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Singular standing-ring solutions of nonlinear partial differential equations

We present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d-dimensional sphere, where d>1. The asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one-dimensional equation that become singular at a point. We provide a detailed numerical investigation of these new singular solutions for the following equations: The nonlinear Schrodinger equation, the biharmonic nonlinear Schrodinger equation, the nonlinear heat equation and the nonlinear biharmonic heat equation.


Introduction
In this study, we consider nonlinear evolution equations of the form Examples for such equations are the nonlinear Schrödinger equation, the biharmonic nonlinear Schrödinger equation, the nonlinear heat equation, and the biharmonic nonlinear heat equation. It is well known that these equations admit solutions that become singular at a point. Recently, it was discovered that the nonlinear Schrödinger equation with a quintic nonlinearity admits solutions that become singular on a d-dimensional sphere [1,2,3,4], see Figure 1. Following [2], we refer to these solutions as singular standing-ring solutions.
The main goal of this study is to present a general framework for constructing singular standingring solutions of nonlinear evolution equations of the form (1). In order to understand the basic idea, let us assume that equation (1) admits a singular standing-ring solution. Then, near the singularity, equation (1) reduces to the one-dimensional equation u t (t, r) = F [u, u rr , u rrrr , · · · ], r = |x|.
Hence, equation (2) "should" admit a solution that becomes singular at a point. Conversely, if the one-dimensional equation (2) admits a solution that becomes singular at a point, then equation (1) "should" admit a standing-ring singular solution. Moreover, the asymptotic profile and blowup rate of the standing-ring solutions of (1) "should" be the same as those of the corresponding solution of the one-dimensional equation (2). Figure 1: Illustration of a two-dimensional singular standing-ring solution at times t 0 < t 1 < t 2 .
The insets show the radial profile of u.
The above argument is obviously very informal. Nevertheless, in what follows we will provide numerical evidence in support of the relation between standing-ring singular solutions of (1) and singular solutions of the one-dimensional equation (2). iψ t (t, x) + ∆ψ + |ψ| 2σ ψ = 0, ψ(0, x) = ψ 0 (x), where x ∈ R d and ∆ = ∂ x 1 x 1 + · · · + ∂ x d x d , is one of the canonical nonlinear equations in physics, arising in various fields such as nonlinear optics, plasma physics, Bose-Einstein condensates, and surface waves. One of the important properties of the NLS is that it admits solutions which become singular at a finite time, i.e., The NLS is called subcritical if σd < 2. In this case, all solutions exist globally. In contrast, solutions of the critical (σd = 2) and supercritical (σd > 2) NLS can become singular in finite time.
Until a few years ago, the only known singular NLS solutions were peak-type. In the critical case σd = 2, it has been rigorously shown [5] that peak-type solutions are self-similar near the singularity, i.e., ψ ∼ ψ R , where ψ R (t, r) = 1 the self-similar profile R(ρ) attains its global maximum at ρ = 0, and the blowup rate L(t) is given by the loglog law In the supercritical case (σd > 2), the rigorous theory is far less developed. However, formal calculations and numerical simulations [6] suggest that peak-type solutions of the supercritical NLS collapse with the self-similar ψ S profile, i.e., ψ ∼ ψ S , where |S(ρ)| attains its global maximum at ρ = 0, and the blowup rate is a square-root, i.e., In the last few years, new singular solutions of the NLS were discovered, which are ring-type [7,2,1,4,3]. In [2], we showed that the NLS with d > 1 and 2 d ≤ σ ≤ 2 admits singular ring-type solutions that collapse with the ψ Q profile, i.e., ψ ∼ ψ Q , where ψ Q (t, r) = 1 and The self-similar profile Q attains its global maximum at ρ = 0. Hence, r max (t) is the ring radius and L(t) is the ring width, see Figure 2.
The ψ Q ring solutions can be classified as follows, see Figure 3: A. In the subcritical case (σd < 2), all NLS solutions globally exist, hence no singular ring solutions exist. B. The critical case σd = 2 corresponds to α = 1. Since r max (t) = r 0 L(t), these solutions undergo an equal-rate collapse, i.e., the ring radius goes to zero at the same rate as L(t). The blowup rate of these critical ring solutions is a square root. C. The supercritical case 2/d < σ < 2 corresponds to 0 < α < 1. Therefore, the ring radius r max (t) = r 0 L α (t) decays to zero, but at a slower rate than L(t). The blowup rate of these ring solutions is . D. The supercritical case σ = 2 corresponds to α = 0. Since r max (t) ≡ r 0 , the solution becomes singular on the d-dimensional sphere |x| = r 0 , rather than at a point. The blowup rate of these solutions is given by the loglog law (5). E. The case σ > 2 was open until now.

