A uniqueness result for a simple superlinear eigenvalue problem

We study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein-Rutmann arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.


Introduction
Nonlinear analogues to linear eigenvalue problems arise in many branches of the sciences and often model coherent structures in spatially extended dynamical systems. Examples include traveling waves for Hamiltonian lattices (see below), breathers in discrete systems [FW98,Wei99], nonlocal aggregation models [BHW13,HO15], and chimera states of the Kuramoto equation [OMT08,Ome18], Assuming a one-dimensional continuum space variable, the simplest scalar equation involves a localized convolution kernel a and a superlinear function f with f (0) = 0. The problem is to find solution pairs (σ, u) consisting of an eigenvalue σ ∈ R and a nontrivial eigenfunction u : R → R which might be of periodic or homoclinic type. The existence of solutions to nonlinear eigenvalue problems such as (1) can be established by different approaches and this has already been done for special choices or certain classes of a and f . Prominent candidates are the Crandall-Rabinowitz theory [CR71], local bifurcation analysis [Ome13,ST19], and taylor made fixed point arguments [BHW13]. Further methods that have been successfully applied in the context of lattice waves include critical point techniques [Pan05], constrained optimization [FW94,FV99,Her11,HM20], and asymptotic methods [MA94,IJ05,Jam12].
The uniqueness problem, however, is notoriously difficult in the superlinear case and we are not aware of any such result except for local uniqueness proofs as given in [FP99,HM19b] for certain asymptotic regimes. In the sublinear case we refer to [Rab71], and for 1-homogeneous nonlinearities f there exists generalizations of the Krein-Rutmann theorem, see for instance [Mah07,Ara18]. In the scalar case, however, such functions f are necessarily linear.
In this paper we restrict our considerations to bilinear functions f which are piecewise linear with two slopes. This simplifying assumption allows us to transform (1) into the linear eigenvalue problem for a modified convolution operator with cut-off parameter ξ. We further suppose that both the kernel a and the eigenfunction u are nonnegative, even, and unimodal. This shape constraint provides an existence and uniqueness result for the linear substitute problem thanks to a variant of the Krein-Rutmann theorem. Moreover, it ensures that the nonlinear relation between σ and ξ is bijective.
Assumptions and Results As illustrated Figure 1, we consider functions f that are piecewise linear on the interval [0, ∞). More precisely, we set f (r) := ζ r for r ∈ [0, θ] , and assume that the three free parameters ζ, θ, η comply with the superlinearity condition It is also essential for our uniqueness result that the kernel a : R → R is integrable and belongs to the cone which contains all bounded L 2 -functions that are even, nonnegative, and unimodal. We suppose further that a is differentiable and strictly unimodal according to because this simplifies the presentation. However, we always discuss how more general results can be obtained by relaxing the precise formulation of theorems or enhancing the technical arguments in their proofs.
Assumption 1 (properties of the convolution kernel ). The convolution kernel a belongs to U and and satisfies additionally (4).
A prototypical example for Assumption 1 is the Gaussian Figure 2: Numerical profiles u σ (gray) and f (u σ ) (black) for several values of σ, the regular kernel (5), and nonlinearity parameters ζ = 0, θ = 0.6, η = 2.5. The gray box represents the interval I ξσ which is defined by u σ (±ξ σ ) = θ. Notice the different plot regions for top and bottom row. are compactly supported and violate (4). Both kernels are nonetheless interesting since they are naturally related to traveling waves in Fermi-Pasta-Ulam-Tsingou chains and certain discrete scalar conservation laws, see [HM19a,Her12] and references therein. For the kernel (6), the eigenvalue problem (1) with bilinear function (2) has already been solved in [TV14] by a combination of Fourier and numerical methods, see the more detailed comments at the end of §3.
Our findings for the nonlinearity (2) and kernels as in Assumption 1 can be summarized as follows.
Main Result 2 (existence and uniqueness under shape constraint).
1. There exists a unique one-parameter family of nonlinear eigenfunction u σ ∈ U parametrized by ζ < σ < ζ + η. The statement of some results must be modified in this case since the strict unimodality condition (4) is not satisfied.
