Quasi-periodic solutions of the equation v_{tt}-v_{xx}+v^3=f(v)

We consider 1D completely resonant nonlinear wave equations of the type v_{tt}-v_{xx}=-v^3+O(v^4) with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutions turn out to be, at the first order, the superposition of a traveling wave and a modulation of long period, depending only on time.


Introduction
This paper deals with a class of one-dimensional completely resonant nonlinear wave equations of the type where f : R → R is analytic in a neighborhood of v = 0 and f (v) = O(v 4 ) as v → 0.
In the recent paper [12], M. Procesi proved the existence of small-amplitude quasi-periodic solutions of (1) of the form v(t, x) = u(ω 1 t + x, ω 2 t − x), where u is an odd analytic function, 2π-periodic in both its arguments, and the frequencies ω 1 , ω 2 ∼ 1 belong to a Cantor-like set of zero Lebesgue measure. It is assumed that f is odd and f (v) = O(v 5 ), see Theorem 1 in [12]. These solutions v(t, x) correspond -at the first order -to the superposition of two waves, traveling in opposite directions: where ω 1 , ω 2 = 1 + O(ε). Motivated by the previous result, we study in the present paper the existence of quasi-periodic solutions of (1) having a different form, namely v(t, x) = u(ω 1 t + x, ω 2 t + x). ( Moreover we do not assume f to be odd. First of all, we have to consider different frequencies than in [12]. Precisely, the appropriate choice for the relationship between the amplitude ε and the frequencies ω 1 , ω 2 turns out to be where b ∼ 1/2, ε ∼ 0. This choice leads to look for quasi-periodic solutions v(t, x) of (1) of the form v(t, x) = u εt, (1 + bε 2 )t + x , where (b, ε) ∈ R 2 , 1+bε 2 ε / ∈ Q. On the contrary, taking in (3) frequencies ω 1 = 1 + ε, ω 2 = 1 + aε as in [12], no quasi-periodic solutions can be found, see Remark in section 2. We show that there is no loss of generality passing from (3) to (5), because all the possible quasi-periodic solutions of (1) of the form (3) are of the form (5), see Appendix B.
Searching small amplitude quasi-periodic solutions of the form (5) by means of the Lyapunov-Schmidt method, leads to the usual system of a range equation and a bifurcation equation.
The former is solved, in a similar way as [12], by means of the standard Contraction Mapping Theorem, for a set of zero measure of the parameters. These arguments are carried out in section 4.
In section 5 we study the bifurcation equation, which is infinite-dimensional because we deal with a completely resonant equation. Here new difficulties have to be overcome. Since f is not supposed to be odd, we cannot search odd solutions as in [12], so we look for even solutions. In this way, the bifurcation equation contains a new scalar equation for the average of u, see [C-equation] in (12), and the other equations contain supplementary terms.
To solve the bifurcation equation we use an ODE analysis; we cannot directly use variational methods as in [3], [4], [6] because we have to ensure that both components r, s in (12) are non-trivial, in order to prove that the solution v is actually quasiperiodic.
Next we prove its non-degeneracy (Lemmas 2,3,4); these computations are the heart of the present work. Instead of using a computer assisted proof as in [12], we here employ purely analytic arguments, see also [11] (however, our problem requires much more involved computations than in [11]). In this way we prove the existence of quasi-periodic solutions of (1) of the form (5), see Theorem 1 (end of section 5).
¿From the physical point of view, this new class of solutions turns out to be, at the first order, the superposition of a traveling wave (with velocity greater than 1) and a modulation of long period, depending only on time: Finally, in section 6 we show that our arguments can be also used to extend Procesi result to non-odd nonlinearities, see Theorem 2.
We also mention that recently existence of quasi-periodic solutions with n frequencies have been proved in [16]. The solutions found in [16] belong to a neighborhood of a solution u 0 (t) periodic in time, independent of x, so they are different from the ones found in the present paper.
where f is analytic in a neighborhood of v = 0 and f (v) = O(v 4 ) as v → 0. We look for solutions of the form (3), for (ω 1 , ω 2 ) ∈ R 2 , ω 1 , ω 2 ∼ 1 and u 2π-periodic in both its arguments. Solutions v(t, x) of the form (3) are quasi-periodic in time t when u actually depends on both its arguments and the ratio between the periods is irrational, ω1 ω2 / ∈ Q. We set the problem in the space H σ defined as follows. Denote T = R/2πZ the unitary circle, ϕ = (ϕ 1 , ϕ 2 ) ∈ T 2 . If u is doubly 2π-periodic, u : T 2 → R, its Fourier series is Let σ > 0, s ≥ 0. We define H σ as the space of the even 2π-periodic functions u : The elements of H σ are even periodic functions which admit an analytic extension to the complex strip {z ∈ C : |Im(z)| < σ}.
(H σ , · σ ) is a Hilbert space; for s > 1 it is also an algebra, that is, there exists a constant c > 0 such that see Appendix A. Moreover the inclusion H σ,s+1 ֒→ H σ,s is compact.
We fix s > 1 once and for all.
We note that all the possible quasi-periodic solutions of (1) of the form (3) are of the form (5) if we choose frequencies as in (4), see Appendix B. So we can look for solutions of (1) of the form (5), v(t, x) = u εt, (1 + bε 2 )t + x , without loss of generality. For functions of the form (5), problem (1) is written as (1) can be written as The main result of the present paper is the existence of solutions u (b,ε) of (7) for (b, ε) in a suitable uncountable set (Theorem 1).
Remark. If we simply choose frequencies ω 1 = 1 + ε, ω 2 = 1 + aε as in [12], we obtain a bifurcation equation different than (12). Precisely, it appears 0 instead of −b(2 + bε 2 ) s ′′ in the left-hand term of the Q 2 -equation in (12); so we do not find solutions which are non-trivial in both its arguments, but only solutions depending on the variable ϕ 1 . This is a problem because the quasi-periodicity condition requires dependence on both variables.

