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Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line

We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse Born approximation can be used to recover some essential information about the unknown coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients.

1. Introduction. We consider a quasi-linear differential operator of order four on the line defined by Q 4 u := u (4) + q 1 (x, |u|)u + q 0 (x, |u|)u, (1) where the complex-valued coefficients q 1 and q 0 are from function spaces defined later. These coefficients depend on the spacial variable x and they are also allowed to depend on the modulus of the function u. A linear counterpart of Q 4 , L 4 u := u (4) + q 1 (x)u + q 0 (x)u, has previously gained attention from several authors. In 1988 Iwasaki [4,5] studied the inverse problem of finding the unknown coefficients q 0 and q 1 as a Riemann-Hilbert boundary value problem. The coefficients q 0 and q 1 were assumed to be realvalued with an exponential type of decay at infinity. Under the assumptions that the operator has no spectral or non-spectral singularities and no negative eigenvalues he was able to provide a uniqueness theorem for the inverse problem when given the so-called reflection and connection coefficients R + and C + for all k = 0 in the rays arg(k) = 0 and arg(k) = π 4 , respectively. More recently, in 2008 Aktosun and Papanicolaou [1] studied the operator L 4 with time-evolving coefficients. In that work the time-dependence in several related scattering coefficients was discussed. This linear operator L 4 has also been studied in higher dimensions [11,13]. In [12] the operator L 4 was generalized by adding a second-order perturbation. The method of Born approximation was studied for this linear operator and it was proved that Born approximation can be used to recover the local singularities of the coefficients.
The motivation to study nonlinear operators of order four can be found, for instance, in theory of vibrations of beams and the study of elasticity. For example, by looking for the time-harmonic solutions U (x, t) = u(x)e −iωt to the nonlinear beam equation (cf. [7]) x U + mU + |U | p−1 U = 0, m > 0, we arrive at the equation where the left-hand-side is of type (1) with q 1 ≡ 0. Albeit real-valued, the following simple model of a suspension bridge from [2] also uses a fourth-order nonlinear equation to model the downward deflection u(x, t). The bridge is understood as a beam of length L with hinged ends. The deflection u(x, t) is subject to three forces with the homogeneous Navier boundary conditions x u(0, t) = ∂ 2 x u(L, t) = 0, where γ, k and W are constants called Young's modulus, the spring constant and forcing constant, respectively. The function f (x, t) is an external forcing term and u + = max{0, u}.
The present paper concerns scattering problems for the operator Q 4 and we study only one special set of solutions to the equation Q 4 u = k 4 u. More precisely, the direct scattering problem for Q 4 can be formulated by the equation where u sc must be outgoing in some sense. Instead of studying directly the equation (2) we apply the standard theory of ordinary differential equations to obtain an integral equation. For k > 0 the kernel of the integral operator (or a fundamental solution to operator L 0 := d 4 dx 4 − k 4 ) is given by This kernel is outgoing in the sense that it satisfies the radiation condition Note that due to (3) our operator inherits some Schrödinger-like properties and this allows us to use similar techniques as those used to study the nonlinear Schrödinger operator [8,9]. By convolving this fundamental solution formally with (2) we obtain the integral equation |u(y, k)|)u (y, k) + q 0 (y, |u(y, k)|)u(y, k))dy.
As it turns out (see Section 3), the solution to this integral equation has the asymptotic representation where a(k) and b(k) are called the transmission and reflection coefficients, respectively. They are defined by for sufficiently large k > 0. This asymptotic representation can now be regarded as a radiating solution to (4). It turns out that for our purposes it is enough to study only the reflection coefficient for sufficiently large k > 0.
In this paper we consider the inverse problem of recovering the potential functions q 1 and q 0 by the Born approximation method. The reflection coefficient b(k) is used to define the inverse Born approximation h B as the Fourier transform of ik 3 2 √ 2π b(k/2). We show that this approximation recovers the jumps and local singularities in the combination h(x) := − 1 2 q 1 (x, 1)+q 0 (x, 1), where the derivative is taken with respect to x. Our main result is that the difference h B −Re(h) defines a continuous function (if the imaginary part is smooth enough) and hence the singularities of h B coincide with those in h.
