A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton

In this paper we prove that the only blowup solutions to the focusing, quintic nonlinear Schr{\"o}dinger equation with mass equal to the mass of the soliton are rescaled solitons or the pseudoconformal transformation of those solitons.


Introduction
The one dimensional, focusing, mass-critical nonlinear Schrödinger equation is given by This equation is a special case of the Hamiltonian equation (1.2) iu t + u xx + |u| p−1 u = 0, u(0, x) = u 0 (x), p > 1.
If u(t, x) is a solution to (1.2), then is a solution to (1.2) with appropriately rescaled initial data. Furthermore, The scaling symmetry in (1.3) controls the local well-posedness theory of (1.1). In that case, p = 5 and s p = 0.
Theorem 1. The initial value problem (1.1) is locally well-posed for any u 0 ∈ L 2 .
(2) If u 0 L 2 is small then (1.1) is globally well-posed, and the solution scatters both forward and backward in time. That is, there exist u − , u + ∈ L 2 (R) such that If u does not blow up forward in time, then sup(I) = +∞ and u scatters forward in time.
(5) Time reversal symmetry implies that the results corresponding to (3) and (4) also hold going backward in time.
Remark 1. It is very important to emphasize that throughout this paper, blow up in positive time may be in finite time or infinite time, unless specified otherwise. The same is true for blow up in negative time.
See [Tao06] for different notions of a solution.

and
(1.23) d 2 dt 2 x 2 |u(t, x)| 2 dx = 16E(u(t)), imply scattering for (1.1) with initial data in u 0 ∈ H 1 (R) ∩ Σ = {u : x 2 |u(x)| 2 dx < ∞}, u 0 L 2 < Q L 2 . More recently, [Dod15] and [Dod16b] proved that (1.1) is globally well-posed and scattering for any initial data u 0 ∈ L 2 , u 0 L 2 < Q L 2 . The proof used the concentration compactness result of [Ker06] and [TVZ08] which states that if u(t) is a blowup solution to (1.1) of minimal mass, and if t n is a sequence of times approaching sup(I), and if u blows up forward in time on the maximal interval of existence I, then u(t n , x) has a subsequence that converges in L 2 , up to the symmetries of (1.1). Using this fact, [Dod15] proved that if u is a minimal mass blowup solution to (1.1), then there exists a sequence t ′ n → sup(I), for which E(v n ) ց 0, where v n is a good approximation of u(t ′ n , x), acted on by appropriate symmetries. Since (1.21) implies that the only u with mass less than Q 2 L 2 and zero energy is u ≡ 0, and the small data scattering result implies that the zero solution is stable under small perturbations, there cannot exist a minimal mass blowup solution to (1.1) with mass less than Q 2 L 2 .
See [BLP81], [BL78], [Str77a], and [Kwo89] for existence and uniqueness of a ground state solution in general dimensions. Also observe that by the Pohozaev identity, Up to the scaling (1.3), multiplication by a modulus one constant, and translation in space, Q is the unique minimizer of the energy functional with mass Q L 2 . See [CL82] and [Wei86].
It is straightforward to verify that (1.20) solves (1.24), and that e it Q solves (1.1). Since e it Q L 6 is constant for all t ∈ R, e it Q blows up both forward and backward in time. Furthermore, the pseudoconformal transformation of e it Q(x), is a solution to (1.1) that blows up as t ց 0, and scatters as t → ∞. Note that the mass is preserved under the pseudoconformal transformation of e it Q.
It has long been conjecture that, up to symmetries of equation (1.1), the only non-scattering solutions to (1.1) are the soliton e it Q and the pseudoconformal transformation of the soliton, (1.26). Partial progress has been made in this direction.
Theorem 2. If u 0 ∈ H 1 , u 0 L 2 = Q L 2 and the solution u(t) to (1.1) blows up in finite time T > 0, then u(t, x) is equal to (1.26), up to symmetries of (1.1).
Proof. This result was proved in [Mer92] and [Mer93], and was proved for the focusing, mass-critical nonlinear Schrödinger equation in every dimension.
For the mass-critical nonlinear Schrödinger equation in higher dimensions with radially symmetric initial data, [KLVZ09] proved Theorem 3. If u 0 L 2 = Q L 2 is radially symmetric, and u is the solution to the focusing, mass-critical nonlinear Schrödinger equation with initial data u 0 , and u blows up both forward and backward in time, then u is equal to the soliton, up to symmetries of the mass-critical nonlinear Schrödinger equation.
