Almost Conservation Laws for KdV and Cubic NLS. UROP Final Paper, Summer 2018

Transcription

1 Almost Conservation Laws for KdV and Cubic NLS UROP Final Paper, Summer 2018 Zixuan Xu Mentor: Ricardo Grande Izquierdo Project suggested by: Gigliola Staffilani Abstract In this paper, we explore the method of almost conservation laws proposed in [1] for integrable systems. In particular, we consider the Korteweg-de Vries initial value problem and give a rigorous proof on how the algorithm that generates these almost conservation laws can be used to recover infinitely many conserved integrals that make the KdV an integrable system. In addition, we applied the same idea of almost conservation laws to the cubic nonlinear Schrödinger equation.

2 1 Introduction As presented in [1], the method of almost conservation laws can be used to extend local well-posedness results of this equation to global-wellposedness ones, which the authors apply to the KdV equation, " Bt u ` B x u ` 1B 2 xpu 2 q 0, (1) upx, 0q u 0 pxq This method is based on studying some norms like Iu H s, where I is a functional given by a certain Fourier multiplier, and u is the solution of our equation. In practice, it requires a careful analysis of how conservation laws are proved in frequency space, in order to gain understanding on how different frequencies interact. A simple example of this idea is the following: consider the L 2 conservation law for the solution of the KdV equation above. One can use Plancherel to write: u t pxq 2 dx pu t pξ 1 q pu t pξ 2 q dξ 1 dξ 2 ξ 1`ξ 2 0 By using the Fourier transform of the equation, one can get the following identity d dt u t pxq 2 dx i i ξ 1`ξ 2 0 ξ 1`ξ 2`ξ 3 0 pξ 3 1 ` ξ 3 2q pu t pξ 1 q pu t pξ 2 q dξ 1 dξ 2 pξ 1 ` ξ 2 q pu t pξ 1 q pu t pξ 2 q pu t pξ 3 q dξ 1 dξ 2 dξ 3 (2) The first term is clearly zero, and the second term becomes zero after writing ξ 1 ` ξ 2 ξ 3 and symmetrizing. As explained in [1] and [2], an interesting idea in order to prove these global results is to define a hierarchy of modified energies EI i ptq for the solution of our dispersive equation that are comparable to the norm u H s that we are trying to control, but whose increments decrease as generations evolve. Whether such a hierarchy can be found depends on the cancellations that occur on a more general type of integrals resembling (2) above. However, these cancellations are not yet well-understood, and therefore we will start with a simpler choice of the multiplier I, with which one may recover the classical conservation laws. A guess for what this choice would be and the proof for the first three conservation laws appears in [2], and in this paper we will provide the full proof. 1

3 Another goal of this paper is to study the analogous case of the cubic NLS equation " ibt u ` u u 2 u, upx, 0q u 0 pxq for x P T, which also enjoyes infinitely many conserved quantities, and to try to accomplish the same objectives as with the KdV equation. In Section 2 we provide some definitions and introduce some notation that will be useful throughout the paper. In Section 3 we provide the main results and proofs. In Section 4 we set out the analogous problem for the cubic NLS equation. And finally, in Section 5 we discuss possible future directions of research on these topics. The results presented in this paper were developed in collaboration with Ricardo Grande Izquierdo. 2 Preliminary Definitions Definition 2.1. Throughout this paper, we define the spatial Fourier transform of fpxq to be Fpfqpξq : fpξq : e ixξ fpxqdx R Note that with this definition of the Fourier transform, we have { B n xfpxq piξq n fpξq. In this paper, we consider the conservation laws of the KdV equation, defined as the following Definition 2.2. We define the KdV equation to be the following: # B t u ` B 3 xu ` upb x uq 0 upx, 0q u 0 pxq We then define the operator I. Definition 2.3. Suppose u is a solution to the KdV equation, then we define an operator I, usually called Fourier multiplier, such that xiupξq mpξqûpξq. 2

