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Singular solutions of the L^2-supercritical biharmonic Nonlinear Schrodinger equation

We use asymptotic analysis and numerical simulations to study peak-type singular solutions of the supercritical biharmonic NLS. These solutions have a quartic-root blowup rate, and collapse with a quasi self-similar universal profile, which is a zero-Hamiltonian solution of a fourth-order nonlinear eigenvalue problem.


Introduction
The focusing nonlinear Schrödinger equation (NLS) iψ t (t, x) + ∆ψ + |ψ| 2σ ψ = 0, where x = (x 1 , . . . , x d ) ∈ R d and ∆ = d j=1 ∂ 2 j is the Laplacian, has been the subject of intense study, due to its role in various areas of physics, such as nonlinear optics and Bose-Einstein Condensates (BEC). It is well-known that the NLS (1) possesses solutions that become singular in a finite time [SS99].
In recent years, there has been a growing interest in extending NLS theory to the focusing biharmonic nonlinear Schrödinger equation (BNLS) iψ t (t, x) − ∆ 2 ψ + |ψ| 2σ ψ = 0, where ∆ 2 is the biharmonic operator. The BNLS (2) is called "L 2 -critical", or simply "critical" if σd = 4. In this case, equation (2) can be rewritten as iψ t (t, x) − ∆ 2 ψ + |ψ| 8/d ψ = 0, ψ(0, x) = ψ 0 (x) ∈ H 2 (R d ). (3) Correspondingly, the BNLS with 0 < σd < 4 is called subcritical, and the BNLS with σd > 4 is called supercritical. This is analogous to the NLS, where the critical case is σd = 2. BNLS solutions preserve the power (L 2 norm) P (t) ≡ P (0), P = ||ψ|| 2 2 , and the Hamiltonian In [BAKS00], Ben-Artzi, Koch and Saut proved that when σ is in the H 2 -subcritical regime the BNLS (2) is locally well-posed in H 2 . Global existence and scattering of BNLS solutions in the H 2 -critical case σ = 4/(d − 4) were studied by Miao, Xu and Zhao [MXZ09] and by Pausader [Pau09b]. The latter work also showed well-posedness for small data. The H 2 -critical defocusing BNLS was studied by Miao, Xu and Zhao [MXZ08] and by Pausader [Pau07,Pau09a]. The above studies focused on non-singular solutions. In this work, we study singular solutions of the BNLS in H 2 , i.e., solutions that exist in H 2 (R d ) over some finite time interval t ∈ [0, T c ), but for which lim t→Tc ψ H 2 = ∞. The first study of singular BNLS solutions was done by Fibich, Ilan and Papanicolau [FIP02], who proved the following results: The simulations in [FIP02] suggested that there exist singular solutions for σd = 4 and σd > 4, and that these singularities are of the blowup type, namely, the solution becomes infinitely localized. However, in contradistinction with NLS theory, there is currently no rigorous proof that solutions of the BNLS can become singular in either the critical or the supercritical case. Most subsequent research of singular BNLS solutions focused on the critical case. Chae, Hong and Lee [CHL11], showed that radial singular solutions of (3) have a power-concentration property. In [BFM10b], we showed that radial singular solutions are quasi self-similar. We also proved, without assuming radial symmetry, that the blowup rate is bound by a quartic-root, the powerconcentration property and the existence of the ground-state of (5). The two latter properties were also proved by Zhu, Zhang and Yang [ZZY10]. In [BFM10b], we also provided informal analysis and numerical evidence that peak-type singular solutions of the critical BNLS collapse with a quasi self-similar R B profile at a blowup rate which is slightly faster than the quartic-root bound.
In this work, we use asymptotic analysis and numerics to find and characterize peak-type singular solutions of the supercritical BNLS. We find that their properties mirror those of the supercritical NLS. Ring-type singular solutions of the supercritical BNLS were studied in [BFG10,BFM10a].