Singular standing-ring solutions of the NLS
One of the goals of this paper is to study singular ring-type solutions for σ > 2 (case E). The most natural guess is that these solutions also blowup with the ψ Q profile. Since α < 0 for σ > 2, see (7c), ψ Q should be an expanding ring, i.e., lim t→Tc r max = ∞. In this study we show that although such expanding rings do not violate power conservation, ψ Q ring solutions cannot exist for σ > 2. Rather, singular rings solutions of the NLS (4) with σ > 2 are standing rings.
The blowup profile and rate of standing-ring solutions can be obtained using the following informal argument. In the ring region of a standing-ring, ψ rr ∼ 1 L 2 and 1 r ψ r ∼ 1 L·rmax . Therefore, the d−1 r ψ r term in equation (4) becomes negligible compared with ψ rr as t → T c . Hence, near the singularity, equation (4) reduces to the one-dimensional NLS 1 1 Throughout this paper, we denote the solution of the one-dimensional NLS by φ, and its spatial variable by x.  A: subcritical case. B: critical case, with equal-rate collapse [7]. C: 2/d < σ < 2, shrinking rings [2]. D: σ = 2, standing rings [2,1,4]. E: the case σ > 2, which is considered in this study.
Therefore, the blowup profile and blowup-rate of standing-ring solutions of the NLS (4) with d > 1 and σ ≥ 2 are the same as those of singular peak-type solutions of the one-dimensional NLS with the same σ. Specifically, standing-ring solutions are self-similar in the ring region, i.e., ψ ∼ ψ F , where ψ F is, up to a shift in r, the asymptotic peak-type profile of the one-dimensional NLS, i.e., and φ S is given by (6a) with d = 1. In addition, the blowup rate L(t) of ψ F is the same as the blowup rate of φ S , see (6c), and is equal to a square-root, i.e., Moreover, κ(σ) is universal, i.e., it depends on σ, but not on the dimension d or the initial condition ψ 0 . Numerically, we observe that ψ F is an attractor for a large class of radially-symmetric initial conditions, but is unstable with respect to symmetry-breaking perturbations.

Singular standing vortex solutions of the NLS
The two-dimensional NLS iψ t (t, x, y) + ∆ψ + |ψ| 2σ ψ = 0, ψ(0, x, y) = ψ 0 (x, y), admits vortex solutions of the form ψ(t, x, y) = A(t, r)e imθ , where m ∈ Z. In [3], we presented a systematic study of singular vortex solutions. In particular, we showed that there exist singular vortex solutions that collapse with the asymptotic profile ψ Q · e imθ , when ψ Q is given by (7). The blowup rates of these solutions are the same as those of ψ Q in the non-vortex case. Therefore, the ψ Q · e imθ vortex solutions can be classified as follows: A. In the subcritical case (σ < 1), all NLS solutions globally exist, hence no singular vortex solutions exist. B. In the critical case σ = 1, these solutions undergo an equal-rate collapse at a square root blowup rate. C. The supercritical case 1 < σ < 2 corresponds to 0 < α < 1. In this case, the ring radius decays to zero at a slower rate than L(t) and the blowup rate is given by (8) where p = 1 1+α = σ 2 . D. The supercritical case σ = 2 corresponds to α = 0. Therefore, the solution becomes singular on a circle. The blowup rate is given by the loglog law (5). E. The case σ > 2 was open until now.
In this study, we show numerically that there exist singular standing-vortex solutions of the two-dimensional NLS with σ > 2 (case E) with the asymptotic profile e imθ ψ F . Moreover, the blowup rate of these singular standing vortices is the same as that of the standing-ring solutions in the non-vortex case, i.e., is given by (9). Therefore, these results extend the ones obtained in the non-vortex case.

Singular solutions of the biharmonic nonlinear Schrödinger equation
Let us consider the focusing biharmonic nonlinear Schrödinger equation (BNLS) equation where ∆ 2 is the radial biharmonic operator. Here, singularly formation is defined as lim In the subcritical case σd < 4, all BNLS solutions exists globally [8]. Numerical simulations [8,9] indicate that in the critical case σ = 4/d and the supercritical case σ ≥ 4/d, the BNLS admits singular solutions. At present, however, there is no rigorous proof that the BNLS admits singular solutions, whether peak-type or ring-type. Peak-type singular solutions of the BNLS (10) were recently studied numerically in [9]. The blowup rate of these solutions is slightly faster than p = 1/4 in the critical case (1/4+loglog?), and is equal to p = 1/4 in the supercritical case.
The ψ Q B solutions can be classified as follows (see Figure 4): A. In the subcritical case (σd < 4), all BNLS solutions globally exist, hence no collapsing ring solutions exist.
Thus, up to the change σ −→ 2σ, this classification is completely analogous to that of singular ring solutions of the NLS (see Figure 3). In this work we show numerically that this analogy carries through to the regime σ > 4. Thus, the BNLS with σ > 4 and d > 1 admits singular standing-ring solutions. Near the standing-ring peak, equation (10) reduces to the one-dimensional BNLS Therefore, the blowup profile and blowup-rate of standing ring solutions of the BNLS (10) with d > 1 and σ ≥ 4 are the same as those of collapsing peak solutions of the one-dimensional BNLS with the same value of σ. Thus, the results for ring solutions of the BNLS with σ > 4 are completely analogous, up to the change σ −→ 2σ, to those for singular standing-ring solutions of the NLS with σ > 2.