4 As shown in the introduction, the symmetrization describes the nonlinear interaction of the frequencies of the solution u to the KdV equation. Through this method, we keep track of the various pieces of û. Now we introduce some notation that will be frequently used in the rest of the paper. Definition multiplier m is a function m : R Ñ C. -multiplier is symmetric if mpξq mpσpξqq for all σ P S k. The symmetrization of a k-multiplier is rms sym pξq 1 ÿ mpσpξqq k! σps k -multiplier generates the k-linear functional via the integration Λ k pmq mpξ 1,..., ξ k qûpξ i q...ûpξ k q dξ 1... dξ k where tpξ 1,..., ξ k q : ξ 1 ` ` ξ k 0u. Remark. This notation does not refer to integrating over a k-dimensional set of measure zero, instead it refers to k 1 integrals, i.e. k 1 ÿ Λ k pmq mpξ 1,..., ξ k qûpξ i q...ûpξ k 1 qûp ξ i qdξ 1 dξ k 1 R R By observation, we find that Iuptq 2 L 2 mpξ 1 qmpξ 2 qûpξ 1 qûpξ 2 qdξ 1 dξ 2 Λ 2 pmpξ 1 qmpξ 2 qq A 2 Hence we have the following useful proposition, which can be found in [1]. Proposition 2.1. Suppose u satisfies the KdV equation, and m is a symmetric k-multiplier and Λ k pmq is the k-linear functional generated by m, then d dt Λ kpmq Λ k pα k mq i k 2 Λ k`1p mpξ 1,..., ξ k`1 qq. where mpξ 1,..., ξ k`1 q pξ k ` ξ k`1 q mpξ 1,...,, ξ k 1, ξ k ` ξ k`1 q i 1 Proof. d dt Λ kpmq d mpξ 1,..., ξ k qûpξ 1 q ûpξ k qdξ 1 dξ k dt A k mpξ 1,..., ξ k q d dt pûpξ 1q ûpξ k qqdξ 1 dξ k 3

5 Substitute in the equation then we have Since d dt Λ kpmq kÿ i 1 kÿ i 1 B t û piξq 3 û 1 2 iξ p u 2 piξ i q 3 mpξ 1,..., ξ k qûpξ 1 q ûpξ k q 1 2 piξ iqûpξ 1 q pu 2 pξ i q ûpξ k qmpξ 1,..., ξ k q pu 2 pξ i q ûpξ i q ûpξ i q R ûpξ i ξ k`1 qûpξ k`1 qdξ k`1 By symmetry on the set we obtain the desired identity. With the above notation, we now define the chain of modified energies following the notation in [1], Definition 2.5. We recursively define the chain of energies E k ptq as the following: E 2 I ptq Iuptq 2 L 2 Λ 2pmpξ 1 qmpξ 2 qq where uptq is the solution to the KdV equation and I is an operator such that { Ifpξq mpξq fpξq, then E k I ptq E k 1 I ptq ` Λ k pσ k q, where σ k is an operator chosen such that d dt Ek I ptq Λ k`1 pm k`1 q. Note that by Proposition 2.1, we have the explicit form M k`1 i k 2 rσ kpξ 1,..., ξ k 1, ξ k ` ξ k`1 q pξ k ` ξ k`1 qs sym It was proposed in [1] that by setting the operator I Bx, n i.e. mpξq piξq n and putting E 2 ptq Iu 2 L 2 Λ 2 pmpξ 1 qmpξ 2 qq, we can recover all the conservation laws of the KdV equation. It was also shown explicitly in [1] that the first two conservation laws can be recovered by the proposed method, yet a rigorous proof was not given. Before illustrating this construction with an example, we first note some properties of the symmetrization operation that will help with the calculations involved in the example. The proof of the following lemmas are straight-forward. 4