Summary of results
We analyze singular solutions of the focusing L 2 -supercritical and H 2 -subcritical BNLS, i.e., when We assume radial symmetry, i.e., that ψ = ψ(t, r), where r = |x|. In this case, equation (2) reduces to where is the radial biharmonic operator. The paper is organized as follows. In Section 2, we show that the supercritical BNLS admits the explicit self-similar singular solutions where the blowup rate of L(t) is exactly a quartic-root WKB analysis of the large-ρ behavior of S B shows that it belongs to L 2+2σ , but not to L 2 . Since lim t→Tc ψ S B 2+2σ = ∞, ψ S B is a singular solution in L 2+2σ , but not in H 2 . To the best of our knowledge, this is the first time that explicit singular solutions of the BNLS are presented. In Section 2.1 we show that the zero-Hamiltonian solutions of (9) satisfy the boundary condition In analogy with the supercritical NLS, we conjecture that for any d, σ and ν, there is unique admissible solution S B admis. (ρ), which has a zero Hamiltonian and is monotonically decreasing. This solution is attained for a unique κ = κ admis. (σ, d, ν) > 0. While a rigorous existence proof for the S B profile remains open, we provide numerical support for the existence of the admissible solutions.
In Section 3 we consider H 2 singular solutions. Using informal asymptotic analysis and the analogy with the supercritical NLS, we conjecture that these solutions undergo a quasi self-similar collapse with the ψ S B profile, where S B is the unique admissible solution S B admis. . The blowup rate of these solutions is given by L(t) ∼ κ admis. (T c − t) 1/4 . These characteristics are confirmed numerically, in simulations of both the one-dimensional and the two-dimensional BNLS.
The numerical simulations of the BNLS were performed using the IGR/SGR method [RW00,DG09], see [BFM10a] for further details. The numerical solution of the nonlinear fourth-order ODE for S B is obtained using a modified Petviashvili (SLSR) method, which is described in the appendix. The code is available online at http://www.math.tau.ac.il/∼fibich/publications.html The results of this study are based on asymptotic analysis and numerical simulations, but not on rigorous analysis. These results show that there is a striking analogy between collapse of peak type solutions in the supercritical NLS and the supercritical BNLS. We note that the rigorous theory for singular solutions of the supercritical NLS is much less developed than that for the critical NLS. Indeed, a rigorous proof of the blowup rate and blowup profile of the supercritical NLS was obtained very recently, and only in the slightly-supercritical regime 0 < σd − 2 ≪ 1 [MRS09]. We hope that this study will motivate a similar rigorous treatment of the supercritical BNLS.

Explicit singular solutions
Let us look for explicit self-similar solutions of the supercritical BNLS (2). Since the BNLS is invariant under the dilation symmetry r → r L , t → t L 4 , ψ → 1 L 2/σ ψ, where L is a constant, this suggests a self-similar solution of the form . (10) Since S B is only a function of ρ, equation (11) must be independent of t. Therefore, there exists a real constant κ such that Likewise, since S B is only a function of ρ, then τ ′ (t)L 4 (t) ≡ ν. Hence, Substituting (12) and (13) in (11) shows that the equation for S B is Since S B is radially-symmetric and decays at infinity, it should satisfy the boundary conditions Equation (14) has the two parameters ν and κ. Note, however, that whereS B (ρ;κ) is the solution of (14) with ν = 1, i.e., Equation (16) can be viewed as a nonlinear eigenvalue problem with the eigenvalue κ and eigenfunction S B . By analogy with the supercritical NLS [KL95,Bud01], we make the following conjecture: Conjecture 3. Let σ be in the L 2 supercritical and H 2 -subcritical regime (6). Then, there exists a solution S B (ρ),κ to equation (16), such thatS B ≡ 0 andκ > 0.
Hence, we have the following result: Lemma 4. Assume that Conjecture 3 holds, and let S B (ρ; κ, ν) be a nontrivial solution of (14). Then, is an explicit solution of the BNLS equation (2).