2. Formal asymptotic expansions suggest the following limit behavior: (a) The eigenfunctions u σ converge as σ ζ to a nontrivial limit profile u ζ ∈ U with but the details are different for ζ = 0 and ζ > 0.
The rigorous part of our main result is proven in Proposition 7 while the nonrigorous asymptotic analysis is presented in the appendix. We also refer to Figures 2, 3, and 4 for numerical simulations with ζ = 0 and emphasize that our results do not exclude the existence of further eigenfunctions outside the cone U. In addition to the trivial nonuniqueness due to the shift invariance of (1), we expect the existence of families of periodic solutions. Moreover, for rapidly decaying kernels one might think about multi-bump solutions, which can be imagined as superposition of finitely many and well separated single-bump solutions having unimodal profile.
Proof strategy and organization of paper The central ideas for the proof of our main results can be sketched as follows. Assuming smoothness of u we deduce u ∈ N , where contains all odd functions that attain nonpositive values on (0, ∞) (usually, elements of −N and N are called positive and negative, respectively). Moreover, differentiating (1) we find with where ξ is the value at which u attains the value θ. This reads In the special case of ζ = 0, the necessary condition (8) reads and involves the effective derivative profile v = χ ξ · u (11) as well as Formula (10) implies the linear eigenvalue problem where is a compact and symmetric operator, which exhibits nice invariance properties. In particular, adapting the strong version of the Krein-Rutmann theorem we can show that the eigenspace to the largest eigenvalue of A ξ is simple and generated by an element of N , see Proposition 4. This observation yields in combination with (9) the existence and uniqueness of a family of solutions to (1) which is naturally parametrized by ξ and can be computed efficiently. Moreover, a closer inspection of the linear eigenvalue problem (13) as carried out in Proposition 5 reveals that the mapping ξ → λ → σ is invertible so that ξ can be replaced by σ. This change of parameters is the nonlinear part of our approach.
Another observation extends the existence and uniqueness result to the general case ζ > 0. More precisely, we identify in Proposition 6 a modified kernelã ∈ U with Rã (x) dx = 1 such that the implication holds with µ = σ −1 ζ. This enables us to transform the general case ζ > 0 into the special case ζ = 0 by replacing the kernel a withã and changing σ. A similar argument applies on the level of the derivatives because (8) can be written asλ u =ã * (χ ξ · u ).
2 Linear eigenvalue problem for the effective derivative profile In this section we study the eigenvalue problem of the operator A ξ from (14) in the space L 2 odd (R) which contains all functions that are square-integrable and odd. Notice that the representation formulas and hold for any v ∈ L 2 odd (R). Lemma 3 (properties of the modified convolution operator). For any ξ ∈ (0, ∞), the operator A ξ , maps L 2 (R) into itself, respects the even-odd parity, and is both compact and self-adjoint. Moreover, the convex cones N and U are invariant under the action of A ξ .
Proof. The first assertions can be derived from (14) using standard arguments for convolution and multiplication operators. Moreover, since a is even and unimodal, the invariance of N and U follows from (16) and the analogous formula in L 2 even (R).
Linear uniqueness result As first main auxiliary result we prove that the largest eigenvalue of A ξ is simple and spanned by a unique normalized eigenfunction in N . Such uniqueness results are usually inferred from the strong version of the Krein-Rutmann theorem but the classical formulation requires an order preserving operator that maps the cone of positive (or negative) elements in its interiors, see for instance [DL90, appendix to chapter IIX]. In our case, however, the cone N has no inner points and the compact support of χ ξ implies for every v ∈ N that A ξ v belongs to the topological L 2 -boundary of N . The assertions of the Krein-Rutmann Theorem hold notwithstanding. The crucial idea is that any eigenfunction that corresponds to the largest eigenvalue of A ξ must belong to the smaller cone and this observation finally enables us to adapt classical arguments. A similar strategy has been used in [BFH14], which combines a variant of the operator A ξ with nonlinear fixed point arguments to prove the existence (but not the uniqueness) of unimodal and compactly supported solutions to the equation where g is a given increasing monomial. This equation describes stationary solutions to a nonlinear biological aggregation model and the link to Krein-Rutmann arguments is that v = satisfies the linearized equation g ( ) v = a * v on the compact support of .
