Wellposedness for a (1+1)-dimensional wave equation with quasilinear boundary condition

We consider the linear wave equation $V(x) u_{tt}(x, t) - u_{xx}(x, t) = 0$ on $[0, \infty)\times[0, \infty)$ with initial conditions and a nonlinear Neumann boundary condition $u_x(0, t) = (f(u_t(0,t)))_t$ at $x=0$. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where $f\colon\mathbb{R}\to\mathbb{R}$ is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where $f$ is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at $x=0$ have enough regularity to solve the initial value problem with their own initial data.

Our interest in (1) stems from the fact that it appears in the context of electromagnetics as an exact reduction of a nonlinear Maxwell system.We recall the Maxwell equations in the absence of charges and currents with the electric field E, the electric displacement field D, the polarization field P, the magnetic field B, and the magnetic induction field H. Particular properties of the underlying material are modelled by the specification of the relations between E, D, P on one hand, and B, H on the other hand.Here, we assume a magnetically inactive material, i.e., B = µ 0 H, but on the electric side we assume a material with a Kerr-type nonlinear behaviour, cf.[1], Section 2.3, given through P(E) = ε 0 χ 1 (x)E + ε 0 χ NL (x)g(|E| 2 )E with x = (x, y, z) ∈ R 3 and |•| the Euclidean norm on R 3 .For simplicity we assume that χ 1 , χ NL are given scalar valued functions instead of the more general situation where they are matrix valued.The scalar constants ε 0 , µ 0 are such that c = (ε 0 µ 0 ) −1/2 is the speed of light in vacuum.Local existence, wellposedness and regularity results for the general nonlinear Maxwell system have been shown on R 3 by Kato [3] and on domains by Spitz [7,8].
Then the quasilinear vectorial wave-type equation ( 2) turns into the scalar equation ( 4) for U = U(x, t), where V (x) = µ 0 ε 0 (1 + χ 1 (x)) − κ −2 and Γ(x) = µ 0 ε 0 χ NL (x).Note that ( 4) is an exact reduction of the Maxwell problem, from which all fields can be reconstructed.E.g., the magnetic induction B can be retrieved from ∇ × E = −∂ t B by time-integration and it will satisfy ∇ • B = 0 provided it does so at time t = 0.By assumption the magnetic field is given by H = 1 µ 0 B and it satisfies ∇ × H = ∂ t D. It remains to check that the displacement field D satisfies the Gauss law ∇ • D = 0 in the absence of external charges.This follows directly from the constitutive equation D = ε 0 (1 + χ 1 (x))E + ε 0 χ NL (x)g(|E| 2 )E and the assumption of the polarized form of the electric field in (3).
In the extreme case where Γ(x) = 2δ 0 (x) is a multiple of the δ-distribution at 0 and where U(x, t) = u t (x, t) for an even function u(x, t) = u(−x, t), by removing one time derivative (4) becomes with f (s) := g(s 2 )s.Clearly (1) is the initial value problem for (5).
Problem (5) with f (s) = ±s 3 has been considered in [4].Under specific assumptions on the linear potential V the existence of infinitely many breathers, i.e., real-valued, time-periodic, spatially localized solutions of (5), was shown.Typical examples of V were given in classes of piecewise continuous functions having jump discontinuities.Under different assumptions on V and Γ, but still including δ-distributions, problem (5) was considered in [2] and real-valued breathers were constructed.Our goal is to study the initial value problem (1) from the point of view of wellposedness, to derive the conservation of momentum and energy, and to verify that known time-periodic solutions from [4] satisfy (1) with their own initial values.Note that the boundary condition in (1) becomes u x (0, t) = ±3u t (0, t) 2 u tt (0, t) in the model case f (s) = ±s 3 .
Hence, (1) is a singular initial value problem which is not covered by typical theories like, e.g., energy methods or monotone operators.Instead, our approach will be to prove existence by making use of the method of characteristics.Uniqueness, wellposedness, global existence, and the conservation of energy and momentum will build upon this.
Our basic assumptions on the initial data u 0 , u 1 are: and in general all function spaces consist of real-valued functions unless the codomain is explicitly mentioned.Motivated by the results from [4] we are interested in the case where the coefficient V may have discontinuities.In particular, we consider piecewise C 1 functions V .
Let I ⊆ R be a closed interval.We call a function φ : I → R piecewise C k if there exists a discrete set D ⊆ I such that φ ∈ C k (I \ D) and the limits φ (j) (x−) and φ (j) (x+) exist for all x ∈ D(φ) and 0 ≤ j ≤ k, although they do not need to coincide.If I is bounded from below (or above), in addition we require φ (j) (min I+) (or φ (j) (max I−)) to exist for all 0 ≤ j ≤ k.Let P C k (I) denote the set of piecewise C k functions on I, and for φ ∈ P C(I) := P C 0 (I) let us denote by D(φ) the set of discontinuities of φ.
For the coefficient V and the nonlinear function f we assume The main theorem of this paper is given next.Theorem 1.1.Assume (A0)-(A3).Then (1) admits a unique and global C 1 -solution.Moreover, (1) is wellposed on every finite time interval [0, T ] with T > 0.
In Proposition 6.1 our concept of continuous dependence on data is stated precisely.In the above result the assumption (A3) is crucial.For a decreasing homeomorphism f the result of Theorem 1.1 does not hold, see Remark 1.7.Since we have already used the notion of a C 1 -solution, we are going to explain it in detail next.As the notion of a C 1 -solution will also be used for subdomains of [0, ∞) × [0, ∞) we first define the notion of an admissible domain.
for almost all x.We denote the relative interior of Ω by In order to explain the notion of a C 1 -solution let us first mention that we cannot expect that a solution of (1) has everywhere second derivatives u tt or u xx .This is essentially due to the nonlinear boundary condition and the discontinuities of second derivatives which propagate away from x = 0.However, if we denote by c(x the inverse of the x-dependent wave speed, then we can factorize the wave operator as It is then reasonable for a C 1 -solution to have almost everywhere a mixed second directional derivative ∂ 2 ν,µ with directions ν = (1, −c(x)) and µ = (1, c(x)).This is the basis for the following definition.