The following notations are used throughout this text. We use C > 0 as a generic constant when it is not important to keep track of its precise value. The symbol L p (Ω) is used to denote the p-based Lebesgue space over the set Ω ⊂ R. The space of continuous functions vanishing at infinity is denoted byĊ(R) and the Sobolev spaces W k p (R), for k ∈ N, are the spaces of those functions whose weak derivatives up to order k are in L p (R). In the case where p = 2 and s ∈ R we say that is finite. Here the Fourier transform pair of f is defined by the formulae This paper is organized as follows. In Section 2 we establish the existence of solutions u = u 0 + u sc to (4) with the property that u sc ∈ W 1 ∞ (R). In Section 3 we study the asymptotic behaviour of u at x → −∞ and define the reflection coefficient b(k). We also motivate the definition of the inverse Born approximation h B . Finally, Section 4 concerns the inverse problem of finding the jumps and singularities of h. We also present a numerical example to demonstrate the Born approximation visually.
2. Existence and uniqueness. In this section we prove that under quite general assumptions the integral equation (4) has a unique solution u, when k > 0 is large enough. In the linear case [12] a Neumann-type series could be used to construct the solution, but the nonlinearity disables this approach in the case of operator Q 4 . Instead, we apply Banach fixed point theorem to prove the existence and uniqueness of the direct scattering problem. Our first assumption 1 is similar to that in [9]. Assumption 2.1. Let us assume that the coefficients q j , j = 0, 1, have the following properties 1. There exists functions α j ∈ L 1 (R) such that for all a > 0 we find C j (a) > 0 with the property that |q j (x, s)| ≤ C j (a)α j (x), for all 0 ≤ s ≤ a. 2. The coefficients q j have the Lipschitz property in the second variable, that is, there exists β j ∈ L 1 (R) such that for all a > 0 we find C j (a) > 0 with the property that |q j (x, Suppose u is a solution to (4) and denote Then + q 0 (y, |u sc + u 0 |)(u sc + u 0 )) dy (6) and by the Leibniz integral rule y) (q 1 (y, |u sc + u 0 |)(u sc + u 0 ) + q 0 (y, |u sc + u 0 |)(u sc + u 0 )) dy. Now solving (6) for u sc is equivalent to solving (4). To do this, pick any ρ > 0 and consider the closed ball B ρ (0) : where · p denotes the usual L p -norm on the line.
Proof. Let us denote We will use Banach fixed-point theorem (see for example [14]) to prove this theorem. Since B ρ (0) is a closed subspace in the complete metric space W 1 ∞ (R), it suffices to show that T : B ρ (0) → B ρ (0) and that T is a contraction.
Let u ∈ B ρ (0). From Assumption 2.1 we get Using this fact we may estimate when k ≥ 1. Adding the above inequalities and taking the supremum yields and by Assumption 2.1 where Derivatives are estimated similarly, and we obtain the estimate and is a contraction when k ≥ k 0 . Remark 1. In the sequel we will let ρ > 0 denote a fixed constant, as in Theorem 2.2. In addition to the existence and the uniqueness of the solution, the proof of Banach's fixed-point theorem (see e.g., [14]) gives us an estimate in terms of the first iteration. Starting the iteration scheme from u (0) for all j = 0, 1, . . ., when k ≥ k 0 . The second inequality above follows from the definition of the operator T since we have 2k 2 and similarly for the derivatives.
3. Asymptotic behaviour of u at −∞. Having obtained a solution u to (4) with u sc ∈ W 1 ∞ (R), we study its asymptotic behaviour. Let us assume here that To motivate the definition of the inverse Born approximation in Section 4 we first formally define when k ≤ −k 0 . These functions are the unique solutions to Here the principal part of the above integral operator has the same behaviour as the Lippmann-Schwinger equation does.
Then as x → −∞ we find that u has the asymptotic representation Due to Corollary 1 it is reasonable to approximate u ≈ u 0 and u ≈ iku 0 for large values of k > 0. Correspondingly, Assumption 2.1 gives for both j = 0, 1 that |q j (y, |u sc + u 0 |) − q j (y, 1)| ≤ C j (ρ)β j (y)|u sc |, so, roughly speaking, q j (y, |u|) ≈ q j (y, 1) for large k > 0. By integrating by parts we can approximate for large k > 0.