Proposition 4 (variant of the Krein-Rutmann theorem). For any 0 < ξ < ∞, the largest eigenvalue λ ξ of A ξ in L 2 odd (R) is simple and the corresponding eigenspace is spanned by a unique eigenfunction v ξ ∈ N ξ with v ξ 2 = 1.
Proof. Variational setting : The operator A ξ admits only real eigenvalues thanks to Lemma 3. It is also the Gâteaux derivative of the functional which is well-defined on L 2 odd (R) since Young's inequality implies Moreover, the largest eigenvalue λ ξ of A ξ can be characterized variationally via where any maximizer corresponds to a normalized eigenfunction to λ ξ and vice versa. Existence of eigenfunction in N : Any v ∈ L 2 odd (R) admits a unique and disjoint-support splitting and using Assumption 1, Lemma 3, as well as (14) we verify Now suppose that v ∈ L 2 odd (R) is a maximizer for (18) with v − = 0 and v + = 0. We then have while the homogeneity of F ξ along with (20) yields The combination of the two latter results with (19) and the normalization condition v 2 2 = 1 implies equality in both parts of (21). We conclude that v − and v + are eigenfunctions to the maximal eigenvalue λ ξ that belong to N .
Regularity by refined cone analysis : We next show that any eigenfunction v ∈ N to λ ξ belongs in fact to the smaller cone N ξ . The strict unimodality and the evenness of a imply Here we used λ ξ > 0, which holds because (16) and (22) provide Uniqueness of normalized eigenfunctions : We first show uniqueness within N as follows. Any two normalized eigenfunctions v 1 and v 2 to λ ξ lie in N ξ and the properties of this cone imply the existence of a parameter 0 < s < ∞ such that v 1 − s v 2 still belongs to N but no longer to N ξ (because of v 1 (0) = s v 2 (0) or v 1 (x) = s v 2 (x) for at least one 0 < x ≤ ξ). This, however, means s = 1 as well as v 1 = v 2 because otherwise v 1 − s v 2 would be nontrivial eigenfunction in N that violates the above regularity result. Secondly, suppose there exists a nontrivial and normalized eigenfunction v that belongs to neither N or −N . Our results derived so far imply that v − / v − and v + / v + are identically maximizers of (18) and hence a contradiction to the assumption on v. In summary, we have shown that the eigenspace to λ ξ in L 2 odd (R) is one-dimensional and spanned by a unique normalized function v ξ ∈ N ξ .
Parameter dependence The second building block for our nonlinar uniqueness result in §3 are the following properties of the Krein-Rutmann quantities. Related numerical simulations are presented in Figure 5.
Proposition 5 (ξ-dependence of λ ξ and v ξ ). The map ξ → λ ξ from Proposition 4 is continuous and strictly increasing with Moreover, the corresponding map ξ → v ξ is L 2 -continuous.
Proof. Monotonicity : Let 0 < ξ 1 < ξ 2 < ∞ be fixed. Since v ξ 1 from Proposition 4 is supported in I ξ 1 , we have and the optimization problem (18) ensures the nonstrict monotonicity λ ξ 1 ≤ λ ξ 2 . However, v ξ 1 cannot be an eigenfunction of A ξ 2 because the properties of a imply that A ξ 2 v ξ 1 has a larger support than v ξ 1 ∈ N ξ 1 . We thus obtain λ ξ 1 = λ ξ 2 . Continuity and asymptotics for small ξ : Suppose that (ξ n ) n∈N is a given sequence with By weak compactness we can extract a (not relabeled) subsequence such that the normalized eigenfunctions v ξn converge weakly to some limit v ∈ L 2 odd (R), where the weak closedness of N ensures v ∈ N . The properties of convolution operators combined with χ ξn → χ n imply that A ξn v ξn converges strongly to A ξ v and this guarantees -thanks to the eigenvalue equation for v ξn -the strong convergence of v ξn . In particular, we obtain and hence λ ≤ λ ξ . On the other hand, (18) ensures λ ξ ≤ λ since holds for anyṽ ∈ L 2 odd (R). We thus get λ = λ ξ and Proposition 4 combined with (23) yields v = v ξ . In summary, we have shown for any convergent sequence (ξ n ) n∈N that λ ξ is the unique accumulation point of the sequence λ ξn n∈N (which is bounded due to the monotonicity with respect to ξ), and this gives rise to the claimed continuity of λ ξ . Our arguments also yields the strong convergence v ξn → v ξ and analogously we derive ξ n → 0 from λ ξn → 0 because A ξn v ξn → 0 holds along any weakly convergent subsequence.