Definition 1.3 (solution).
A function u ∈ C 1 (Ω) on an admissible domain Ω is called a C 1solution to (1) if the following hold: Problem (1) has a momentum given by and an energy given by where The conservation of momentum and energy is stated next.
Another common notion of solution for (1) is the notion of a weak solution, which we only give for Ω = [0, ∞) 2 .The fact that a C 1 -solution to (1) is also a weak solution to (1) holds true an will be proven in Proposition 5.2 in Section 5.

Definition 1.6 (weak solution)
Remark 1.7.Due to assumption (A3) we have only considered increasing functions f .If we instead allow f : R → R to be a decreasing homeomorphism, then (1) will not be wellposed in general and can have multiple solutions.Consider for example the cubic term f (y) = −y 3 with constant potential V = 1 and homogeneous initial data: By direct calculation one can show that the right-traveling wave x ≥ t is a nontrivial solution to (8).In fact, u is a C 1 -solution of (∂ x + ∂ t )u = 0.But (8) also has the trivial solution u = 0, or u(x, t) = ±u p (x, t − τ ) for any τ ≥ 0. However, due to the continuity of f −1 , one can still show existence of solutions to (1) in the case where f grows at least linearly, cf.(A4).This follows from the arguments in Sections 3 and 4. Theorem 1.4 also holds when f is decreasing, but now the quantity F (y) tends to −∞ as y → ±∞, so that (7) does not give rise to estimates on u.Lastly, also in this case C 1 -solutions to (1) are weak solutions.
In addition to the problem being posed on the positive real half-line x ∈ [0, ∞), we can also consider the same quasilinear problem posed on a bounded domain x ∈ [0, L] where we impose a homogeneous Dirichlet condition at x = L: Both Theorem 1.1 and Theorem 1.4 remain valid when making the obvious adaptations to this setting.
Remark 1.9.For Dirichlet boundary data, momentum is in general not conserved.
The paper is structured as follows.In Section 2 we provide a change of variables which turns the wave operator with variable wave speed in (1) into a constant coefficient operator with a convenient factorization.In Section 3 we collect all results on the linear wave equation that is obtained from the change of variables in Section 2. Section 4 contains the proof of the existence and uniqueness part of the main result of Theorem 1.1 under an extra assumption which will removed in the subsequent Section 5.This section also contains the proof of energy and momentum conservation as stated in Theorem 1.4, and the fact that C 1 -solutions of (1) in the sense of Definition 1.3 are also weak solutions, cf.Proposition 5.2.The wellposedness part of Theorem 1.1 can be found in Section 6.Finally, in Section 7 we verify that the breather solutions from [4] satisfy (1) with their own initial values.The Appendices A and B contain some technical results used in the proofs of the main results.