In this paper we completely resolve this conjecture in one dimension, showing that the only blowup solutions to (1.1) with mass u 0 2 L 2 = Q 2 L 2 are the soliton and the pseudoconformal transformation of the soliton. This result should also hold in higher dimensions, which will be addressed in a forthcoming paper.
The proof of Theorem 4 will occupy most of the paper. Once we have proved Theorem 4, we will remove the symmetry assumption on u 0 , proving Theorem 5. The only solutions to (1.1) with mass u 0 L 2 = Q L 2 that blow up forward in time are the family of soliton solutions (1.29) e −iθ−itξ 2 0 e iλ 2 t e ixξ0 λ 1/2 Q(λ(x−2tξ 0 )+x 0 ), λ > 0, θ ∈ R, x 0 ∈ R, ξ 0 ∈ R, ξ 0 ∈ R, and the pseudoconformal transformation of the family of solitons, where Applying time reversal symmetry to (1.1), this theorem completely settles the question of qualitative behavior of solutions to (1.1) for initial data satisfying u 0 L 2 = Q L 2 .
The reader should see [NS11] and the references therein for this result for the Klein-Gordon equation.

Reductions of a symmetric blowup solution
Let u be a symmetric blowup solution to (1.1) with mass u 0 L 2 = Q L 2 . Defining the distance to the two dimensional manifold of symmetries acting on the soliton (1.20) by there exist λ 0 > 0 and γ 0 ∈ R where this infimum is attained. Indeed, Lemma 1. There exist λ 0 > 0 and γ 0 ∈ R such that Proof. Since Q, along with all its derivatives, is rapidly decreasing, , is differentiable, and hence continuous as a function of λ and γ.
Remark 2. This seems very unlikely to the author, since (2.11) is equivalent to the statement that there exists u 0 L 2 = Q L 2 that satisfies Since it is unnecessary to the proof of Theorem 4 to show that (2.12) cannot happen, this question will remain unconsidered in this paper.
Using the weak sequential convergence result of [Fan18], Theorem 4 may be reduced to considering solutions that blow up in positive time for which (2.1) is small for all t > 0.
Theorem 6. Let 0 < η * ≪ 1 be a small, fixed constant to be defined later. If u is a symmetric solution to (1.1) on the maximal interval of existence I ⊂ R, u 0 L 2 = Q L 2 , u blows up forward in time, and then u is a soliton solution of the form (1.27) or a pseudoconformal transformation of a soliton of the form (1.28).
Remark 3. Scaling symmetries imply that (2.1) and the left hand side of (2.16) at a fixed time are equal.
Proof that Theorem 6 implies Theorem 4. Let u(t) be the solution to (1.1) with symmetric initial data u 0 that satisfies u 0 L 2 = Q L 2 . If (2.17) lim then (2.16) holds for all t > t 0 , for some t 0 ∈ I. After translating in time so that t 0 = 0, Theorem 6 easily implies Theorem 4 in this case. However, the convergence theorem of [Fan18] only implies u(t) must converge to Q along a subsequence after rescaling and multiplying by a complex number of modulus one.
Theorem 7. Let u be a symmetric solution to (1.1) that satisfies u 0 L 2 = Q L 2 and blows up forward in time. Let (T − (u), T + (u)) be the maximal lifespan of the solution u. Then there exists a sequence t n → T + (u) and a family of parameters λ n > 0, γ n ∈ R such that If (2.17) does not hold, but there exists some t 0 > 0 such that for every n. After passing to a subsequence, suppose that for every n, t − n < t n < t − n+1 , where t n is the sequence in (2.18) and t − n is the sequence in (2.20). The fact that (2.21) inf γ∈R,λ>0 is upper semicontinuous as a function of t, and is continuous for every t such that (2.21) is small guarantees that there exists a small, fixed 0 < η * ≪ 1 such that the sequence t + n , defined by, satisfies t + n ր sup(I) and (2.23) inf λ>0,γ∈R Indeed, the fact that (2.21) is upper semicontinuous as a function of t implies that is a closed set. Since this set is also contained in a bounded set, it has a maximal element t + n , and t + n ≥ t − n . The fact that (2.21) is upper semicontinuous in time also implies that (2.25) inf λ>0,γ∈R On the other hand, since Remark 4. The constant 0 < η * ≪ 1 will be chosen to be a small fixed quantity that is sufficiently small to satisfy the hypotheses of Theorem 9, sufficiently small such that (2.21) is continuous in time when (2.21) is bounded by η * , sufficiently small such that η * ≤ η 0 , where η 0 is the constant in the induction on frequency arguments in Theorem 12, and so that T * = 1 η * is sufficiently large to satisfy the hypotheses of Theorem 16.