Lyapunov-Schmidt reduction
The operator M b,ε is diagonal in the Fourier basis e mn = e imϕ1 e inϕ2 with eigenvalues −D b,ε (m, n), that is, if u is written in Fourier series as in (6), where the eigenvalues D b,ε (m, n) are given by For ε = 0 the operator is M b,0 = 2 ∂ 2 ϕ1ϕ2 ; its kernel Z is the subspace of functions of the form u(ϕ 1 , ϕ 2 ) = r(ϕ 1 ) + s(ϕ 2 ) for some r, s ∈ H σ one-variable functions, We can decompose H σ in four subspaces setting Thus the kernel is the direct sum Z = C ⊕ Q 1 ⊕ Q 2 and the whole space is H σ = Z ⊕ P . Any element u can be decomposed as u(ϕ) =û 0,0 + r(ϕ 1 ) + s(ϕ 2 ) + p(ϕ 1 , ϕ 2 ) = z(ϕ) + p(ϕ).
We denote · the integral average: given g ∈ H σ , Note that 1 2π 2π 0 e ikt dt = 0 for all integers k = 0, so for all r ∈ Q 1 , s ∈ Q 2 , p ∈ P, u ∈ H σ , and by means of these averages we can construct the projections on the subspaces, Let u = z + p as in (11); we write u 3 as u 3 = z 3 + (u 3 − z 3 ) and compute the cube z 3 = (û 0,0 + r + s) 3 . The operator M b,ε maps every subspace of (10) in itself So we can project our problem (7) on the four subspaces: Now we study separately the projected equations.

The range equation
We write the P -equation thinking p as variable and z as a "parameter", We would like to invert the operator M b,ε . In Appendix C we prove that, fixed any γ ∈ (0, 1 4 ), there exists a non-empty uncountable set B γ ⊆ R 2 such that, for all (b, ε) ∈ B γ , it holds Precisely, our Cantor set B γ is whereB γ is a set of "badly approximable numbers" defined as for every h = m,n =0ĥ mn e imϕ1 e inϕ2 ∈ P . Thus we obtain a bound for the inverse operators, uniformely in (b, ε) ∈ B γ : Applying the inverse operator (M b,ε|P ) −1 , the P -equation becomes We would like to apply the Implicit Function Theorem, but the inverse operator (M b,ε|P ) −1 is defined only for (b, ε) ∈ B γ and in the set B γ there are infinitely many holes, see Appendix C. So we fix (b, ε) ∈ B γ , introduce an auxiliary parameter µ and consider the auxiliary equation Following Lemma 2.2 in [12], we can prove, by the standard Contraction Mapping Theorem, that there exists a positive constant c 1 depending only on f such that, equation (15) admits a solution p (b,ε) (µ, z) ∈ P . Moreover, there exists a positive constant c 2 such that the solution p (b,ε) (µ, z) respects the bound Than we can apply the Implicit Function Theorem to the operator at every point (0, z, 0), so, by local uniqueness, we obtain the regularity: p (b,ε) , as function of (µ, z), is at least of class C 1 .
Notice that the domain of any function p (b,ε) is defined by (16), so it does not depend on (b, ε) ∈ B γ .
In order to solve (14), we will need to evaluate p (b,ε) at µ = ε; we will do it as last step, after the study of the bifurcation equation.
We observe that in these computations we have used the Hilbert algebra property