As ρ → ∞, the nonlinear term in (14) becomes negligible, and (14a) reduces to where We now use WKB to find the large ρ behavior of (18): Lemma 5. Let S lin B (ρ) be a solution of (18). Then, Proof. In order to apply the WKB method, we substitute S lin B (ρ) = exp(w(ρ)), and expand Substituting w 0 (ρ) = αρ p and balancing terms shows that p = 4/3, and that the equation for the leading-order, the O ρ 4/3 terms, is Therefore, The equation for the next order, the O (1) terms, is implying that The next-order terms are O ρ −4/3 = o(1) and can be neglected. We therefore obtain the three solutions S B,2 , S B,3 , and S B,4 . Since (18) is a fourth order ODE, another solution is required. To obtain the fourth solution, we substitute w 0 ∼ β log(ρ) in (18) and obtain that the equation for the leading-order, the O(1) terms, is and that the next-order terms are O ρ −4 = o(1) and can be neglected. The fourth solution is therefore S B,1 .
Equation (18) thus has the two algebraically-decaying solutions, S B,1 and S B,2 , the exponentiallyincreasing solution S B,3 , and the exponentially-decreasing solution S B,4 . The fact that S B,3 increases exponentially as ρ → ∞ is inconsistent with the boundary condition (14b). Therefore, Since σd > 4, the exponent 2 3σ (σd − 1) of S B,2 is larger that the exponent 2 σ of S B,1 , hence .
Direct calculations give that Hence, Lemma 6. Assume that Conjecture 3 holds. Then, ψ S B is an explicit solution of the BNLS equation (2) that becomes singular in L 2+2σ as t → T c .

Zero-Hamiltonian solutions
As is the case of peak-type solutions of the supercritical NLS, a key role is played by the zero-Hamiltonian solutions.
Theorem 7. Let σ be in the L 2 supercritical and H 2 -subcritical regime (6). Let S B be a solution of (14).
Proof. The exponentially increasing solution c 3 S B,3 must vanish, as explained above. Convergence of the Hamiltonian requires that ∆S B ∈ L 2 . Since ∆S B,2 ∈ L 2 , see (21), it follows that c 2 = 0. Since S B,1 ∈ L 2 , then S B ∈ L 2 . To show that (23) is equivalent to demanding that c 2 = 0, we first note that in the H 2 -subcritical regime d − 4 < 4 σ and so γ 0 < 2 3 2 + 2 σ < γ 1 . Next, direct calculation gives that and that where the LHS is the result of the next term in the WKB approximation of S B,1 . Therefore, Since, for γ 0 < γ < γ 1 , it follows that c 2 = 0 if and only if the limit (23a) is satisfied.
The fourth-order nonlinear ODE (14a) requires four boundary conditions. Three boundary conditions are given by (14b), and the fourth condition will be the zero-Hamiltonian condition (23a). Generically, one can expect that for a given ν, this nonlinear eigenvalue problem has an enumerable number of eigenvalues κ (n) with corresponding eigenfunctions S B (n) . As in the case of the supercritical NLS [KL95,Bud01], we conjecture that for any (σ, d, ν) there is a unique admissible solution, which is monotonically decreasing.
Conjecture 9. Let σ be in the L 2 supercritical and H 2 -subcritical regime (6), and let ν > 0. Then, the nonlinear eigenvalue problem posed by equation (14a), subject to the boundary conditions where γ satisfies ( 3 Peak-type H 2 -singular solutions

Informal analysis
As in the supercritical NLS [LPSS88, SKL93], we expect that singular peak-type solutions of the supercritical BNLS undergo a quasi self-similar collapse, so that where ψ S B is the self-similar profile (10). The singular region r ∈ [0, r c ] is constant in the coordinate r. Therefore, in the rescaled variable ρ = r/L(t), the singular region ρ ∈ [0, r c /L(t)] becomes infinite as L(t) → 0. This is in contradistinction with the critical-BNLS case, where the singular region ρ ∈ [0, ρ c ] is constant in the rescaled variable ρ, but shrinks to a point in the original coordinate r [BFM10b].