Theorem 8 (Upper semicontinuity of the distance to a soliton). The quantity is upper semicontinuous as a function of time for any t ∈ I, where I is the maximal interval of existence for u. The quantity (2.28) is also continuous in time when (2.28) is small.
Proof. Choose some t 0 ∈ I and suppose without loss of generality that For t close to t 0 , let Since e i(t−t0) Q solves (1.1), Equations (2.30), (2.31), and Strichartz estimates imply that for J ⊂ R, t 0 ∈ J, t,x (J×R) . Local well-posedness of (1.1) combined with Strichartz estimates implies that u L 4 t L ∞ x (J×R) = 1 on some open neighborhood J of t 0 . Therefore, for ǫ(t 0 ) L 2 small, partitioning J into finitely many pieces, Therefore, Then there exists a sequence t ′ For t ′ n sufficiently close to t 0 , repeating the arguments giving (2.33) and (2.34) with t ′ n as the initial data gives a contradiction.
Making a profile decomposition of u(t + n , x), the fact that u is a minimal mass blowup solution that blows up forward in time and t + n ր sup(I) implies that there exist λ(t + n ) > 0 and γ(t + n ) ∈ R such that Moreover, observe that (2.18) and (2.33) directly imply that for all t ∈ [0, sup(Ĩ)), whereĨ is the interval of existence of the solutionũ to (1.1) with initial datã u 0 , andũ blows up both forward and backward in time. However, Theorem 6 and (2.41) imply thatũ must be of the form (1.28). Such a solution scatters backward in time, which contradicts the fact thatũ blows up both forward and backward in time. Therefore, Theorem 6 implies that (2.20) cannot hold for any symmetric solution to (1.1) with mass u 0 L 2 = Q L 2 , so by Theorem 6, any symmetric solution to (1.1) that blows up forward in time must be of the form (1.27) or (1.28).

Decomposition of the solution near Q
Turning now to the proof of Theorem 6, make a decomposition of a symmetric solution close to Q, up to rescaling and multiplication by a modulus one constant. This result is classical, see for example [MM02], although here there is an additional technical complication due to the fact that u need not lie in H 1 .
Theorem 9. Take u ∈ L 2 . There exists α > 0 sufficiently small such that if there exist λ 0 > 0, γ 0 ∈ R that satisfy Furthermore, Remark 5. Since e iγ is 2π-periodic, the γ in (3.3) is unique up to translations by 2πk for some integer k.
Therefore, in Theorem 6, there exist functions Proof. Suppose J = [a, b] is an interval that satisfies x (J×R) ≤ 1, and J ⊂ [0, sup(I)). Suppose without loss of generality that λ(a) = 1 and γ(a) = 0. Also, suppose for now that u(a) Ḣ1 < ∞. Strichartz estimates and local well-posedness theory imply that Since λ(a) = 1 and γ(a) = 0, Then, by direct calculation and the fact that Q is smooth and rapidly decreasing, iQ 3 ) L 2 by an identical calculation. Then by the proof of Theorem 9, λ(t) and γ(t) are Lipschitz as a function of time for t close to a, and by the Lebesgue differentiation theorem, λ and γ are differentiable almost everywhere for t near a.
Following [Mer01], the decomposition in Theorem 9 gives a positivity result.

A long time Strichartz estimate
Having shown that it is enough to consider solutions to (1.1) that are close to the family of solitons, and that there is a good decomposition of solutions that are close to the family of solitons, the next task is to obtain a good frequency localized Morawetz estimate. The proof of the frequency localized Morawetz estimate will occupy sections four, five, and six.
The proof of scattering in [Dod15] for (1.1) when u 0 L 2 < Q L 2 utilized a frequency localized Morawetz estimate. There, the Morawetz estimate was used to show that E(P n u(t n )) → 0 along a subsequence, where P n is a Fourier truncation operator that converges to the identity in the strong L 2 -operator topology. Then the Gagliardo-Nirenberg inequality, (1.19), and the stability of the zero solution to (1.1) implies that u ≡ 0. In the case that u 0 L 2 = Q L 2 , [Fan18] and [Dod20] proved that E(P n u(t n )) → 0 along a subsequence, so the almost periodicity of u implies that u(t n ) converges to a rescaled version of Q.