Asymptotics for large ξ : Direct computations for the piecewise constant and compactly supported test functionṽ ξ withṽ yield ṽ ξ 2 = 1 as well as  (22) is not satisfied for some values of ξ, the proof and/or the precise statement of Proposition 4 needs to be modified as discussed in the text. See also Figure 5 for the corresponding Krein-Rutmann eigenvalues. and arbitrary v 0 ∈ N . Moreover, using F ξ (− sgn |v|) ≥ F ξ (v) we deduce that λ ξ is not only the largest eigenvalue but also the spectral radius of A ξ and hence the exponential convergence of v n as n → ∞. This approximation scheme is often called Power Method and a straight forward discretization (fine but equidistant spatial grid and Riemann sums instead of integrals) has been used to produce the numerical results in this paper. Alternatively, one can write the sequence as where λ n will converge as n → ∞ to the Krein-Rutmann eigenvalue λ ξ . A similar improvement dynamics can also be applied to the nonlinear eigenvalue problem (1), but the convergence is more subtle due to the lack of uniquness results for general functions f . We refer to [HM20] for numerical examples and a more detailed discussion of the analytical properties.
Generalizations We finally discuss the case of less regular kernels a. The crucial ingredient to the regularity step v ∈ N ⇒ A ξ v ∈ N ξ in the proof of Proposition 4 is (22). This condition hinges on the strict unimodality of a as in (4) and is not satisfied for all a ∈ U, see Figure 6 for an illustration. For kernels like the tent map (6), a recursive argument reveals the implication v ∈ N ⇒ A n ξ v ∈ N ξ for all sufficiently large n ∈ N, and this guarantees that all assertions in Proposition 4 remain valid since the Krein-Rutmann eigenfunction still belongs toÑ ξ . This, however, is no longer true for the kernel (7). Instead, we have to distinghuish between the following three parameter regimes: 1. 0 < ξ ≤ 1 4 implies that A ξ v vanishes for any v ∈ L 2 odd (R), so A ξ restricted to L 2 odd (R) is actually the trivial operator and we have λ ξ = 0, see the third panel in Figure 5. 2. In the case of 1 4 < ξ < 1 2 , there exists a unique and normalized Krein-Rutmann eigenfunction v ξ , which does not belong to N ξ but to a modified subcone of N with v 1 2 − ξ < 0 , v(x) < 0 for 1 2 − ξ < x ≤ ξ , v(x) = 0 for 0 < x < 1 2 − ξ and x > ξ , see the first column in Figure 4.
In summary, if a is given by (7), then Proposition 4 must be formulated more carefully and the analogue to Proposition 5 stipulates the one-to-one correspondence between ξ ∈ ( 1 4 , ∞) and λ ξ ∈ (0, 1). Similar statements apply to any other kernel a ∈ U that admits a constant plateau near x = 0.

Nonlinear eigenfunctions with unimodal profile
We are now able to prove our main result from §1 in two steps. First we show that the general case with given kernel a and ζ > 0 is equivalent to the special case ζ = 0 for a modified kernelã.
Here, the transformed kernel is well-defined, depends on σ, and complies with Assumption 1, while the simplified nonlinearitỹ represents (2) with ζ = 0.
revealsā ∈ U thanks to Lemma 3, which in turn ensures that L respects the even-odd parity and mediates the implication (15). We further havẽ as well asã The latter formula provides viaã (x) < 0 for all x > 0 the strict unimodality ofã. Transformation of the nonlinear problem : Let (σ, u) be a given solution to (1). Our definitions implyσ and hence the validity of (24) thanks to the existence of L −1 . Similarly, the reverse implication follows by applying L to both sides of (24).