A change of variables
It will be convenient to normalize the wave speed to 1.To achieve this, we introduce a new variable z = κ(x) = x 0 1 c(s) ds, and thus a new coordinate system (z, t).Avoiding new notation we denote the functions V, c, u, u 0 , u 1 transformed into this new coordinate system again by V, c, u, u 0 , u 1 .The relation between the two coordinate systems is given by From now on until the end of Section 5, we will exclusively work with the coordinate system (z, t).As before we denote the points where c is discontinuous by D(c) and the points where c z is discontinuous by D(c z ).
Formally the initial value problem (1) transforms into where we need to take into account that u x = 1 c u z is continuous (and not u z itself) and that the differential equation does not hold at the discontinuities of c and c z .A detailed definition of the solution concept is given below in Definition 2.3.
We begin by rephrasing Definitions 1.2 and 1.3 for the new coordinate system.Definition 2.1 (admissible domain).We call a set where h ≡ +∞ or h : [0, ∞) → R is Lipschitz continuous with Lipschitz constant 1.We denote its relative interior by Next we introduce function spaces that capture the condition of the continuity of 1 c u z .Definition 2.2 (x-dependent function spaces).Let the transformation between (x, t) and (z, t)coordinates be given by κ(x, t) )} where we understand u to be a function of (z, t) variables, and Similarly, for an interval I ⊆ [0, ∞) we define where again we understand v to be a function of z.

Definition 2.3 (solution).
A function u ∈ C 1 (x,t) (Ω) on an admissible domain Ω is called a C 1 -solution to (10) if the following hold:

Auxiliary results on the linear part
In this section we gather some auxiliary results and estimates on the linear wave equation.These will prove useful for the study of the nonlinear boundary condition.All results of this section hold under the assumptions (A0)-(A3).
We first note that the wave equation has finite speed of propagation; if we know its behavior at time t 0 on an interval [z 0 − r, z 0 + r], then we can defer its accurate behavior on the space-time triangle with corners (z 0 − r, t 0 ), (z 0 + r, t 0 ) and (z 0 , t 0 + r).Definition 3.1.For (z 0 , t 0 ) ∈ R 2 and r > 0 we denote the triangle with corners (z 0 − r, t 0 ), (z 0 + r, t 0 ) and (z 0 , t 0 + r) by its base projected onto the z-axis is given by P z ∆(z 0 , t 0 , r) = [z 0 − r, z 0 + r] with projection P z (z, t) := z.Similarly, we define left and right half triangles whose bases are given by Recall the solution formula for the 1-dimensional wave equation: , and g ∈ L ∞ (∆) is continuous outside a set L consisting of finitely many lines of the form {z = const}.Then the function belongs to C 1 (∆) and is the unique C 1 -solution of the problem [6,Theorem 9.41].As a consequence, any of the two factorizations of the wave operator ) can be used and yields the same solution.
By combining the above Theorem 3.2 with a fixed point argument, we can treat the initial value problem for c(z) u z on sufficiently small triangles ∆.In order to have a slightly more general situation available we work with a piecewise continuous function has a unique solution u ∈ C 1 (∆) in the sense of Theorem 3.2 with g = −λu z and L = D(λ)×R.We denote this solution by Φ(u 0 , u 1 ) := u.Remark 3.5.If additionally u 0 , u 1 are odd around z = z 0 and λ is odd around z = z 0 , then the solution of ( 11) is odd around z = z 0 .To see this, notice that under these assumptions the odd reflection of the solution u of (11) again solves (11) -but with the opposite factorization of the wave operator.Hence, by Remark 3.3 and uniqueness of solutions, u coincides with its odd reflection.
Proof of Corollary 3.4.W.l.o.g.we assume (z 0 , t 0 ) = (0, 0).Let u ∈ C 1 (∆).Then by Theorem 3.2 u is a solution if and only if holds for (z, t) ∈ ∆.Taking the derivative w.r.t.z we obtain We consider (13) as a fixed point problem for u z ∈ C(∆).If we denote the right-hand side of (13) by T (u z )(z, t), then clearly T maps C(∆) into itself.Furthermore, one has so that by Banach's fixed-point theorem there exists a unique solution u z of (13).With the help of u z we define u as in (12) and thus get the claimed result.
In the setting of the above proof, we can obtain estimates on the solution u.First, if we set q := r λ ∞ , then by Banach's fixed-point theorem we have we also obtain Combining these estimates, we get the following result.
Corollary 3.6.In the setting of Corollary 3.4, the following estimates hold with q := r λ ∞ : In particular, there exists a constant C = C(r, λ ∞ ) such that the operator-norm of the linear solution operator Φ : to the solution of (11), satisfies Recall that in Definition 2.3 we required uz c to be continuous.Since c may have jumps, e.g. at z 0 , we also need to treat the jump condition We prepare this in the following lemma by adding to (11) the inhomogeneous Dirichlet condition u(z 0 , t) has a unique C 1 -solution u : ∆ + → R in the sense of Theorem 3.2 with g = −λu z and L = D(λ) × R. We denote this solution by Φ + (b, u 0 , u 1 ) := u.The assertion also holds for the right half triangle ∆ − := ∆ − (z 0 , t 0 , r) with corresponding solution operator Φ − .
Proof.Note that the function G b defined on ∆ + by Note that v 0 (z 0 ) = v 1 (z 0 ) = 0 by assumption.If we extend the functions v 0 , v 1 , and λ in an odd way and G b in an even way around z = z 0 , we can consider the problem where ∆ := ∆(z 0 , t 0 , r) and B := P z ∆.Arguing as in the proof of Corollary 3.4, we see that due to the Banach fixed-point theorem, (17) has a unique solution, which must be odd, cf.Remark 3.5.Now, on one hand the solution of ( 17) solves (after restriction to ∆ + ) (16) and, on the other hand, after odd extension around z = z 0 every solution of ( 16) solves (17).This shows existence and uniqueness for (16) and hence for (14).
Remark 3.8.One can show that there exists a constant C = C(r, λ ∞ ) such that When treating the nonlinear problem (1), the operators Φ ± play an important role and the estimate in Remark 3.8 will be used.However, we need to investigate the dependency of Φ ± on the datum b more precisely.This will be achieved next in the case where u 0 = u 1 = 0.
Lemma 3.9 (Estimate on Φ ± in the case u 0 = u 1 = 0).Let ∆ ± , and λ be as in Lemma 3.7 with where m := max{t 0 , t − |z − z 0 |}, α := 2 4−q λ ∞ , and β := Proof.We only give the proof in the "+"-case and for (z 0 , t 0 ) = (0, 0).We revisit the proof of Lemma 3.7 where Φ + is defined.From (13) we know that v z satisfies We denote the term on the right-hand side by T (v z )(z, t) and already know that T is Lipschitz continuous with constant q < 1. Therefore we may write the solution as v z := lim n→∞ T n (0) and thus have to study v (n) z := T n (0).The claimed inequality for u z will follow once we have shown that vanishes for |z| ≥ t then also v z ) vanishes on this set.So in the following we may assume |z| < t.We will only consider z ≥ 0 as z < 0 can be treated similarly.For z ≥ 0 and t > z the expression m = max{t − |z|, 0} simplifies to m = t − z.We begin by estimating the terms which are independent of v The remaining two summands are treated by Summing up all four estimates, we obtain It remains to verify C 1 ≤ α and C 2 ≤ β.In fact, using t, z ≤ r, we obtain where the equalities hold by definition of α and β, respectively.