In fact, [Fan18] and [Dod20] proved more, that E(P u(t)) → 0 in an averaged sense on an interval [0, T ] ⊂ I. The operator P is fixed on a fixed time interval, but P converges to the identity in the strong L 2 -operator topology as T → sup(I). The proof of Theorem 6 will argue that if E(P u(t)) goes to zero in a time averaged sense, then u must be equal to the soliton, if the solution is global. If the solution blows up in finite time, then u must equal a pseudoconformal transformation of the soliton.
An essential ingredient in this proof is an improved version of the long time Strichartz estimates in [Dod16b]. The proof will make use of the bilinear estimates of [PV09], which were also used in the two dimensional problem, [Dod16a].
Eventually, the proof of Theorem 6 will make use of long time Strichartz estimates on an interval is the function given by (3.30). However, to avoid obscuring the main idea, it will be convenient to consider the case when λ(t) = 1 first, since the generalization to the case (4.1) is fairly straightforward.
Suppose without loss of generality that a = 0 and b = T . Choose to be small constants, suppose for all t ∈ J, and choose η 1 ≪ η 0 sufficiently small so that and therefore, . When i ∈ Z, i > 0, let P i denote the standard Littlewood-Paley projection operator. When i = 0, let P i denote the projection operator P ≤0 , and when i < 0, let P i denote the zero operator.
Theorem 12. The long time Strichartz estimate, holds with implicit constant independent of T .
Proof. This estimate is proved by induction on j. Local well-posedness arguments combined with the fact that 1, and when i = 0, (4.10) (P ≥i u)(P ≤i−3 u) = 0. Therefore, 1.
This is the base case.
Remark 6. The implicit constant in (4.11) does not depend on T or η 1 .
To prove the inductive step, recall that by Duhamel's principle that if J = [(a−1)2 3k−3i , a2 3k−3i ], then for any t 0 ∈ J, By (4.4) and the fact that where sup v is the supremum over all such v supported on P i satisfying v V 2 ∆ (J×R) = 1. See [HHK09] for a proof.
By Hölder's inequality, Next, when i > 4, since U 2 ∆ ⊂ U 4 ∆ , (4.4) and (4.14) imply (4.18) . When i ≤ 4, the fact that for any a ∈ Z, 1, the fact that the Fourier inversion formula and Hölder's inequality imply (4.20) and the fact that (4.4) implies, after rescaling Similar calculations can be made for the terms Therefore, x . By (4.20), (4.21), and the Sobolev embedding theorem, . Making a similar calculation, it only remains to compute, using Definition 1, (4.24), and (4.26), Also, by V 2 ∆ ⊂ U 9/4 ∆ and the Sobolev embedding theorem, where sup v0 is over all v 0 L 2 = 1 supported in Fourier space on the support of P i . Therefore, we have finally proved, (4.33) To complete the proof of Theorem 12, it only remains to prove Indeed, assuming that (4.34) is true, (4.33) becomes Then taking a supremum over 0 ≤ i ≤ j, which by induction on j, starting from the base case (4.11), proves Theorem 12.
The bilinear estimate (4.34) is proved using the interaction Morawetz estimate (see [PV09] and [Dod16a]). To simplify notation, let where v 0 L 2 = 1 andv 0 is supported on the Fourier support of P j for some j ≥ i. Then take the Morawetz potential, Let F (u) = |u| 4 u. Then u ≤i−3 solves the equation Then by the fundamental theorem of calculus, Bernstein's inequality, the Fourier support ofvu ≤i−3 , v 0 L 2 = 1, and the fact that u L 2 = Q L 2 , (4.42) Also note that so it is not too important to pay attention to complex conjugates in the proceeding calculations. First, by (4.26), . Now consider the term, Since by Fourier support arguments Following (4.16)-(4.28), Again following (4.16)-(4.28), Finally, observe that the Fourier support of (4.52) is on frequencies |ξ| ≥ 2 i−6 . Therefore, integrating by parts, . A similar calculation gives the estimate (4.54) The terms may be analyzed in a similar manner.
Theorem 12 may be upgraded to take advantage of the fact that u is close to the soliton.