From Hamiltonian conservation it follows that H [ψ S B ] is bounded, because otherwise the nonsingular region would also have an infinite Hamiltonian. Therefore, from Theorem 7 it follows that H [S B ] = 0.
In Lemma 8 we saw that the zero-Hamiltonian solutions of (14a) are in L 2+2σ , but not in L 2 . Hence, ψ S B ∈ L 2 . From power conservation, however, it follows that if ψ 0 ∈ H 2 , then ψ ∈ L 2 . As in the NLS case, see [BP92], this "contradiction" can be resolved as follows.
3. In particular, S B (ρ) = R B (ρ). 4. The blowup rate of singular peak-type solutions is exactly a quartic root, i.e., 5. The coefficient κ of the blowup rate of L(t) is equal to the value of κ of the admissible solution S B , i.e., In particular, κ is universal (i.e., it does not depend on the initial condition).
In Section 3.2 we provide numerical evidence in support of Conjecture 12.
In both cases, the solutions blowup at a finite time, see Figure 1.
To check whether the solutions collapse with the self-similar profile (28), the solution was rescaled according to Comparing this rescaling with (28b) shows that it implies that |S B (0)| = S B ∞ = 1. This requirement can always be satisfied with a proper choice of ν, see (15). Figure 2A shows the rescaled solutions at the focusing levels L = 10 −4 and L = 10 −8 , and the rescaled solution S B (ρ) of (28).    Table 1: Position of circles in Figure 3.
The three curves are indistinguishable, showing that the solution is self-similar with the S B profile, and not with the R B profile. As additional evidence, Figure 3 shows that as ρ → ∞, the self-similar profile of ψ decays as ρ −2/σ , which is in agreement with the decay rate of S B,1 (ρ).
We next verify that the solution converges to the asymptotic profile for r ∈ [0, r c ], i.e., for ρ ∈ [0, r c /L(t)]. To do this, we plot in Figure 4 the rescaled solution at focusing levels of 1/L = 10, 100, 1000, 10000, as a function of log(r/L). The curves are indistinguishable at r/L = O(1), but bifurcate at increasing values of r/L. These "bifurcations positions" are marked by circles in Figure 4, and their r/L values are listed in Table 1. The "bifurcation positions" are linear in 1/L, indicating that the region where ψ ∼ ψ S B is indeed ρ ∈ [0, r c /L(t)], which corresponds to r ∈ [0, r c ].
In order to compute the blowup rate p, we performed a least-squares fit of log(L) with log(T c −t), see Figure 5. The resulting values are p ≈ 0.2502 in the d = 1, σ = 6 case, and p ≈ 0.2504 in the d = 2, σ = 3 case. Next, we provide two indications that the blowup rate is exactly 1/4, i.e., that L(t) ∼ κ 4 T c − t, κ > 0.
1. If the blowup rate is exactly a quartic root, then L 3 L t → − κ 4 4 < 0. Indeed, Figure 6A shows In the case d = 2, σ = 3, L 3 L t → −0.171, implying that Since L 3 L t converges to a finite, negative constant, this shows that the blowup rate is exactly 1/4.
2. According to Lemma 10, if lim t→Tc L 3 L t < 0, the self-similar profile S B (ρ) does not satisfy the standing-wave equation (26), but rather is a solution to the problem (28c), as is clearly demonstrated in Figure 2.
We recall that calculation of the profile S B requires knowing the numerical values for ν and κ, see Appendix A. The value of κ was obtained previously from the limit lim t→Tc (−4L 3 L t ) 1/4 . We approximate the value of the coefficient ν from Indeed, Figure 6B shows that in both cases L 4 (t) dτ dt quickly converge to ν numeric (d = 1, σ = 6) = 0.36187, ν numeric (d = 2, σ = 3) = 0.22826.
