in a resonant setting and if it is slightly perturbed.
 3.
The most interesting dynamics is described for the 1:1:1:1 resonance in Sect. 2.6 and the detuned resonance in Sect. 2.7. The resonance zones vanish (have size \(o(1)\)) as \(\varepsilon \rightarrow 0\) and so are free boundary layers in the sense of singular perturbation theory. In these zones the resonances produce locally stable and unstable periodic solutions with corresponding stable and unstable manifolds. Intersection of invariant manifolds associated with the periodic solutions in the resonance zones are in Hamiltonian mechanics the main source of chaos. For small values of \(\varepsilon \) this will enable the possibility of boundary layer chaos in the cubic KleinGordon equation, it may be more prominent if \(\varepsilon \) increases.
 4.
Changing the boundary conditions or the shape of the domain will of course change our results. However, the setup of our analysis is typical for such new problems. Symmetries, for instance considering circle or ring domains, may simplify the analysis. New phenomena may arrive when studying nonconvex domains.
Notes
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