Remark 7. If C is the implicit constant in (4.62), then for η 0 ≪ 1 sufficiently small, The same argument can also be made when λ(t) ≥ 1 η1 for all t ∈ J.
and η −2 1 T = 2 3k , then The same argument could also be made for λ(t) having a different lower bound, by rescaling λ(t) to λ(t) ≥ 1 η1 , computing long time Strichartz estimates, and then rescaling back.

Almost conservation of energy
Since (3.31) implies that ǫ(t) L 2 is continuous as a function of time, the mean value theorem implies that under the conditions of Theorem 14, there exists t 0 ∈ [a, b] such that The next step in proving Theorem 6 is to control as a function of ǫ(t 0 ) L 2 . Theorem 11 would be a very useful tool for doing so, except that while Q lies in H s (R) for any s > 0, ǫ need not belong to H s (R) for any s > 0. Therefore, Theorem 11 will be used in conjunction with the Fourier truncation method of [Bou98]. See also the I-method, for example [CKS + 02].
be an interval such that Proof. By the mean value theorem, there exists t 0 ∈ J such that Next, decompose the energy. LetQ refer to a rescaled version of Q, that is, It is also convenient to splitǫ into real and imaginary parts, As in Theorem 11, by (3.3), (5.7) SinceQ is smooth and rapidly decreasing, E(Q) = 0, and λ(t 0 ) ≥ 1 η1 , Bernstein's inequality implies that Next, integrating by parts and using (3.48), the smoothness of Q, and Bernstein's inequality, Next, by Hölder's inequality, since λ(t 0 ) ≥ 1 η1 , (5.10) By the Sobolev embedding theorem, (5.11) Next compute the change of energy.
Remark 8. For these computations, it is not so important to distinguish between u andū.
Case 1: k 1 , k ′ 1 ≤ k + 6: Once again, if k 1 , k ′ 1 ≤ k + 6, then the right hand side of (5.23) is zero. Case 2: k 1 or k ′ 1 ≥ k + 6, eight terms are ≤ k: In the case that k 1 or k ′ 1 ≥ k + 6, and eight of the terms in (5.23) are at frequency ≤ k, then by Fourier support properties the final term should be at frequency ≥ k + 3. The contribution in this case is bounded by Case 3: k 1 or k ′ 1 ≥ k + 6, two terms are ≥ k: The contribution of the case that k 1 or k ′ 1 ≥ k + 6, two additional terms in (5.23) are at frequency ≥ k, and the other seven terms are at frequency ≤ k is bounded by Case 4: k 1 or k ′ 1 ≥ k + 6 and at least three additional terms in (5.23) are at frequencies ≥ k.
This case may be reduced to a case where at least four terms in (5.23) are at frequency ≥ k, and at least four terms are at frequency ≤ k + 9. To see why, notice that all five terms in F (P ≤k+9 u) are at frequency ≤ k + 9, so if four or five of the terms in P ≤k+9 F (u) are at frequency ≥ k, then we are fine.
If exactly, three terms in P ≤k+9 F (u) are at frequency ≥ k, then take the two terms in P ≤k+9 F (u) that are at frequency ≤ k to be terms at frequency ≤ k + 9. Meanwhile, since at least four terms are at frequency ≥ k, so in (5.26) there is one term at frequency ≥ k and two more terms at frequency ≤ k + 9. If exactly two terms in P ≤k+9 F (u) are at frequency ≥ k, then there are three terms that are at frequency ≤ k. In that case, so in (5.27) there are two terms at frequency ≥ k and one term at frequency ≤ k + 9. If one term in P ≤k+9 F (u) is at frequency ≥ k, then there are four terms in P ≤k+9 F (u) at frequency ≤ k. Then there must be at least three more in F (P ≤k+9 u), so If no terms in P ≤k+9 F (u) are at frequency ≥ k, then there must be four in F (P ≤k+9 u), so The contribution of all the different subcases of case four, (5.26)-(5.29), may be bounded by This proves Theorem 15.
Corollary 1. If Proof. The proof uses Theorem 15, Theorem 11, rescaling, and the fact that Q is smooth and all its derivatives are rapidly decreasing.

A frequency localized Morawetz estimate
The next step will be to combine long time Strichartz estimates with almost conservation of energy to prove a frequency localized Morawetz estimate adapted to the case when λ(t) does not vary too much.