These values are within 1%-3% from the values of 1.037 and 0.909 we obtained directly from the BNLS simulations, see (30).
Finally, we verified that the value of κ in the blowup rate (28d) is universal. We solve the BNLS in the case d = 1, σ = 6 with the initial condition ψ 0 (x) = 2e −x 4 . In this case, the calculated value of κ(d = 1, σ = 6) is κ = lim t→Tc 4 −4L t L 3 ≈ 1.037, which is equal, to first 3 significant digits, to the previously obtained value, see (30a), for the initial condition ψ 0 (x) = 1.6e −x 2 . Similarly, in the case d = 2, σ = 3, we solve the equation with the initial condition ψ 0 (x) = 3e −x 4 . The calculated value of κ(d = 2, σ = 3) is κ = lim t→Tc A Numerical calculation of the S B profile In order to solve equation (28c), we first define its linear part, which is the fourth-order linear differential operator L [S] under the BCs, see equation (24), The nonlinear ODE (28c) is therefore rewritten as For given numerical values of ν and κ, we wish to calculate the ground state of the nonlinear boundary-value problem (34). In order to do so, we modify the SLSR method for the calculation of the ground-state of the NLS [Pet76, PS01, AM05] and BNLS [BFM10b] as follows. We consider the fixed-point iterative scheme for the solution of (34). In the standard application of the SLSR method, L is a differential operator of constant coefficients, and its inversion is easily performed using the Fourier transform. In our case, L is a variable-coefficient operator, and the Fourier Transform cannot be used. Therefore, we discretize the operator L using finite differences, see Appendix A.1, and invert it using the LU decomposition. We observe numerically that generically, the iterations (35) converge to zero for a small initial guess and diverge to infinity for a large initial guess. To avoid this divergence, we rescale the approximate solutions at each iteration, so that they satisfy the integral relation: which follows from multiplication of (35) by S. Here, < · > denotes the standard inner product f, g = f * gρ d−1 dρ. Following a similar argumentation as in [BFM10b], we obtain that the iterations are In our simulation, this method converged for every value of ν and κ that we tried. The numerical values of ν was obtained from the on-axis phase of the BNLS simulation solutions, as explained in Section 3.2. In order to obtain a prediction of κ, we recall that the specific choice (29) of the blowup rate L(t) implies that |S B (0)| = 1. We therefore use the SLSR solver to search for the value of κ for which |S B (0) = 1|.
A.1 Discretization of L Using half-integer grid the O h 2 centered-difference discretizations of the radial biharmonic operator D 2 ρ and of the firstderivative, the approximation at the interior nodes is The stencil is five-nodes wide, so two ghost-nodes are needed at each boundary. In order to enfold the ghost-nodes at ρ = 0, we relate them to the interior nodes, using the symmetry of the solution S(ρ) = S(−ρ), so that This relation is substituted in the discretization of the equation at ρ 0 and ρ 1 . At the other boundary ρ = R max we use the approximate form of the solution obtained from the WKB approximation, i.e., we require S n = c 1 S B,1 (ρ n ) + c 4 S B,4 (ρ n ), n = N − 2, N − 1, N, N + 1, . . . . Some care should be taken when choosing the parameters R max and N . On the one hand, we use the closed-form approximations for S B,1 and S B,4 that become more accurate for R max ≫ 1.

In matrix form, this becomes
On the other hand, since S B,4 has a super-exponentially decreasing term e −ρ 4/3 , choosing toolarge a value of R max leads to numerical instabilities. Finally, in order to resolve the rapidoscillations e iρ 4/3 of S B,4 , the grid-size N must be chosen such that ρ 1/3 h ρ = O R 4/3 max /N ≪ 1, hence that N ≥ O R 4/3 max . The grid-size N , however, cannot be arbitrarily large, since the condition number of L is O N 4 .
In the simulations presented in this study, we used an extension of above approach to a fourthorder approximation, and set R max = 160 and N = 32000.