6 Lemma 2.1. (Linearity) Suppose m : R k Ñ C, n : R k Ñ C are two k- multipliers, then we have rm ` ns sym rms sym ` rns sym Lemma 2.2. Suppose m : R k Ñ C is a k-multiplier and σ P S k, then rmpξqs sym rmpσpξqqs sym Lemma 2.3. Let m, n be two k-multipliers, if m is symmetric, then rmns sym mrns sym We now provide a detailed example to explicitly illustrate how the chain of modified energies recover a conservation law of the KdV equation. Example 2.1. In the case where n 3, we have E 2 ptq B 3 xu 2 L 2 Λppiξ 3 1qpiξ 3 2qq mpξq iξ 3 Then σ 3 rξ7 1s 3rξ 3 1s 7 3 rξ2 1ξ 2 2s then M 4 i pξ 1 qξ 1 ` m 2 pξ 2 qξ 2 ` m 2 pξ 3 ` ξ 4 qpξ 3 ` ξ 4 q 2 rm2 s 3ξ 1 ξ 2 i 2 rξ7 1 ` ξ2 7 ` pξ 3 ` ξ 4 q 7 s 3ξ 1 ξ 2 7i 3 prξ5 1s ` 3rξ1ξ 4 2 s ` 5rξ1ξ 3 2sq 2 35i 3 rξ3 1ξ2s 2 σ 4 M4 35 α 4 12 rξ 3 1ξ 2 2s rξ 3 1s Note that we have the following identities p4 3 rξ3 1srξ 2 1s 1 3 rξ5 1sq rξ 1ξ 2 s ξ 5 1 ` ξ 5 2 ` ξ 5 3 ` pξ 4 ` ξ 5 q 5 ξ 5 1 ` ξ 5 2 ` ξ 5 3 pξ 1 ` ξ 2 ` ξ 3 q 5 5pξ 1 ` ξ 2 qpξ 1 ` ξ 3 qpξ 2 ` ξ 3 qpξ 2 1 ` ξ 2 2 ` ξ 2 3 ` ξ 1 ξ 2 ` ξ 1 ξ 3 ` ξ 2 ξ 3 q ξ 3 1 ` ξ 3 2 ` ξ 3 3 ` pξ 4 ` ξ 5 q 3 ξ 3 1 ` ξ 3 2 ` ξ 3 3 pξ 1 ` ξ 2 ` ξ 3 q 3 3pξ 1 ` ξ 2 qpξ 1 ` ξ 3 qpξ 2 ` ξ 3 q 5

7 Therefore we have Note that M 5 2iσ 4 pξ 1, ξ 2, ξ 3, ξ 4 ` ξ 5 q pξ 4 ` ξ 5 q 2i pξ 1ξ 2 ` ξ 1 ξ 3 ` ξ 2 ξ 3 pξ 1 ` ξ 2 ` ξ 3 q 2 qpξ 1 ` ξ 2 ` ξ 3 q rξ 1ξ 2 ξ 3 s rξ 1 ξ 2 ξ 3 s 1 3 rξ 1ξ 2 pξ 1 ` ξ 2 qs 2 3 rξ2 1ξ 2 s 1 6 rξ3 1s i.e. rξ 1 ξ 2 ξ 3 s 1 30i α 5 Therefore σ 5 M5 35 α Notice that E 5 ptq E 2 ptq`λpσ 3 q`λpσ 4 q`λpσ 5 q is exactly the conservation law according to [4] ppbxuq upb2 xuq 2 ` u2 pb x uq u5 qdx C R 3 Results We now provide a proof for the correspondence between the chain of modified energies and the conservation laws. We first start with three useful lemmas. Lemma 3.1. Suppose Λ k ppq 0, then p 0 on the set. Proof. By definition Λ k ppq ppξ 1,..., ξ k qûpξ 1 q ûpξ k qdξ 1 dξ k Suppose by contradiction that p 0, using Plancherel s theorem and taking the inverse Fourier transform, we have Λ k ppq p 1 pu 0, u 1,..., u degppq qdx 0 R where p 1 is a homogeneous differential polynomial of u of degree k, and u j denotes the j-th derivative of u. Then since this is true for every time t, 6

8 this implies that R p1 dx is a conservation law, hence it suffices to argue that R p1 dx cannot be a conservation law of the KdV equation. Note that every conservation law C n can be written as C n R P n dx nÿ Λ k pp k q where P n is a differential polynomial of u homogenous of rank n according to [4]. The rank of each term in the form u a 0 0 u a 1 1 u a l l is defined to be r : k 2 lÿ p1 ` 1 2 jqa j j 0 Note that for the polynomial p 1, each monomial has ř l j 0 a j k. Suppose p 1 is a conservation law of rank n, n ě k, then for each monomial in p 1, we must have ř l j 0 j a j 2pn kq, j ď degppq, therefore the choice of a j must corresponds to a proper subset of the monomials in C n. Since the conservation law of rank n for the KdV equation is unique, p 1 cannot be a conservation law, contradiction. Hence we can conclude that p 0 on the set. Lemma 3.2. Suppose that for all solutions of the KdV equation Λ n ppq ` Λ m pqq 0, n ă m. Then Λ n ppq 0, Λ m pqq 0. In particular, p 0 and q 0 on the set A n and A m, respectively. Proof. Suppose that u is a solution of KdV, then (with the corresponding scaling of the initial data) the following is also a solution of KdV for all λ P R u λ pt, xq λ 2 upλ 3 t, λxq Then taking the Fourier transform with respect to x, we obtain xu λ pt, ξq λ 3 pupλ 3 t, λ 1 ξq If Λ n ppq ` Λ m pqq 0 holds for all solutions of KdV, it holds for u λ and note that Λ n ppq ppξ 1,..., ξ n qxu λ pξ 1 q xu λ pξ n qdξ 1 dξ n A n Then by change of variables sending ξ i to λξ i we obtain Λ n ppq p 1 pξ 1,..., ξ n qûpξ 1 q ûpξ n qdξ 1 dξ n A n 7