In the second step we finally establish our existence and uniqueness result for nonlinear eigenfunctions.
Uniqueness : Now suppose that 0 = u ∈ U solves (1) for ζ = 0. This gives 0 < u(0) = u ∞ (otherwise f (u) would vanish identically) and there exists ξ ∈ (0, ∞) with u(ξ) = θ. By differentiating (1) with respect to x -and using the notations (11), (12) -we establish the formulas (10) and (13)  Comments Notice that the transformed kernelã in (25) depends also on ζ and satisfies the analogue to the crucial condition (22) even if the kernel a is not strictly unimodal due to a compact support or constant plateaus, see Figure 7 for an illustration. Consequently, all results in this section cover for ζ > 0 the kernels (6) and (7) as well, and our comments at the end of §2 on how to generalize Propositions 4 and 5 are relevant for ζ = 0 only. The transformed kernelã also features prominently in [TV14], which studies the bilinear eigenvalue problem for the tent map kernel by different methods. In our notations, the main ideas can be described as follows. The functionw withw represents the inhomogeneity, is continuous (especially at x = ±ξ), and satisfies for all x ∈ R. The linear problem for the effective derivative profile (11) can hence be written analogously to (13) as and [TV14] solves this equation under the consistency relation ξ 0 v(x) dx = θ by combining two ingredients. The fixed point problem (29) is solved with parameter ξ by Wiener-Hopf methods. This requires detailed information on the Fourier transform ofã and direct computations reveal as well as a(k) k 2 = 4 sin 2 (k/2) for the kernel (6). Moreover, the value of ξ is determined numerically by means of a scalar nonlinear equation equivalent to (26). This approach works well since a is a nice known function and provides power series expressions for v and u, which can be used to derive intricate but almost explicit approximation formulas. We finally mention that nonnegative eigenfunctions (being unimodal or not) cannot exist for σ ≥ ζ + η since (1) implies where the last estimate is actually strict for any nontrivial u. Similarly, using we can disprove the existence of nonnegative eigenfunctions for σ ≤ ζ under the additional assumption u ∈ L 1 (R). Notice that the eigenfunction from Proposition 7 are integrable due to (24) and becausẽ f (u σ ) is always compactly supported. More precisely, u σ decays as fast as a or the transformed kernel a for ζ = 0 and ζ > 0, respectively. A similar argument has been applied in [HM20], which proves the existence and the localization (but not the uniquness) of nonlinear eigenfunctions for more general functions f in a nonlinear variational setting.

A Formal asymptotic analysis
We characterize the limiting behavior of the nonlinear eigenfunction u σ from Proposition 7. We always start with the special case ζ = 0 and discuss the necessary modification in the case ζ > 0 afterwards. The natural quantity for the formal asymptotic analysis with ζ = 0 are the compactly supported functions which determine u σ via We rescale w σ according to because this allows us to work on the fixed interval I := [−1, +1], and notice that Moreover, the nonlinear eigenvalue equation (1) combined with χ ξσ · u σ = θ + λ σ w σ and σ = η λ σ yields where the operator is the rescaled counterpart of A ξ from (14).
Small eigenvalues for ζ = 0 Proposition 5 shows ξ σ ≈ 0 and by Taylor expansion of (33) we get The eigenvalue equation (32) thus implies the approximate identity for some constant c σ and any x ∈ [−1, +1]. Computing the integrals we find  and equating the coefficients -first in front of x 2 and afterwards in front of 1 -we identify the scaling relations 3 a(0) c σ ξ σ ≈ θ , where λ σ satisfies (26). Moreover, the function approaches a Dirac distribution as σ → 0, see also Figure 8, so (31) implies that u σ converges to a multiple of a, where the scaling factor is consistent with u σ (ξ σ ) = θ. In summary, we expect the following behavior for small σ.
Conjecture 8 (asymptotics for ζ = 0 and σ ≈ 0). Suppose that a that is sufficiently smooth at the origin with a (0) < 0. Then, we have as well as where χ is the indicator function of I. Moreover, holds at least in the sense of pointwise convergence.