Singular solutions of the nonlinear heat equation and the biharmonic nonlinear heat equation
The d-dimensional nonlinear heat equation (NLHE) and the d-dimensional nonlinear biharmonic heat equation (NLBHE) admit singular solutions for any σ > 0 [10,11]. To the best of our knowledge, until now all known singular solutions of (12) and of (13) collapse at a point (or at a finite number of points [12]). In this study, we provide numerical evidence that the NLHE (12) and the NLBHE (13) admit singular standing-ring solutions. The blowup profile and blowup rate of these solutions are the same as those of singular peak-type solutions of the corresponding one-dimensional equation.

Critical exponents of singular ring solutions
In Figure 5A we plot the blowup rate parameter p of singular ring solutions of the NLS, see (8). As σ increases from 2/d to 2−, p increases monotonically from 1/2 to 1−. When σ = 2, the blowup rate is given by the loglog law (5), i.e., p = 1/2 with a loglog correction. Finally, p = 1/2 for σ > 2.
Since lim the blowup rate has a discontinuity at σ = 2. Surprisingly, the blowup rate is not monotonicallyincreasing with σ. For example, a ring solution of the NLS with σ = 1.8 blows up faster than a ring solution of the NLS with σ = 2.2.
The above results show that the critical exponent of singular ring solutions of the NLS is σ = 2: The blowup rate is discontinuous at σ = 2, and the blowup dynamics changes from a shrinkingring (σ < 2) to a standing-ring (σ ≥ 2), see Figure 5C. We can understand why σ = 2 is a critical exponent using the following argument. Standing-ring solutions are 'equivalent' to singular peak solutions of the one-dimensional NLS with the same nonlinearity exponent σ. Since σ = 2 is the critical exponent for singularity formation in the one-dimensional NLS, it is also a critical exponent for standing-ring blowup.
In the case of the NLHE and BNLHE equations, there is no critical exponent of singular ring solutions. Indeed, these equations admit standing-ring solution for any σ > 0, precisely because there is no critical exponent for singularity formation in the corresponding one-dimensional equations. For σ ≥ 4, α B ≡ 0 (standing rings).

Paper outline
The paper is organized as follows. In Section 2 we review the theory of singular peak-type solutions of the supercritical NLS, and conduct a numerical study of the one-dimensional case. In Section 3.1 we prove that standing-ring blowup can only occur for σ ≥ 2, and show that the blowup profile and blowup-rate of singular standing ring solutions of the supercritical NLS with σ > 2 and d > 1 are the same as those of peak-type solutions of the one-dimensional NLS equation. In Section 3.2 we confirm these results numerically. We then show numerically that the singular standing-ring profile ψ F is an attractor for radially-symmetric initial conditions (Section 3.3), but it is unstable with respect to symmetry-breaking perturbations (Section 3.4). In Section 4 we show analytically and numerically that expanding ψ Q ring solutions do not exist for σ > 2. Section 5 extends the results to singular vortex solutions. In Section 6 we study singular peak-type solutions of the one-dimensional supercritical BNLS. In Section 7 we show that singular ring solutions of the supercritical BNLS with σ > 4 are standing-rings, whose blowup profile and blowup-rate are the same as those of peak-type solutions of the one-dimensional BNLS. In Section 8 we show that singular standing-ring solutions of the nonlinear heat equation exist for any σ > 0, and that their blowup profile and blowup-rate are the same as those of peak-type solutions of the one-dimensional NLHE. In Section 9 we show that singular standing-ring solutions of the nonlinear biharmonic heat equation exist for any σ > 0, and that their blowup profile and blowup-rate are the same as those of peak-type solutions of the one-dimensional BNLHE. The numerical methods used in this study are briefly described in Section 10.

Theory review
Let us consider the one-dimensional supercritical NLS In contrast to the extensive theory on singularity formation in the critical NLS, much less is known about the supercritical case. Previous numerical simulations and formal calculations (see, e.g., [6,Chapter 7] and the references therein) suggested that peak-type singular solutions of the supercritical NLS (14) collapse with a self-similar asymptotic profile φ S , .i.e., φ ∼ φ S , where The blowup rate L(t) of these solutions is a square root, i.e., where κ > 0. In addition, the self-similar profile S is the solution of In general, solutions of (17) are complex-valued, and depend on the parameter κ and on the initial condition S(0). Solutions of (17) whose amplitude |S| is monotonically-decreasing in ξ, and which  (17), as function of σ.
have a zero Hamiltonian, are called admissible solutions [6]. For each σ, equation (17) has a unique admissible solution (up to a multiplication by a constant phase e iα ). This solution is attained for specific real values of κ and S(0), which we denote as Moreover, numerical simulations and formal calculations suggest that: 1. The self-similar profile of singular peak-type solutions of the NLS (14) is an admissible solution of (17). 2. The constant κ of the blowup rate (16) is universal (i.e., is independent of the initial condition ψ 0 ), and is equal to κ S (σ).