Remark 11. Due to the presence of derivatives in it is convenient to dispense with theQ(x) andǫ(t, x) notation and return to the Q and ǫ notation. We understand that P ≤k+9 Q( x λ(t) ) denotes the frequency projection after rescaling, not a rescaled projection. A rescaled projection appears in (6.44).
For terms of order ǫ 3 and higher, it is not too important to pay attention to complex conjugates, since these terms will be estimated using Hölder's inequality.
Since both the left and right hand sides of (6.2) are scale invariant, the same argument also holds for an interval J where (6.59) A ≤ λ(t) ≤ AT 1/100 , for any A > 0.
7. An L p s bound on ǫ(s) L 2 when p > 1 Transitioning to s variables, under the change of variables (3.30), Theorem 16 and Corollary 2 imply that if [a, a + T ] ⊂ [0, ∞) is an interval on which Theorem 16 implies good L p s integrability bounds on ǫ(s) L 2 under (2.16), which is equivalent to Theorem 17. Let u be a symmetric solution to (1.1) that satisfies u L 2 = Q L 2 , and suppose and ǫ(0) L 2 = η * . Then with implicit constant independent of η * when η * ≪ 1 is sufficiently small. Furthermore, for any j ∈ Z ≥0 , let By definition, s 0 = 0, and the continuity of ǫ(s) L 2 combined with Theorem 7 implies that such an s j exists for any j > 0. Then, for each j, with implicit constant independent of η * and j ≥ 0.

Monotonicity of λ
Next, using a virial identity from [MR05], it is possible to show that λ(s) is an approximately monotone decreasing function.
Theorem 18. For any s ≥ 0, let Then for any s ≥ 0, Then we can show that u is a soliton solution to (1.1), which is a contradiction, since λ(s) is constant in that case. The proof that (8.3) implies that u is a soliton uses a virial identity from [MR05]. Using (3.31), compute Indeed, by direct computation, Then by (3.37), (3.38), (7.5), and the fundamental theorem of calculus, Therefore, there exists s ′ ∈ [s − , s + ] such that Since s ′ ≥ 0, there exists some j ≥ 0 such that s j ≤ s ′ + T * < s j+1 . Using the proof of Theorem 17, in particular (7.28), Then by Theorem 16, (8.7) implies and therefore by definition of s j+1+J , (8.10) sj+1+J s ′ ǫ(s) L 2 ds 1.
9. Almost monotone λ(t) The almost monotonicity of λ implies that when sup(I) = ∞, u is equal to a soliton solution, and when sup(I) < ∞, u is the pseudoconformal transformation of the soliton solution. Proof. For any integer k ≥ 0, let Then by (8.2), for all s ∈ I(k). By (3.30), the fact that sup(I) = ∞ implies that Therefore, there exists a sequence k n ր ∞ such that and that |I(k n )| ≥ |I(k)| for all k ≤ k n .
It only remains to show that in the case that sup(I) < ∞, u is a pseudoconformal transformation of the soliton. If one could show that the energy of u 0 is finite, then this fact would follow directly from the result of [Mer93]. Similarly, if one could generalize the result of [Mer93] to data that need not have finite energy, then the proof would also be complete.
We do not quite prove this fact. Instead, suppose without loss of generality that sup(I) = 0, and Then decompose ).
In particular, Doing some algebra, This is clearly the pseudoconformal transformation of a soliton. This finally completes the proof of Theorem 6.

A non-symmetric solution
When there is no symmetry assumption on u, there is no preferred origin, either in space or in frequency. As a result, two additional group actions on a solution u must be accounted for, translation in space, and the Galilean symmetry, This gives a four parameter family of soliton solutions to (1.1), given by (1.29). Making the pseudoconformal transformation of (1.29) gives a solution in the form of (1.30).
In this section we prove Theorem 5, that the only non-symmetric blowup solutions to (1.1) with mass u 0 2 L 2 = Q 2 L 2 belong to the family of solitons and pseudoconformal transformation of a soliton. To prove this, we will go through the proof of Theorem 4 in sections two through nine, section by section, generalizing each step to the non-symmetric case. There are several steps for which the argument in the symmetric case has an easy generalization to the non-symmetric case, after accounting for the additional group actions (10.1) and (10.2). There are other steps for which the non-symmetric case will require substantially more work.