4. Proof of Theorem 1.1 In this section, we will prove the existence and uniqueness part of the main Theorem 1.1 under the additional assumption that f grows at least linearly, i.e., for some A, B > 0 we have In Section 5 we will show how to remove this assumption.The wellposedness part of Theorem 1.1 will be completed in Section 6.
We will again use that the wave equation has finite speed of propagation so that we may argue locally.To be more specific, we will work on the following types of triangular domains: • A jump triangle is a triangle ∆ = ∆(z 0 , 0, r) with base B = P z ∆ ⊆ (0, ∞), where z 0 ∈ D(c) and B intersects D(c) in no other point.These are useful for the study of the jump condition uz(z+,t) c(z+) = uz(z−,t) c(z−) .• A boundary triangle is a half-triangle ∆ + = ∆ + (0, 0, r) with base B + = P z ∆ + = [0, r] where B + does not intersect D(c).These are used to study the nonlinear Neumann condition uz c(0) = (f (u t )) t .• A plain triangle is a triangle ∆ = ∆(z 0 , 0, r) with base B = P z ∆ ⊆ (0, ∞) not intersecting D(c).These are used to cover the remaining space.
Proof.This follows immediately from Corollary 3.4 and Corollary 3.6.
holds for all t ∈ [0, r].Using (18), we can write (19) as We denote the right-hand side by T (b)(t) and show now that where β is the constant from Lemma 3.9.If we choose µ > β, then Ψ is a strict contraction so that b = Ψ(b) has a unique solution by Banach's fixed-point theorem.Using Remark 3.8, the fixed-point theorem also shows that b linearly and continuously depends on u 0 and u 1 .Moreover, boundedness of the linear solution operator Φ then follows from (18).Proof.As in the previous lemma, we write b(t) = u(0, t), Then u is a solution on ∆ + if and only if u = Φ + (b, u 0 , u 1 ) and df (u t (0, t)) dt = u z (0, t) c(0) .
We may rewrite the latter equation as , where b can be reconstructed from d via b d (t) := u 0 (0) + t 0 f −1 (d(τ )) dτ we are left with solving Therefore, it suffices to show that (20) with initial datum d(0) = f (u 1 (0)) has a unique solution.
Uniqueness: Assume that d, d are solutions to (20) that coincide up to time t ⋆ ≥ 0, but not at time t n for some t n ≥ 0 with where β is the constant from Lemma 3.9.
Clearly, the claim holds true for t = t ⋆ , and thus by continuity for t close to t ⋆ .Assume the claim is false.Then there exists some minimal and hence On the other hand, setting b := b d − b d we have Combining these facts, we find which contradicts (21).So the claim holds.
Letting ε go to 0, we obtain for any t ≥ t ⋆ .Fubini implies that the term on the right-hand side is negative for t ∈ (t ⋆ , t ⋆ + 1 β ), a contradiction.
Existence: Let D, µ > 0. Consider the set which is a convex and compact subset of C([0, r]), as well as the operator We choose D := max{|f −1 (u 1 (0))|, 1}, so that K is nonempty as it contains the constant function To check that T maps into K, we need to verify that for any d ∈ K one has Therefore T maps K into itself if we choose Hence existence follows by applying Schauder's fixed-point Theorem.
With these auxiliary results finished, we are able to prove the main theorem.
Step 1 -Constructing a solution: Denote by C the set containing all jump, boundary and plain triangles where the heights r have to satisfy r cz c ∞ < 1.As we have just shown in the previous three lemmata, (10) admits a unique solution on each ∆ ∈ C. Since C is closed with respect to finite intersection, we obtain a solution u of (10) on By restriction, we therefore obtain a solution u (1) of ( 1) on [0, ∞) × [0, h] for any 0 < h < h.
If t inf = 0, this holds because both u and ũ satisfy the same initial conditions.If t inf > 0, by assumption we have u(z, t) = ũ(z, t) for z ∈ B ε and t < t inf as (z, t) ∈ ∆ and therefore also u t (z, t) = ũt (z, t), so that the claim is obtained by taking the limit t → t inf .
If we choose ε small enough, then ∆ ε is a jump (if z ∞ ∈ D(c)), boundary (if z ∞ = 0) or plain triangle (otherwise).By the previously established uniqueness results on these triangles, u and ũ must coincide on ∆ ε .But since t n ≥ t inf for all n, we have (z n , t n ) ∈ ∆ ε for n sufficiently large, so that u(z n , t n ) = ũ(z n , t n ).This cannot be since (z n , t n ) ∈ N. Remark 4.4 (Modifications for the bounded domain version).In order to capture the homogeneous Dirichlet boundary condition for the bounded domain version of the theorem, we also need to consider "Dirichlet" triangles ∆ − with center z 0 = L. Problem (1) is well-defined on the domain ∆ − assuming r cz c ∞ < 1.In fact the solution on "Dirichlet" triangles is simply given by u = Φ − (0, u 0 , u 1 ).We can then proceed as in the above proof to show existence and uniqueness of solutions.