Simulations
To the best of our knowledge, the theory of supercritical peak-type collapse which is presented in Section 2.1, was tested numerically only for d ≥ 2. Since this theory is not rigorous, and since we will make use of these results in Sections 3 and 4, we now confirm numerically the above theoretical predictions for the one-dimensional supercritical NLS (14).
We verified numerically that for each σ, there exists a unique admissible solution of (17). The corresponding values of κ S and S 0 as a function of σ are shown in Figure 6. For example, for σ = 3, the parameters of the admissible solution of equation (17) are We now solve the one-dimensional NLS (14) with σ = 3 and the Gaussian initial condition φ 0 (t = 0, x) = 2e −2x 2 . We first show that the NLS solution collapses with the self-similar profile φ S , see (15). To do that, we rescale the solution according to (20) Next, we consider the blowup rate of φ. To do that, we first assume that and find the best fitting κ and p, see Figure 7B. In this case κ ≈ 1.713 and p ≈ 0.5007, indicating that the blowup rate is square-root or slightly faster. In order to check whether L is slightly faster than a square root, we compute the limit lim t→Tc LL t , see [7]. Recall that for the square-root blowup rate (16), while for a faster-than-a-square root blowup rate LL t goes to zero. Since lim Tc→t LL t = −1.384, see Figure 7C, the blowup rate of φ is square-root (with no loglog correction), i.e., In particular, there is an excellent match (to 4 digits) between κ blowup−rate 1D ≈ 1.664 extracted from the blowup rate of φ, see (22) and the parameter κ S (σ = 3) of the admissible S(ξ; σ = 3) profile, see (19).

Singular standing-ring solutions of the supercritical NLS
Let us consider singular solutions of the supercritical NLS In this Section we show that equation (23) has singular standing-ring solutions for σ ≥ 2. Since the case σ = 2 was already studied in [2,1,4], we mainly focus on the case σ > 2.

Analysis
The following Lemma shows that standing-ring blowup can only occur for σ ≥ 2: and Then, σ ≥ 2.
Proof. The power of the collapsing part ψ F is be the amount of power that collapses into the standing-ring singularity. We say that a singular standing-ring solution ψ undergoes a strong collapse if P collapse > 0, and a weak collapse if P collapse = 0.
Corollary 2. Under the conditions of Lemma 1, ψ F undergoes a strong collapse when σ = 2, and a weak collapse when σ > 2.
Proof. This follows directly from the proof of Lemma 1.
Let us further consider singular standing-ring solutions of the NLS (23). Following the analysis in [2, section 3.2], in the ring region of a standing-ring solution, i.e., for r − r max = O(L), Therefore, the d−1 r ψ r term in equation (23) becomes negligible compared with ψ rr as t → T c .
Hence, near the singularity, equation (23) reduces to the one-dimensional supercritical NLS (14), i.e., where φ is a peak-type solution of the one-dimensional NLS (14). Therefore, we predicted in [2] that the blowup dynamics of standing ring solutions of the NLS (23) with d > 1 and σ = 2 is the same as the blowup dynamics of collapsing peak solutions of the one-dimensional critical NLS with σ = 2, as was indeed confirmed numerically in [2] and analytically in [1,4]. Similarly, we now predict that the blowup dynamics of standing-ring solutions of the NLS (23) with d > 1 and σ > 2 is the same as the blowup dynamics of collapsing peak solutions of the supercritical one-dimensional NLS (14) with the same nonlinearity exponent σ: is given by (24a). 2. The self-similar profile ψ F is given by where φ S (t, x), see (15), is the asymptotic peak-type profile of the one-dimensional NLS (14) with the same σ. In particular, F = S is the admissible solution of equation (17) with 3. The blowup rate of ψ is a square root, i.e., where κ S (σ),is the parameter κ = κ S of the self-similar profile S, see (18).
Conjecture 3 implies that the parameter κ of the blowup-rate (26) of ψ is equal to the parameter κ of the blowup rate (16) of φ. In particular, κ depends on the nonlinearity exponent σ, but is independent of the dimension d and of the initial condition ψ 0 .

Simulations
We solve the NLS (23) with d = 2 and σ = 3 for the initial condition and observe that the solution blows-up with a ring profile. In Figure 8 we plot the ring radius r max (t) = arg max r |ψ| as a function of the focusing factor 1/L(t), as the solution blows up over 10 orders of magnitude. Since lim t→Tc r max (t) = 5.0011, the ring is standing and is not shrinking or expanding. We now test Conjecture 3 numerically item by item. 1. In Figure 7A we plot the rescaled solution at 1/L = 10 4 and 1/L = 10 8 , and observe that the two lines are indistinguishable. Therefore, we conclude that the standing-ring solutions blowup with the self-similar ψ F profile (24a). 2. To verify that the self-similar blowup profile ψ F is, up to a shift in r, the asymptotic blowup peak-profile φ S of the one-dimensional NLS (14), we superimpose in Figure 9A the self-similar profile of the solution of the one-dimensional NLS (14) from Figure 7A and the admissible solution S(x, σ = 3), and observe that, indeed, the four curves are indistinguishable. 3. Figure 9B shows that Therefore, the blowup rate is a square-root or slightly faster. Figure 9C shows that lim Tc→t LL t ≈ −1.385, indicating that the blowup rate is square-root (with no loglog correction), i.e., Thus, there is an excellent match between the parameter κ = κ S (σ = 3) ≈ 1.664 of the admissible S profile, see (19), the value of κ blowup−rate 2D ≈ 1.664 extracted from the blowup rate of the solution of the two-dimensional NLS, and the value of κ blowup−rate 1D ≈ 1.664 extracted from the blowup rate of the solution of the one-dimensional NLS, see (22).