4. Inverse problem. We are now ready to turn our attention to the inverse problem of finding the jump discontinuities and local singularities of the combination q 0 (x, 1)− 1 2 q 1 (x, 1). By using Fourier inversion and our formal equality b(k) = b(−k) for k < 0 on the approximation we are motivated to propose the following definition.
. for some C > 0 (depending on ρ and the coefficients) which is independent of k and j when k ≥ k 0 . Proof. We start by estimating q 1 (y, |u|)u − q 1 (y, |u j |)u j dy =: I 0 + I 1 .
By Assumption 2.1 we have In the same manner, by using Assumption 2.1 and Corollary 1 we can say that for some constant C > 0. Combining the estimates for I 0 and I 1 gives the claim.
Proof. By integrating by parts in the definition of b 0 (k/2), we obtain for k ≥ 2k 0 that Let χ [a,b] (k) denote the characteristic function of the interval [a, b]. By using Definition 4.1 we obtain as the Fourier transform of a compactly supported distribution. Furthermore, h ∈ L 1 (R) gives that F −1 (h) ∈ L ∞ (R) and χ [0,2k0] F −1 (h) ∈ L 1 (R). Invoking the Riemann-Lebesgue lemma shows that h must vanish at infinity. Next, recall that in the sense of distributions [6], where χ [0,∞[ is the Heaviside function, p.v. 1 x is the principal value distribution of 1 x and δ 0 is the delta distribution. The Fourier transform of the product Finally, calculating the above convolution by using (10) gives which simplifies to the claimed representation.
Proof. We consider only the left integral of (13), the right one can be analysed similarly. We start with large x > 0 and split the integral as Since Φ is integrable both of these terms vanish as x → +∞ uniformly in k ≥ k 0 . On the other hand, if x < 0 is large in absolute value, then we have directly that e −kx Only the integrability of Φ uniformly in k ≥ k 0 is used in the above proof. This means that we can let Φ depend on k ≥ k 0 as long as it is dominated by some integrable function which does not depend on k. Now we are ready to prove our main result, that is, the precise calculation of the second term of the inverse Born sequence. Heuristically, our aim is to show that most parts of b 1 (k/2) for k ≥ 2k 0 are O(k −5 ), which by (11) shows that the parts of h B,1 corresponding to them belong to H s (R) ⊂Ċ(R) for all 1 2 < s < 3 2 . Additionally, the use of Assumption 4.5 results in a linearization that yields precisely h B,0 and some additional terms that can be considered separately by using properties of the Fourier transform.
Proof. When k ≥ 2k 0 we can write where (see Remark 1) It is easy to see that Consider first the integral I 2 . By definition Here I 2 is estimated by for some C > 0. Then by using the explicit form of (u (1) sc ) we write Since |q 1 (y, |u 1 |)| ≤ C 1 (1 + ρ)α 1 (y) then the second integral H 2 = O(k −5 ). Next we apply (12) to expand where H 1 denotes the first integral and the rest are O(k −5 ). The integral H 1 contains some convenient symmetry. Let us denote Then Q k is (due to G k ) antisymmetric i.e. Q k (y, z) = −Q k (z, y) and is dominated by some function in L 1 (R × R) uniformly in k ≥ 2k 0 . This allows us to conclude that It remains to consider the integral I 1 . We use (12) to expand where J denotes the second integral of this expansion. As in the proof of Lemma 4.3 we have when k ≥ 2k 0 . To conclude that J yields a continuous term in h B,1 we split the integral into two parts Then J 1 can further be split as Next the inner integral in J 1 is divided into two parts with respect to the region of integration on ] − ∞, y] and ]y, ∞[ and then integrated by parts with respect to z to get where J 1 denotes the first integral and In the last step we used the fact that h 1 ∈ W 1 1 (R) ⊂ L 1 (R) ∩ L ∞ (R) to conclude that the last two integrals are O(k −5 ). Because Q ∈ L ∞ (R) then Assumption 4.5 about q * 1 (·, 1) gives that q * 1 (·, 1)Q ∈ L 1 (R) ∩ L p (R) and the Hausdorff-Young inequality (see, e.g., [3]) implies that k 4 J 1 (k) ∈ L r ([2k 0 , ∞[) for some 1 < r < ∞. Now Hölder's inequality gives that Next, dividing the inner integral in J 1 into two parts and then integrating by parts with respect to z gives The application of Lemma 4.6 in second and fourth integrals shows that all of the above integrals are O(k −5 ). The term J 2 can be calculated similarly. We first split As before, in J 2 we split the inner integral into two parts and integrate by parts with respect to z to get By the same argument as for J 1 we have that The term J 2 can be calculated similarly as J 1 to obtain another O(k −5 ) term.