We believe that the assertions of Conjecture 8 can be derived by standard arguments but notice that (35) and (36) do not cover the kernels (6) and (7) since these are not smooth and satisfy a (0) = −∞ and a (0) = 0, respectively. Nonetheless, a modified asymptotic analysis should reveal that (37) is still satisfied. For the tent map kernel, a similar convergence result has been derived in [TV14] for ζ > 0 and η → ∞. This limit has much in common with the anticontinuum or high-energy limit studied in [HM19b].
Small eigenvalues for ζ > 0 In this case, we have to replace the kernel a by the transformed kernel from Proposition 6, which we now denote byã σ as it depends on σ, and σ byσ = σ − ζ. The problem is thatã σ does not converge strongly as σ ζ but its amplitude gets smaller while its effective width approaches ∞, see Figure 7 for an illustration. The precise analysis depends on the singular scaling behavior ofã σ . For sufficiently nice kernels we can suppose that the limits Figure 9: Numerical simulation with ζ = 1.0, θ = 0.6, η = 2.5 for the Gaussian kernel (5). The computations are performed with the nonlinear improvement dynamics described in [HM20] and the profiles u σ (gray) and f (u σ ) (black) are shown for several values of σ. Conjecture 9 predicts that u σ converges as σ ζ to the constant function with value θ, but this limit is hard to capture numerically. are well defined, and direct computations for the Gaussian kernel (5) provide the values κ 0 = 1 as well as κ 2 = 2 ζ(3/2)/ √ π in terms of the zeta function. In consistency with Proposition 6 we base our analysis of which solves an equation similar to (32). The scaling relations encoded by (38) still allow us to Taylor expandã σ even though the corresponding cut-off parameter ξ σ can no longer assumed to be small. In fact, repeating the arguments from above we find as well as Equating coefficients gives 2 3 κ 2 ξ 3 σ ≈ η −1 , 4 3 κ 0cσ ξ σ (σ − ζ) 1/2 ≈ θ as well as and in summary we find a limiting behavior that differs considerably from Conjecture 8. In particular, (σ − ζ) 1/2w σ converges to a piecewise smooth limit function with integral κ −1 0 θ, whose convolution with the small and slowly varying functionã σ approximates u σ up to a small prefactor. See also Figure 9.
Conjecture 9 (asymptotics for ζ > 0 and σ ≈ ζ). Suppose that the kernel a is sufficiently regular so that (38) is satisfied. Then, we have in the sense of pointwise convergence.
The different limit behaviors for ζ = 0 and ζ > 0 can also be understood heuristically as follows. For ζ = σ = 0, there exists a plethora of nonlinear eigenfunction in U, namely any function u ∈ U which satisfies f (u) ≡ 0 due to u ∞ = u(0) ≤ θ. In the case of ζ > 0, however, σ = ζ combined with 0 ≤ u ≤ θ reduces (1) to u = a * u, but the only fixed points of the convolution operator are the constant functions.
Large eigenvalues We start again with ζ = 0 but it turns out that our formal asymptotic results cover the case ζ > 0 as well. Since ξ σ is now large, we restate (33) as A ξ w (x) = ξ(x+1) ξ(x−1) a y w x + ξ −1 y dy and employ the formal asymptotic expansion ξ(x−1) a y w x + ξ −1 y dy .
We further emphasize that the scaling relations (41) are consistent with our numerical simulations in Figures 2 and 3. The analytical justification of Conjecture 10 and the underlying approximation (39) is much harder than the rigorous derivation of (35) and (36) because the Taylor expansion is now applied to w σ and requires uniform estimates of w σ p for at least one p ∈ [1, ∞]. A similar problem concerns the linear eigenvalue problem from §2. In order to ensure the consistent approximation v ξ (x) ≈ −(ξ) −1/2 χ ξ (x) sin ξ −1 πx for the normalized eigenfunction from Proposition 4, we have to guarantee that v ξ p ≤ C p ξ 1 p − 5 2 holds for all large ξ and some constant C p independent of ξ. We are, however, not ware of a corresponding reference. Moreover, the limit ξ → ∞ is rather intricate due to a huge number of nearby eigenvalues that discretize the continuous spectrum of the limit operator A ∞ w = a * w.