Robustness of ψ F and universality of κ
The initial condition (27) in Figures 8 and 9 is different from the asymptotic profile ψ F . Since the solution ψ blows up with the asymptotic profile ψ F , this indicates that ψ F is an attractor. The initial condition (27), however, is already ring-shaped. Therefore, we now show that initial conditions which are not ring-shaped can also blowup with the ψ F profile. In [13,14], we developed a nonlinear Geometrical Optics (NGO) method which showed that high-power super-Gaussian initial conditions evolve into a ring profile. To see this, in Figure 10 we solve the NLS (3) with d = 2 and σ = 3, and the super-Gaussian initial condition ψ 0 (r) = 2e −r 4 , and observe that the NLS solution, indeed, evolves into a ring. Since lim t→Tc r max (t) = 0.33 , see Figure 11A, this singular solution is a standing ring. Therefore, we see that initial conditions which are not rings can also blowup with the ψ F standing ring profile.
We now consider the blowup rate of the above solution, since lim t→Tc LL t = −1.384 , see Figure 11B, this implies that This value of κ blowup−rate 2D identifies with the one obtained for the ring-type initial condition (27), see (29). We thus see that the parameter κ of the blowup rate (26) is, indeed, independent of the initial condition.

Instability with respect to symmetry-breaking perturbations
In Section 3.3 we saw that the standing-ring asymptotic profile ψ F is an attractor for a large class of radially-symmetric initial conditions. In general, NLS solutions with a ring structure are stable under radial perturbation, but unstable under symmetry-breaking perturbations [7,2,3]. We now show that ψ F is also unstable with respect to symmetry-breaking perturbations. To see that, let us consider the two-dimensional NLS iψ t (t, r, θ) + ψ rr + 1 r ψ r + 1 r 2 ψ θθ + |ψ| 2σ ψ = 0,  with the initial condition ψ 0 (r, θ) = f (r) (1 + εh(θ)) .
We chose f (r) so that when ε = 0, the solution blows up with the ψ F profile at r = r max . We now consider the case 0 < ε ≪ 1. Since for a standing ring the 1 r ψ r term becomes negligible compared with ψ rr , see Section 3.1, equation (30) can be approximated in the ring-peak region (r ≈ r max ) with iψ t + ψ rr + 1 r 2 max ψ θθ + |ψ| 2σ ψ = 0. This is the two-dimensional focusing NLS. Therefore, the solution will localize at local maximum points in the (r, θ) plane, thereby breaking the radial symmetry.
To see this numerically, we solve the two-dimensional NLS (30) with σ = 3 and with the initial condition This initial condition is the standing-ring initial condition (27), with an O(0.01) small bump at θ = 0 , see Figure 12A. As predicted, as the solution self-focuses, it localizes around the small initial bump at θ = 0 (see Figure 12B and 12C), resulting in breakup of radial symmetry.

Existence of non-standing ring solutions for σ > 2 ?
Lemma 1 does not exclude the possibility that there exist non-standing rings for σ > 2. The main reason that this question arises is as follows. In [2] we discovered ring solutions of the supercritical NLS for d > 1 and 2 d < σ ≤ 2 of the form ψ ∼ ψ Q , see (7). Therefore, it is natural to attempt to extrapolate these results to the regime σ > 2. Since α < 0 when σ > 2, the ring radius r max (t) goes to infinity as t → T c , hence ψ Q is an expanding-ring profile for σ > 2, if such a solution exists. Note that although the ring is expanding to an infinite radius, the power of the collapsing part ψ Q remains bounded, as Therefore, these expanding rings, if they exist, do not violate power conservation.
In [2] we solved the NLS (23) with σ = 2.1 > 2 and d = 2 and the ring initial condition ψ 0 = 4 √ 3 sech(2(r − 5)) . The solution turned out to be a singular standing ring, rather than an expanding one. Moreover, the blowup profile was different from ψ Q . In retrospect, this NLS solution was a standing-ring with the ψ F profile, see Section 3. Nevertheless, this still leaves open the question of whether there exist expanding-ring ψ Q solutions for σ > 2.