To finish the proof we combine the results into the formula and Lemma 4.3 on the first term b 0 gives the claimed explicit representation and an additionalĊ-term. Since k 3 J 1 (k) and k 3 J 2 (k) are integrable outside the origin the application of the Riemann-Lebesgue lemma shows that the corresponding parts of h B,1 are continuous and vanish at infinity. Finally, using (11) to the O(k −5 )-part of b 1 completes the proof.
Proof. Since b(k) = b j (k) + O(1/k j+4 ), then Definition 4.1 and (11) imply for j = 1 that h B (x) = h B,1 (x) (mod H s (R)) for all s < 3 2 . Then Lemma 4.7 gives the claimed representation. The preceding theorem already gives us some information about the unknown coefficients q 0 and q 1 . The principal value integral appearing in Theorem 4.8 is precisely the Hilbert transform of the imaginary part of h. By using the mapping properties of the Hilbert transform (see, e.g., [10]) one can recover information about h even if the coefficients are complex-valued. If Im(h) has some regularity in the sense that Im(h) ∈ H r (R), r > 1 2 , we can use the Fourier transform and Sobolev embedding theorems to prove the following result, which justifies our claim about the solution to the inverse problem of recovering jumps and singularities of h. Remark 2. The nonlinearity is only present in this recovery at point 1. Suppose that q 1 (x, 1) = q 0 (x, 1) ≡ 0. Then u 0 (x, k) = e ikx solves equation (4) uniquely and reflection coefficient b(k) ≡ 0, that is, there is no reflection. This shows that, e.g. nonlinearities of type q(x, |u|) = f (x)(1 − |u| p ), p > 0, give trivial results regardless of f (x). Also scaling u 0 by a constant c > 0 and making appropriate changes to the assumptions through-out the text changes the reconstruction to correspond to the function h(x, c).
Numerical example. Finally, to illustrate the method we give a numerical example. We consider the operator Q 4 u = u (4) + q 1 |u| 2 u + q 0 |u| 2 1 + |u| 2 u with two nonlinearities, first-order cubic nonlinearity and zero-order saturation type nonlinearity, both of which have applications in physics. Here we choose the scatterers and note that q 1 ∈ W 1 1 (R), q 0 ∈ L 1 (R), where q 0 is pure imaginary when it is non-zero.
With these potentials we are able to calculate the required Hilbert transform in closed form, thus we expect to recover the (discontinuous and singular) combination The numerical computation is done by following [12]. Figure 1 presents the unknown coefficients with the red line indicating the real coefficient q 1 and the black dashed line indicating the imaginary part q 0 . Figure 2 depicts the numerically reconstructed combination of the coefficients in the red line compared with the actual unknown value (15) plotted in the black dashed line. This example shows that numerically we are able to recover the jumps and singularities in nonlinear complex coefficients quite accurately.
Conclusion. The direct and inverse scattering problems for the first-order quasilinear perturbation of the one-dimensional biharmonic operator in the frequency domain with singular coefficients (in the space coordinate) were considered. It is assumed that the nonlinearities depend on the modulus of the wave and that they are complex-valued. The linear case and many well-known (in physics) types of nonlinearities are included in the considerations. Under some additional regularity conditions (Lipschitz with certain Taylor-type expansions) for the nonlinearities the classical inverse scattering Born approximation is justified for this nonlinear operator of order 4. Note that in principle one could consider many types of data, namely (at least) the transmission, reflection and connection coefficients in both directions ±∞ on the line. For our purposes only part of this data alone is sufficient for recovering essential information about the unknown coefficients of our operator. More precisely, the Born approximation only requires the reflection coefficient (which corresponds to the asymptotic of the reflected wave at −∞) to be known for arbitrarily high frequencies. Under this limited data we proved the recovery of the jump discontinuities and infinite singularities of the coefficients in the space coordinate when the nonlinearity is at point 1. These results generalize the well-known results for the linear and nonlinear Schrödinger operator on the line. The considered method (Born approximation) also has natural generalization for the multi-dimensional case.