Energy, Momentum, and Completion of Theorem 1.1
We recall that the energy of ( 1) is given by where We now show that both quantities are time-invariant.
Part 1: Energy.With ν being the outer normal at ∂Ω we calculate The sum ± over the boundary integrals can be simplified to The sum ± of the integrands in the integral over Ω vanishes as can be seen by the following calculation using once more the differential equation Since D(c) and D(c z ) are discrete sets, we find an increasing sequence 0 = a 1 < a 2 < a 3 < . . .
and sum (23) from k = 1 to K. As terms along common boundaries cancel, we obtain or equivalently The estimates established in Corollary 3.6 and the assumptions on the initial conditions u 0 , u 1 show that u t (z, t) and u z (z, t) converge to 0 as z → ∞ uniformly on [t 1 , t 2 ].In the limit K → ∞, we thus obtain Switching back to (x, t)-coordinates, we infer where the last equality is due to Lemma A.1.This shows the claimed energy conservation: .
Part 2: Momentum.We calculate Again we choose Ω = [a k , a k+1 ] × [t 1 , t 2 ], and sum (24) from k = 1 to K. As before all terms along common boundaries cancel, whence we obtain in the limit K → ∞ we find the claimed momentum conservation: .
In Section 4, we required an extra growth condition (A4) on f in order to prove a first version of Theorem 1.1.We now discuss how to exploit the energy conservation to eliminate this extra growth assumption and prove Theorem 1.1 in full generality.
Proof of Theorem 1.1 without additional assumption (A4).Fix T > 0 and let Since F (y) = y 0 f (y) − f (x) dx we see that F (y) → ∞ as y → ±∞.Therefore the set {y : F (y) ≤ C} is contained in the interval [−K, K] for some K > 0. Now consider the cut-off version of f given by which satisfies the growth conditions from Section 4. Therefore, Theorem 1.1 can be applied to (1) with f replaced by f K and we obtain a solution u K on [0, ∞) × [0, T ].Lemma 5.1 gives F K (u K,t (0, t)) ≤ C, so that u K,t (0, t) takes values in [−K, K] where the functions f, F and f k , F k coincide.Hence u K solves the original problem (1) up to time T .
Next, we verify that C 1 -solutions to (1) are indeed weak solutions in the sense of Definition 1.6.Proposition 5.2.A C 1 -solution to (1) is also a weak solution to (1).
Proof.Let u be a C 1 -solution to (1).We have to show that Let Ω ⊆ [0, ∞) × [0, ∞) be a Lipschitz domain such that c is C 1 on Ω. Denoting the outer normal at ∂Ω by ν, we obtain We next show that the sum of the last two integrals equals zero.First, we calculate Let γ : [0, l] → R be a positively oriented parametrization of ∂Ω by arc length.As ν is the outer normal at ∂Ω, the identity γ ′ = (ν 2 , −ν 1 ) ⊤ holds.Hence, as γ is closed.Thus we have shown We also consider a single boundary-type triangle ∆ + with center z 0 = 0 and height r.Writing b(t) := u(0, t), b (n) (t) := u (n) (0, t), d(t) := f (u t (0, t)) as well as d (n) (t) := f (u (n) t (0, t)), as in the proof of Lemma 4.3 we obtain where Lemma 3.9 gives Multiplying with sign d (n) (t) − d(t) and integrating, we obtain .
This shows the uniform convergence of d as n → ∞, and therefore we see that Combined, we find that that u ). Applying this result repeatedly k times, we see that u where k ∈ N is chosen such that k r 2 ≥ T .