Analysis
We now prove that singular ring solutions with the ψ Q profile do not exist for σ > 2.
Lemma 4. When σ > 2, there are no singular NLS solution such that ψ ∼ ψ Q , see (7), and Proof. The result shall follow directly from Lemmas 5 and 6. Proof. We first recall that, as shown by Merle [15], for every singular solution ψ of the supercritical NLS To find the limiting behavior of ∇ψ(t) 2 2 as t → T c , note that by the conservation of the Hamiltonian In addition, where in the last equality we used the value of α given by (7c), and, in particular, that α < 0. Therefore, by (32), (34), (35) Hence, the bound (33) implies that p < 1.
Proof. Substitution of ψ Q , see (7), into the NLS (23) gives the following ODE for Q, Since Q depends only on ρ, each of the time-dependent terms of (36) should go to a constant as t → T c . In particular, C(t) should go the constant as t → T c . Under assumption (32), Since lim t→Tc L(t) = 0, then p > 0, see (32). In addition, for σ > 2, α < 0, see (7c). This implies that c < 0 and in particular c = 0. Since C(t) should goes to a constant as t → T c then, by (37), 2αp + 2p − 2 ≥ 0 . Therefore, 1 − p p ≤ α < 0.

Simulations
The result of Lemma 4 that expanding-ring singular solutions with the profile ψ Q do not exist, is based on formal arguments rather than on a rigorous analysis. Therefore, we now provide a numerical support for this result. To do that, we solve the NLS (23) with d = 2 and σ = 3 and with the expanding ring profile initial condition If a ψ Q solution indeed exists, then ψ would be a singular ring solution whose radius goes to infinity. In Figure 13A we plot the ring radius r max (t) as a function of the focusing factor 1/L, as the solution blows up over 10 orders of magnitude. Initially, the ring radius, indeed, expands from r max (0) = 10 to r max (t) ≈ 12.11. This expansion is due to the defocusing (expanding) phase term e −iαr 2 of the initial condition. However, the ring stops to expand when 1/L ≈ 20, and becomes a singular standing ring with radius r max (T c ) ≈ 12.11. Since the initial condition was an expanding ring, this simulation provides a strong support to the result of Lemma 4.
We now consider the blowup rate of the above solution, Figure 13B shows that lim implying that This value of κ blowup−rate 2D identifies with the one obtained for a ψ F collapse, see (29). Therefore, this simulations provides an additional support to the robustness of ψ F and to the universality of κ (see Section 3.3).
In [3] we proved that if the initial condition is a radially-symmetric vortex, then the solution remains a vortex: Lemma 7. Let ψ be a solution of the NLS (39) with the initial condition ψ 0 (r, θ) = A 0 (r)e imθ . Then, ψ(t, r, θ) = A(t, r)e imθ , where A(t, r) is the solution of Note that the phase singularity at r = 0 implies that A(r = 0) = 0. Hence, all vortex solutions are ring-type solutions. Specifically, all the singular solutions of (41) are ring-typed and not peaktyped.
In [3] we showed by formal calculations and numerical simulations that equation (39) admits singular shrinking-vortex solutions for 1 ≤ σ < 2, and singular standing-vortex solutions for σ = 2. Moreover, we showed that the blowup rate and profile of the standing-vortex solutions is the same as in the two-dimensional non-vortex case. We now show that this is also true for σ > 2, namely, that the analysis conducted in Section 3.1 for non-vortex standing-ring collapse, applies also for singular standing-vortex solutions.

Simulations
We solve equation (41) with m = 1 and σ = 3, with the initial condition In Figure 14 we plot the ring radius r max (t) as a function of the focusing factor 1/L(t), as the solution blows up over 10 orders of magnitude. Since lim t→Tc r max (t) = 5.0011, the vortex is standing. We now test Conjecture 9 numerically item by item.
1. In Figure 15 we plot the rescaled solution, see equation (28), at 1/L = 10 4 and 1/L = 10 8 , and observe that, indeed, the standing ring solution undergoes a self-similar collapse with the profile (25). 2. To verify that the self-similar collapse profile is, up to a shift in r and multiplication by e imθ , the asymptotic collapse profile φ S of the one-dimensional NLS (14), we superimpose the rescaled solution of the one-dimensional NLS (14) from Figure 7A, as well as the admissible solution S(x, σ = 3), onto the rescaled solutions of Figure 15A and observe that, indeed, the four curves are indistinguishable. 3. Figure 15B shows that Therefore, the blowup rate is square root or slightly faster. Figure 15C shows that lim Tc→t LL t ≈ −1.384, indicating that the blowup rate is square-root, i.e., In addition, there is an excellent match between the parameter κ = κ S (σ = 3) ≈ 1.664 of the admissible S profile, see (19), and the value of κ blowup−rate 2D−vortex ≈ 1.664 extracted from the blowup rate of the two-dimensional vortex solution.

Singular peak-type solutions of the one-dimensional supercritical BNLS
In Section 2 we reviewed the theory of singular peak-type solutions of the one-dimensional NLS. In this section, we present the analogous findings for the one-dimensional BNLS. We will make use of these results in the study of singular standing-ring solutions of the BNLS in Section 7.