10.1. Reductions of a non-symmetric blowup solution. Using the same arguments that show that Theorem 4 may be reduced to Theorem 6, Theorem 5 may be reduced to Theorem 20. Let 0 < η * ≪ 1 be a small fixed constant to be defined later. If u is a solution to (1.1) on the maximal interval of existence I ⊂ R, u 0 L 2 = Q L 2 , u blows up forward in time, and Reducing Theorem 5 to Theorem 20 requires the following generalization of Theorem 7, which was proved in Theorem 2 of [Dod20].
Theorem 21. Assume that u is a solution to (1.1) with u 0 L 2 = Q L 2 that does not scatter forward in time. Let (T − (u), T + (u)) be its lifespan, T − (u) could be −∞ and T + (u) could be +∞. Then there exists a sequence t n ր T + (u) and a family of parameters λ n > 0, ξ n ∈ R, x n ∈ R, and γ n ∈ R such that Lemma 1 can easily be generalized to the non-symmetric case, proving that e iγ e ixξ0 λ 1/2 u 0 (λx+ x 0 )−Q L 2 attains its infimum on γ ∈ R, ξ 0 ∈ R, x 0 ∈ R, λ > 0. Theorem 8 is also easily generalized to the non-symmetric case, showing that the left hand side of (10.3) is upper semicontinuous in time and continuous in time when small. Therefore, Theorem 5 is easily reduced to Theorem 20 using the same argument that reduced Theorem 4 to Theorem 6. 10.2. Decomposition of a non-symmetric solution near Q. When a non-symmetric u is close to a soliton, it is possible to make a decomposition of u, generalizing Theorem 9 to account for the additional group actions in (10.1) and (10.2).
10.4. Almost conservation of energy for a non-symmetric solution. It is possible to use the long time Strichartz estimates in Theorem 23 to prove an almost conservation of energy for a non-symmetric solution.
Theorem 24. Let J = [a, b] be an interval such that Then, Proof. Decompose the energy as in Theorem 11. Since E(Q) = 0 and (ǫ 2 , Q x ) = 0, (10.49) Using the bounds on |ξ(t)| λ(t) , the fact that Q and all its derivatives are rapidly decreasing, Fourier truncation, and the mean value theorem implies that (10.48) holds for some t 0 ∈ J. Then, using the long time Strichartz estimates in Theorem 23 and following the proof of Theorem 15 gives Theorem 24.
It is also possible to generalize Corollary 1 to the non-symmetric case.
As in (6.15), (10.66) Theorem 26. Let u be a nonsymmetric solution to (1.1) that satisfies u L 2 = Q L 2 , and suppose (10.67) sup and ǫ(0) L 2 = η * . Then with implicit constant independent of η * when η * ≪ 1 is sufficiently small. Furthermore, for any j ∈ Z ≥0 , let By definition, s 0 = 0, and as in Theorem 17, such an s j exists for any j > 0. Then, for each j, with implicit constant independent of η * and j ≥ 0.
By the triangle inequality, and by Hölder's inequality, It is therefore possible to prove Theorem 26 by induction. Indeed, suppose that for some n > 0, Then by (10.40), ln λ(s)| CJ.
Next, rescaling so that inf s∈[s ′ ,s ′ +J n+1 T * ] λ(s) = 1 η1 , setting ξ(s ′ ) = 0, (10.39) implies which implies that ǫ(s) L 2 belongs to L p s for any p > 1, but not L 1 s . 10.7. Monotonicity of λ in the non-symmetric case. It is possible to use the virial identity from [MR05] to show monotonicity in the non-symmetric case as well. As in the proof of Theorem 19, there exists a sequence k n ր ∞ such that (10.111) |I(k n )|2 −2kn ≥ 1 k 2 n , and that |I(k n )| ≥ |I(k)| for all k ≤ k n .
Let r n be the smallest integer that satisfies which implies that ξ(s) converges to some ξ ∞ as s → ∞. Making a Galilean transformation that maps ξ ∞ to the origin and taking n → ∞, since m ≥ 0, (10.26) implies that E(u 0 ) = 0. Therefore, by the Gagliardo-Nirenberg inequality, u 0 is a soliton.

Acknowledgement
The author was partially supported by NSF Grant DMS-1764358. The author was also greatly helped by many stimulating discussions with Frank Merle at Cergy-Pontoise and University of Chicago, as well as his constant encouragement to pursue this problem. The author would also like to recognize the many helpful discussions that he had with Svetlana Roudenko and Anudeep Kumar Arora, both at George Washington University and Florida International University.