9 where where Similarly p 1 pξ 1,..., ξ n q λ degppq 2n`1 ÿ i ξ a 1 1 ξn an c i λ ř n i 1 a i degppq Λ m pqq q 1 pξqûpξ 1 q ûpξ m qdξ 1 dξ m A m q 1 pξ 1,..., ξ n q λ ÿ degpqq 2m`1 ξ b 1 1 ξm bm d i λ ř m i 1 b i degpqq i Since n ă m, then degppq ă degpqq, and therefore 1 λ pλ npp 1 q ` Λ degppq 2n`1 m pq 1 qq 0 Then we have p ÿ ξ a 1 1 ξn an c i A n λ ř qûpξ n 1q ûpξ i 1 a n qdξ 1 dξ n i degppq`2n 1 i ` pλ ÿ degpqq degppq`2pm nq ξ b 1 1 ξm bm d i A m λ ř qûpξ m 1q ûpξ i 1 b m qdξ 1 dξ m 0 i degpqq i We now consider the case t 0 and reduce the solution to the initial data u 0. By taking λ Ñ 0, the second term vanishes and we have p ÿ ξ a 1 1 ξn an c i A n λ ř qû n 0pξ i 1 a 1 q û 0 pξ n qdξ 1 dξ n 0 i degppq`2n 1 i Note that ř n i 1 a i ď degppq 2n ` 1, therefore we have A n ÿ i s.t. ř n i 1 a i degppq c i ξ a 1 1 ξ an n û 0 pξ 1 q û 0 pξ n qdξ 1 dξ n 0 Since this is true for every initial data, the polynomial ÿ c i ξ a 1 1 ξn an i s.t. ř n i 1 a i degppq must be zero on the set A n. However, this produces a contradiction about the degree of p unless degppq 0, in which case by the same reasoning as above, p 0. Then Λ n ppq 0, and therefore the initial hypothesis becomes Λ m pqq 0. By Lemma 3.1, q 0. 8

10 Lemma 3.3. Suppose that S is a finite subset of positive integers and ř kps Λ kpp k q 0. Then p k 0 on the set for every k P S. Proof. The same scaling trick as in Lemma 3.2 should prove that p k0 0 for k 0 min S k. From then on, one simply iterates the idea, the key fact being that the exponent of λ in each integral is positive, i.e. degpp k q degpp k0 q ` 2pk k 0 q ą 0 for k k 0, which allows every other item to vanish as λ Ñ 0. We are now ready to give the main theorem of this paper Theorem 3.4. Let C n denote the n-th conservation law of the KdV equation, then C n`1 E n`2 I B n x ptq n ě 0 Proof. Note that for n ě 0 C n`1 ppbxuq n 2 ` r n`1 qdx, R where r n`1 is a differential polynomial in u and its first n 1 derivatives with each term no greater than degree n ` 2. Taking the Fourier transform of C n`1, we can write the following zc n`1 n`2 ÿ k 2 rp k pξ 1,..., ξ k qsûpξ 1 q ûpξ k qξ 1 dξ k Note that by the Plancherel theorem, z C n`1 C n`1. By differentiating with respect to time and with Proposition 2.1, we have where n`2 ÿ B t C n`1 p α k rp k pξ 1,..., ξ k qsûpξ 1 q ûpξ k qdξ 1 dξ k k 2 k 2 i r p k pξ 1,..., ξ k`1 qsqûpξ 1 q ûpξ k`1 qdξ 1 dξ k`1 q, `1 p k pξ 1,..., ξ k`1 q p k pξ 1,..., ξ k 1, ξ k ` ξ k`1 q pξ k ` ξ k`1 q. Since B t C n`1 0 and for all 3 ď k ď n 2, each integral on has the 9