Analysis
Let us consider the one-dimensional supercritical focusing BNLS At present, there is no theory for singular peak-type solutions of equation (44). A recent numerical study [9] suggests that peak-type singular solutions of the supercritical BNLS (44) collapse with a self-similar asymptotic profile φ B , .i.e., φ ∼ φ B , where The blowup rate L(t) of these solutions is a quartic root, i.e., where κ B > 0. In addition, the self-similar profile B(ξ) is the solution of Note that the NLS analogue of the B(ξ) profile is not the S(ξ) profile of equation (17), but rather S(ξ) · e i L t 4L x 2 . The analogue of the quadratic phase term in BNLS theory is at present unknown. In general, symmetric solutions of (47) are complex-valued, and depend on the parameter κ B and on the initial conditions B(0) and B ′′ (0). We conjecture that, in analogy with the NLS, the following holds: 1. The nonlinear fourth-order ODE (47) has a unique 'admissible solution' with κ B = κ B (σ), B(0) = B 0 (σ), and B ′′ (0) = B ′′ 0 (σ). 2. This admissible solution is the self-similar profile of the asymptotic peak-type blowup profile φ B , see (45). 3. The value of κ B of the blowup rate (46) is equal to κ B (σ) of the admissible B profile.

Simulations
We solve the one-dimensional BNLS (44) with σ = 6 and the Gaussian initial condition We first show that the BNLS solution blows up with the self-similar profile φ B , see (45). To do that, we rescale the solution according to , The rescaled profiles at focusing levels of 1/L = 10 4 and 1/L = 10 8 are indistinguishable, see Figure 16A, indicating that the solution is indeed self-similar while focusing over 4 orders of magnitude. Next, we consider the blowup rate of the collapsing solution, see Figure 16B. To do that, we first assume that and find the best fitting κ B and p. In this case κ B ≈ 1.020 and p ≈ 0.25017, indicating that the blowup-rate is close to a quartic root. To verify that the blowup rate is indeed p = 1/4, we compute the limit lim t→Tc L 3 L t . Note that for the quartic-root blowup rate (46) while for a faster-than-a-quartic root blowup rate L 3 L t → 0. Since lim Tc→t L 3 L t ≈ −0.2898, see Figure 16C, the blowup rate is a quartic-root (with no loglog correction), i.e.,

Singular standing-ring solutions of the supercritical BNLS
In Section 3 we analyzed singular standing-ring solutions of the NLS with σ > 2. In this section, we derive the analogous results for the biharmonic NLS with σ > 4.

Analysis
Let us consider singular solutions of the focusing supercritical BNLS where is the radial biharmonic operator. The following Lemma, which is the BNLS analogue of Lemma 1, shows that standing-ring collapse can only occur for σ ≥ 4: Lemma 11. Let ψ be a self-similar standing-ring singular solution of the BNLS (10), i.e., ψ ∼ ψ B , where and Then, σ ≥ 4.
Proof. Integration gives ψ B Proof. This follows directly from the proof of Lemma 11.
Let us further consider standing-ring solutions of the BNLS (51). In this case, in the ring region of a standing ring solution, the terms of the biharmonic operator, see (52), behave as Therefore, ∆ 2 r ψ ∼ ∂ 4 r ψ. Hence, near the singularity, equation (51) reduces to the one-dimensional BNLS (44), i.e., where φ is a peak-type solution of the one-dimensional BNLS (44). Therefore, we predicted in [9] and also confirmed numerically that 2. To verify that the self-similar collapse profile ψ B is, up to a shift in r, equal to the asymptotic collapse profile φ B of the one-dimensional BNLS (44), we superimpose the rescaled solution of the one-dimensional BNLS (44) from Figure 16A, onto the rescaled solutions of Figure 18A, and observe that, indeed, the curves are indistinguishable. 3. Figure 18B shows that Therefore, the blowup rate is quartic root or slightly faster. Figure 18C shows that lim ≈ 1.0376 extracted from the one-dimensional BNLS solution, see Section 6.2.

Singular standing-ring solutions of the nonlinear heat equation
The d-dimensional radially-symmetric nonlinear heat equation (NLHE) where u is real and ∆ = ∂ rr + d−1 r ∂ r , admits singular solutions for any σ > 0 [10]. To the best of our knowledge, until now all known singular solutions of (58) collapsed at a point (or at a finite number of points [12]). We now show that the NLHE admits also singular standing-ring solutions. The blowup profile and blowup rate of these solutions are the same as those of singular peak-type solutions of the one-dimensional NLHE with the same σ.
admits singular solutions that collapse with the self-similar peak-type profile v S (t, x) = 1 where see [16]. Note that unlike the one-dimensional NLS, the one-dimensional NLHE admits singular solutions for any σ > 0.

Analysis
Let us consider singular standing-ring solutions of the NLHE (58). Since ∆u ∼ u rr in the ring region, near the singularity equation (58) reduces to the one-dimensional NLHE (59). Therefore, we conjecture that singular standing-ring solutions of the NLHE (58) exist for any σ > 0, and that the blowup profile and blowup rate of these solutions are the same as those of singular peak-type solutions of the one-dimensional NLHE with the same σ.
and v S is given by equation (60).