Breather solutions and their regularity
One can also consider (1) in the context of breather solutions, where a breather is a timeperiodic and spatially localized function.With time-period denoted by T , the time domain becomes the torus T := R/ T and after dropping the initial data, (1) reads In [4] the case of a cubic boundary term f (y) = 1 2 γy 3 (γ ∈ R \ {0}) and a 2π-periodic step potential V : R → R given by where b > a > 0 and θ ∈ (0, 1) was discussed.It was shown that if V satisfies where ω := 2π T is the frequency, then there exist infinitely many weak breather solutions u of (26) with time-period T .A weak solution of (26) is defined next.Definition 7.1.Let f : R → R be an increasing, odd homeomorphism.A weak solution of (26 We require that the trace u(0, •) of u at x = 0 has an integrable weak first-order time derivative in order to give a pointwise meaning to u t (0, t) and, in particular, to define f (u t (0, t)) pointwise almost everywhere.
Note that in the setting of [4], the assumptions of Theorem 7.3 are satisfied with r = 1 3 .In the following, we are going to prove Theorem 7.3 and we will always assume the assumptions of Theorem 7.3.

Fourier decomposition of
We denote by e k (t) := 1 √ T e ikωt the orthonormal Fourier base of L 2 (T) and decompose u in its Fourier series with respect to t: , we see that any solution u of (26) satisfies 0 = Lu and therefore also each ûk is an H 1 ((0, ∞), C)-solution of (27).As V (and therefore also L k ) is given explicitly, we can characterize the space of solutions of (27) as follows.A proof of Proposition 7.4 for k odd can be found in [4,Appendix A2].The nonexistence result for even k can be obtained using similar arguments: For k = 0 the monodromy matrix for L k is the identity matrix so that (27) only has spatially periodic solutions.For k = 0, the solutions of (27) are affine.

We can apply
We next investigate the properties of the maps defined by (29) and (30), which we consider as maps between the fractional Sobolev-Slobodeckij spaces W s,p (T).The definition and all employed properties of the spaces W s,p (T) can be found in Appendix B. In the following we use the suffix "anti" to denote that the space consists of functions which are T 2 -antiperiodic in time.
there is nothing left to show for n = 0. Now assume that the estimate has been shown for some fixed n.Recalling the definition of G b from (15), we have G b even,z (z, t) = − sign(z)b ′ (max{t − |z|, 0}).Notice that G b even,z (z, t) vanishes for |z| ≥ t.Therefore, if v (n) z

Lemma 4 . 1 .
Let ∆ be a plain triangle with base B. Assume r cz c ∞ < 1.Then (10) has a unique C 1 -solution u on ∆ and there exists a constant C = C(r, cz c ∞ ) such that the solution operator

Lemma 4 . 2 .
Let ∆ be a jump triangle with base B. Assume r cz c ∞ < 1.Then (10) has a unique C 1 -solution u on ∆ and there exists a constant C = C(r, cz c ∞ ) such that the solution operator where µ > 0 will be chosen later.So let b, b ∈ X and write b := b − b.Next we estimate