11 form p i k 1r p 2 k 1s ` α k r p k sqûpξ 1 q ûpξ k qdξ 1 dξ k B t C n`1 α 2 rp 2 pξ 1, ξ 2 qsûpξ 1 qûpξ 2 qdξ 1 dξ 2 A 2 n`2 ÿ ` p i k 1 k 3 2 r p k 1s ` α k rp k sqûpξ 1 q ûpξ k qdξ 1 dξ k ` p i n ` 2 r p n`2 sqûpξ 1 q ûpξ n`3 qdξ 1 dξ n`3 A n`3 2 Λ 2 pα 2 rp 2 sq ` n`2 ÿ k 3 Λ k i k 1 2 r p k 1s ` α k rp k sq ` Λ n`3 p i n ` 2 r p n`2 sq 0 2 Notice that α 2 0 on the set A 2, hence the integral A 2 α 2 p 2 pξ 1, ξ 2 qûpξ 1 qûpξ 2 qdξ 1 dξ 2 0. Since the term with the highest degree in C n`1 consists no derivatives of u, i.e. p n`2 is a constant, therefore p n`2 0 on the set A n`3, therefore r p 2 n`2sûpξ 1 q ûpξ n`3 qdξ 1 dξ n`3 0. Hence A n`3 i n`2 n`2 ÿ k 3 Λ k i k 1 2 r p k 1s ` α k rp k sq By Lemma 3.3, for every k 3,..., n ` 2 we have 0 i k 1 2 r p k 1s ` α k rp k s 0 on. (3) By construction of the chain of modified energies EI B k ptq Ek 1 x n I B ptq ` Λ kpσ x n k q EI B 2 ptq Λ 2ppiξ x n 1 q n piξ 2 q n q Λpσ 2 q, where σ k Mk α k and M k i k 1r σ 2 k 1s. By definition we have σ 2 rp 2 s and therefore M 3 σ 2 r p 2 s. Therefore continuing this process and taking into account (3), notice that M k i k 1r p 2 k 1s and we choose σ k such that M k ` α k σ k 0, then σ k rp k s. Remember than the algorithm created a chain of modified energies satisfying: E n`2 I B ptq En`1 x n I B ptq ` Λ n`2pσ x n n`2 q n`2 EI B 2 ptq ` ÿ Λ x n k pσ k q By the explanation above, it is now clear that k 3 C n`1 E n`2 I B ptq n ě 0. x n 10

12 4 Cubic NLS From the results above, it is natural to consider applying the same idea to other integrable equations. Another frequently studied integrable equation is the cubic nonlinear Schrödinger (NLS) equation, which also satisfies infinitely many conservation laws, defined as the following Definition 4.1. We define the cubic nonlinear Schrödinger (NLS) equation as the following # ib t φ ` B 2 xφ ` 2 φ 2 φ 0 φpx, 0q φ 0 pxq According to [3], the conservation laws of the cubic NLS equation are given by the following recursive formula Definition 4.2. The Cubic NLS equation is an integrable system, its conservation laws can be recursively found as follows w n`1 i dw n dx The conserved quantities are C n w 1 φpxq n 1 ÿ pxq ` φpxq 8 8 k 1 φpxqw n pxq Hence the first four conservation laws are C 1 φpxq 2 dx, C 2 w k pxqw n k pxq i φpxqb x φpxqdx C 3 p φpxqb xφpxq 2 ` φpxq 4 qdx C 4 pip φb xφ 3 φ 2 φb x φ 4 φ 2 φbx φqdx Let φ be a solution to the cubic NLS equation, then φ satisfies B t φ ipb 2 xφ ` 2 φ 2 φq Take the fourier transform with respect to space, then B t φ ippiξq 2 φ ` 2 y φ 2 φq iξ 2 φ ` 2i φ φ φ 11