The bifurcation equation
We consider auxiliary Z-equations: we put f µ instead of f ε in (12), We substitute the solution p (b,ε) (µ, z) of the auxiliary P -equation (15) inside the auxiliary Z-equations (18), We have p (b,ε) (µ, z) = 0 for µ = 0, so the term (u 3 − z 3 ) − f µ (u) vanishes for µ = 0 and the bifurcation equations at µ = 0 become We look for non-trivial z =û 0,0 + r(ϕ 1 ) + s(ϕ 2 ) solution of (19). We rescale setting so the equations become In the following we show that, for |λ−1| sufficiently small, the system (21) admits a non-trivial non-degenerate solution. We consider λ as a free real parameter, recall that Z = C × Q 1 × Q 2 and define G : R × Z → Z setting G(λ, c, x, y) as the set of three left-hand terms of (21). Lemma 1. There existσ > 0 and a non-trivial one-variable even analytic function Proof. We prove the existence of a non-trivial even analytic function β 0 which satisfies For any m ∈ (0, 1) we consider the Jacobi amplitude am( · , m) : R → R as the inverse of the elliptic integral of the first kind We define the Jacobi elliptic cosine setting cn(ξ) = cn(ξ, m) = cos(am(ξ, m)), see [1] ch.16, [15]. It is a periodic function of period 4K, where K = K(m) is the complete elliptic integral of the first kind Jacobi cosine is even, and it is also odd-symmetric with respect to K on [0, 2K], that is cn(ξ + K) = −cn(ξ − K), just like the usual cosine. Then the averages on the period 4K are cn = cn 3 = 0.
We have constructed two solutionsū,v of the homogeneous equation; their wronskianū ′v −ūv ′ is equal to 1, so we can write a particular solutionw of the non-homogeneous equation (25) as Every solution of (25) is of the form w = Aū + Bv +w for some (A, B) ∈ R 2 . Since h is even,w is also even, so w is even if and only if A = 0.
An even function w = Bv +w is 2π-periodic if and only if w(ξ + 2π) − w(ξ) = 0, that is, by (30), We removeū(ξ), derive the expression with respect to ξ and from (30) it results zero at any ξ. Then the expression is a constant; we compute it at ξ = 0 and obtain, since hū is odd and 2π-periodic, that w is 2π-periodic if and only if B = 1 V 2 k 2π 0 hv. Thus, given h even 2π-periodic, there exists a unique even 2π-periodic w such that w ′′ + 3β 2 0 + 3 β 2 0 w = h and this defines the operator L, L is linear and continuous with respect to · σ ; it is the Green operator of the equation x ′′ + 3β 2 0 + 3 β 2 0 x = h, so, by classical arguments, it is a bounded operator of H σ,s into H σ,s+2 ; the inclusion H σ,s+2 ֒→ H σ,s is compact, then L : H σ → H σ is compact.

Waves traveling in opposite directions
In this section we look for solutions of (1) of the form (2), for u ∈ H σ . We introduce two parameters (a, ε) ∈ R 2 and set the frequencies as in [12], For functions of the form (2), problem (1) is written as . Thus the problem can be written as L a,ε [u] = −εu 3 + εf ε (u).
By Lyapunov-Schmidt reduction we project the equation (46) on the four subspaces, We repeat the arguments of Appendix C and find a Cantor set A γ such that |D a,ε (m, n)| > γ for every (a, ε) ∈ A γ . Then L a,ε is invertible for (a, ε) ∈ A γ and the P -equation can be solved as in the section 4.
Proposition. Let A, B ∈ Mat 2 (R) be invertible matrices such that AB −1 has integer coefficient. Then, given any u ∈ H σ , the function v(t, x) = u A(t, x) can be written as v(t, x) = w B(t, x) for some w ∈ H σ , that is {u • A : u ∈ H σ } ⊆ {w • B : w ∈ H σ }.
In fact,B γ contains all the irrational numbers whose continued fractions expansion is of the form [0, a 1 , a 2 , . . . ], with a j < γ −1 − 2 for every j ≥ 2. Such a set is uncountable: since γ −1 − 2 > 2, for every j ≥ 1 there are at least two choices for the value of a j . Moreover, it accumulates to 0: if y = [0, a 1 , a 2 , . . . ], it holds 0 < y < a −1 1 , and a 1 has no upper bound. See also Remark 2.4 in [2] and, for the inclusion of such a set inB γ , the proof of Theorem 5F in [14], p. 22.
We prove the following estimate.