Simulations
We solve the NLHE (58) with d = 2 and σ = 3 and the initial condition The solution blows up with a ring profile, see Figure 19. Since lim t→Tc r max (t) = 4.9994 > 0, see Figure 20A, the ring is standing.
In order to show that the solution blows up with the self-similar u S profile (61), we rescale the solution according to where λ(t) and L(t) are given by (60b) and T c is extracted from the numerical simulation. The rescaled profile at 1/L = 10 8 is in perfect agreement with the rescaled u S profile u S,rescaled = 1 see Figure 20B. Therefore, the numerical results provide a strong support to Conjecture 14.
9. Singular standing-ring solutions of the nonlinear biharmonic heat equation The d-dimensional radially-symmetric biharmonic nonlinear heat (BNLHE) equation where u is real and ∆ 2 is the radial biharmonic operator (52), admits singular solutions for any σ > 0 [11]. To the best of our knowledge, all known singular solutions of the BNLHE (65) collapse at a point. We now show that the BNLHE admits singular standing-ring solutions. The blowup profile and blowup rate of these solutions are the same as those of singular peak-type solutions of the one-dimensional BNLHE with the same σ.

Peak-type solutions of the one-dimensional BNLHE (review)
The one-dimensional BNLHE equation admits singular peak-type solutions which collapse with the self-similar peak-type profile v B (t, x) = 1 L 2/σ (t) B(ξ), see [11]. The self-similar profile B(ξ) is not known explicitly. Unlike the one-dimensional BNLS, the one-dimensional BNLHE admits singular solutions for any σ > 0.

Analysis
Let us consider singular standing-ring solutions of the BNLHE (65). Since ∆ 2 u ∼ u rrrr in the ring region, near the singularity the BNLHE (65) reduces to the one-dimensional BNLHE (66). We therefore conjecture that standing-ring solutions of equation (65) exist, and that their blowup profile and blowup rate is the same as those of singular peak solutions of equation (66): Conjecture 15. Let u(t, r) be a singular standing-ring solution of the BNLHE (65). Then, the solution is self-similar in the ring region, i.e.,u ∼ u B for r − r max = O(L), where u B (t, r) = v B (t, x = r − r max (t)) , and v B is given by equation (67).

Simulations
We solve the BNLHE (65) with d = 2 and σ = 3 and the initial condition (62). The solution blows up with a standing-ring profile, see Figure 21A. In Figure 21B we plot the solution, rescaled according to (57), at focusing levels of 1/L = 10 4 and 1/L = 10 8 . The two curves are indistinguishable, indicating that the solution is indeed self-similar. Next, we want to show that the self-similar blowup profile is given by B(ξ), the self-similar profile of peak-type solutions of the one-dimensional NLHE, see (67). To do that, we compute the solution of the one-dimensional BNLHE (66) with σ = 3 and u(t = 0, x) = 2e −x 2 , and superimpose its profile at 1/L = 10 8 in Figure 21B. The three rescaled solutions are indistinguishable, indicating that standing-ring solutions of the BNLHE blowup with the self-similar profile of peak-type solutions of the one-dimensional BNLHE. In addition, we have from the numerical simulations that lim t→Tc L 3 L t ≈ −1.2108 when d = 2, where L(t) := u −σ/2 ∞ , and lim t→Tc L 3 L t ≈ −1.2108 when d = 1, where L(t) := v −σ/2 ∞ . Therefore, the blowup rate of the standing-ring solution of the two-dimensional BNLHE is equal, up to 5 significant digits, to the blowup rate of the singular peak-type solution of the one-dimensional BNLHE, and is given by Thus, the numerical results provide a strong support for Conjecture 15.

Admissible solutions of (17)
The admissible solution S of (17) was calculated using the shooting method of Budd, Chen and Russel [17,Section 3.1]. In this method, one searches in the two-parameter space (S(0), f c ) for the parameters such that the solution of (17) satisfies the admissible solution condition

Solution of the NLS, BNLS, NLHE and BNLHE
In this study, we computed singular solutions of the NLS (4), the BNLS (51), the NLHE (58) and the BNLHE (65). Theses solutions become highly-localized, so that the spatial scale-difference between the singular region r − r max = O(L) and the exterior regions can be as large as O(1/L) ∼ 10 10 . In order to resolve the solution at both the singular and non-singular regions, we use an adaptive grid.
We generate the adaptive grids using the Static Grid Redistribution (SGR) method, which was first introduced by Ren and Wang [18], and was later simplified and improved by Gavish and Ditkowsky [19]. Using this approach, the solution is allowed to propagate (self-focus) until it starts to become under-resolved. At this stage, a new grid, with the same number of grid-points, is generated using De'Boors 'equidistribution principle', see [18,19] for details.
The method in [19] also allows control of the portion of grid points that migrate into the singular region, preventing under-resolution at the exterior regions. In [9], we further extended the approach to prevent under-resolution in the transition region O(L) ≪ r − r max ≪ O(1).
On the sequence of grids, the equations are solved using a Predictor-Corrector Crank-Nicholson scheme.