13 Note that a significant difference between the KdV equation and the cubic NLS equation is that the solutions to the KdV equation are real functions but the solutions to the cubic NLS equation are complex functions. Hence we make the following modification to our previous definitions. Definition 4.3. Suppose m is a k-multiplier for k an even number, we define the k-linear functional generated by m via integration to be the following Λ k pmq mpξ 1,..., ξ k q φpξ1 q φpξk{2 q φpξ k{2`1 q φpξ k qdξ 1 dξ k Similarly, we have the following proposition Proposition 4.1. Suppose φ is a solution to the cubic NLS equation, then where Proof. d dt Λ kpmq ` d dt Λ kpmq Λ k pα k mq ` 2kiΛ k`2 p mpξ 1,..., ξ k`2 qq α k ipξ 2 1 ` ` ξ 2 kq mpξ 1,..., ξ k`2 q mpξ 1,..., ξ k 1, ξ k ` ξ k`1 ` ξ k`2 q mpξ 1,..., ξ k q d φpξ 1 q φpξk{2 q dt φpξ k{2`1 q φpξ k qdξ 1 dξ k ÿk{2 j 1 mpξ 1,..., ξ k q φpξ1 q p iξ 2 j φpξj q ` 2i φ φ φpξj qq φpξ k qdξ 1 ξ k kÿ j k{2`1 mpξ 1,..., ξ k q φpξ1 q p iξ 2 j φpξ j q ` 2i φ φ φpξj qq φpξ k qdξ 1 ξ k kÿ mpξ 1,..., ξ k qpiξj 2 q φpξ1 q φpξk{2 q φpξ k{2`1 q φpξ k qdξ 1 dξ k j 1 ÿk{2 ` mpξ 1,..., ξ k q2i φpξ1 { q φ 2pξ j φq φpξ k qdξ 1 ξ k j 1 kÿ ` mpξ 1,..., ξ k q2i φpξ1 q yφ 2 φpξj q φpξ k qdξ 1 ξ k j k{2`1 Λ k pα k mq ` 2kiΛ k`2 p mpξ 1,..., ξ k`2 qq where α k ipξ 2 1 ` ` ξ 2 kq 12

14 mpξ 1,..., ξ k`2 q mpξ 1,..., ξ k 1, ξ k ` ξ k`1 ` ξ k`2 q Similarly, we provide an example for illustration Example 4.1. In order to recover the conservation law C 3, we set Then Hence E 2 ptq Λ 2 ppiξ 1 q 2 q M 4 2ip4rξ 2 1s ` 6rξ 1 ξ 2 sq 4irξ 2 1s σ 4 M 4 α 4 4irξ2 1s 4irξ 2 1s 1 E 4 ptq E 2 ptq ` Λ 4 pσ 4 q p φpxqb xφpxq 2 φpxq 4 qdx Therefore by similar intuition, we have the following conjecture for the cubic NLS equation, which has not been rigorously proven Conjecture 4.1. Let C k denote the k-th conservation law of the cubic NLS equation, then C k`1 E k`2 I ptq k ě 0, k even where EI 2 ptq φb xφdx k Λ k ppiξ 1 q k q 5 Future Research R In the future, it would be interesting to work out the details for the case of cubic NLS and rigorously prove how the chain of modified energies recover conservation laws. In addition, it would be interesting to consider different multipliers m, for example the multiplier # 1, ξ ă N mpξq N s ξ s, ξ ą 10N in order to obtain some results on extending local well-posedness to global well-posedness. One important obstacle when tackling this problem would be to better understand the cancellations that take place as one computes more elements in the chain of modified energies. Even though we were expecting to advance towards this goal when studying this method applied to the recovery of conservation laws, the proofs we ended up producing don t seem to shed much light on these cancellations and new ideas might be necessary. 13

15 References [1] Staffilani, G., KdV and almost conservation laws, Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), , Contemp. Math., 320, Amer. Math. Soc., Providence, RI, [2] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T.; Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc. 16 (2003), no. 3, [3] Faddeev, L. D. and Takhtajan, L. A., Hamiltonian methods in the theory of solitons, Classics in Mathematics, Springer, Berlin, 2007, Translated from the 1986 Russian original by Alexey G. Reyman. [4] Miura, R. M.; Gardner, C. S.; Kruskal, M. D; Korteweg-de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion, Journal of Mathematical Physics, Volume 9, Number 8, August