Mathematical Physics, Analysis and Geometry 2: 1–24, 1999. 1 © 1999 Kluwer Academic Publishers. Printed in the Netherlands. Square Integrability and Uniqueness of the Solutions of the Kadomtsev–Petviashvili-I Equation LI-YENG SUNG Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. (Received: 20 February 1998; in final form: 27 November 1998) Abstract. We prove that the solution of the Cauchy problem for the Kadomtsev–Petviashvili-I Equation obtained by the inverse spectral method belongs to the Sobolev space Hk.R2/ for k > 0, under the assumption that the initial datum is a small Schwartz function. This solution is shown to be the unique solution within a class of generalized solutions of the Kadomtsev–Petviashvili-I equation. Mathematics Subject Classification (1991): 35Q53. Key words: Kadomtsev–Petviashvili-I equation, inverse spectral method, Cauchy problem, uniqueness of solutions. 1. Introduction The Cauchy problem .qt � 6qqx C qxxx/x D 3qyy; (1.1a) q.x; y; 0/ D q0.x; y/; (1.1b) for the Kadomtsev–Petviashvili-I (KPI) equation (cf. [8]) is solved in [6] by the inverse spectral method, under the assumption that q0 is a small Schwartz function. It is shown in [6] that the solution q.x; y; t/ obtained by the inverse spectral method 1 2 is a C classical solution of (1.1) for .x; y/ 2 R , t D6 0, and t ! q.�; t/ 2 1 2 C ..�1; 0/ [ .0;1/;C0.R //. 1 2 We will show in this paper that t ! q.�; t/ 2 C.R;H .R // and q.x; y; t/ is the unique generalized solution for the forward (respectively backward) prob- 3 2 1 1 2 lem of (1.1) in the class C.T0;1/;H .R // \ C ..0;1/;L .R // (respectively 3 2 1 1 2 C..�1; 0U;H .R // \ C ..�1; 0/;L .R //). The inverse spectral method for KPI is studied formally in [10] and [5]. Rigor- ous aspects of this method have been investigated in [14] and [20]. The version of the inverse spectral method used in this paper is essentially that of [20]. However, the results in [14] and [20] are obtained under the additional assumption that Z dx q0.x; y/ D 0; (1.2) R 2 LI-YENG SUNG which is also assumed in many of the papers that study (1.1) by PDE techniques (cf. [4, 12, 13, 16, 17]). As a consequence of the nonphysical constraint (1.2), the scattering data have decay in all directions, which greatly simplifies the analysis. When (1.2) is not assumed, the analysis is much more subtle due to the lack of decay of the scattering data in certain directions. The implications of the constraint (1.2) are also studied in [1, 2] and [3] using the inverse spectral method. But the fact that the solution obtained by the inverse spectral method (without assuming (1.2)) is the unique solution of (1.1) in a general class of solutions has not been rigorously established until now. Since there is an isomorphism (cf. [9]) between solutions of the KP equation and the Johnson equation (cf. [7]) in the case of rapidly decaying initial data, the results of this paper can also be applied to the Johnson equation. The rest of the paper is organized as follows. The inverse spectral method for (1.1) is described in Section 2, where we give both the solution from the left and the solution from the right. We also recall some relevant results from [6]. Section 3 contains the proof that the inverse spectral solution from the left and from the right are identical. The integrability of the time-dependent Jost function is studied in 1 2 Section 4. We prove in Section 5 that t ! q.�; t/ 2 C.R;H .R //, and establish the uniqueness of solutions in Section 6. For the convenience of the reader we collect here some notation frequently used in this paper. n 0 n (a) S.R / is the space of Schwartz functions in n real variables and S .R / is the � space of tempered distributions equipped with the weak -topology. k 2 2 (b) H .R /, k D 0; 1; 2; : : :, are the L based Sobolev spaces in two real variables, T 1 2 1 k 2 and H .R / D H .R / is equipped with the natural Fréchet space kD0 m p topology. W .R/ is the L based Sobolev space in one real variable. p (c) C.X; Y / is the space of continuous maps from the topological space X into the topological space Y and Cb.X; Y / is the space of bounded continuous maps n from the Banach space X into the Banach space Y . Cb.R / is the space of n n bounded continuous functions on R equipped with the sup-norm and C0.R / n is the subspace of Cb.R / whose members vanish at infinity. 1 (d) Let I be an open interval and Y be a topological vector space. C .I; Y / is the space of continuously differentiable Y -valued functions on I . 1 (e) Let X be a measure space and Y be a Banach space. L .X; Y / is the space of Y -valued Bochner integrable functions on X. O Q (f) The Fourier transforms f and f are defined by Z Z O �i.x�Cy�/ Q �ix� f .�; �/ D dx dy e f .x; y/ and f .�; y/ D dx e f .x; y/: 2 R R � (g) The operators P are defined by Z �1 f .l/ � .P f /.k/ D dl : 2�i R l � .k � 0i/ SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 3 They always act on functions in the k variable in this paper. (h) A . B means A 6 constant � B, where the constant is independent of the variables in A and B. 2. The Inverse Spectral Method We describe in this section the solution of .1:1/ obtained by the inverse spectral method. We consider the following time-dependent Schrödinger equation defined by the initial datum q0: i�y � �xx C q0� D 0; (2.1) and introduce the Jost function �.x; y; k/ by the relation 2 �i.xk�yk / �.x; y; k/ D �.x; y; k/e : (2.2) The equation for � is i�y � �xx C 2ik�x C q0� D 0: (2.3) Taking the Fourier transform in the x variable, we can write .2:3/ as i 2 �Q y.�; y; k/ D i.� � 2k�/�Q .�; y; k/ C .qQ0 � �Q /.�; y; k/; (2.4) 2� where � denotes the convolution in the � variable. C � The Jost function � (resp. � ) is defined for Im k > 0 (resp. Im k 6 0) by � Z Z � y 1 i C 0 0 �Q .�; y; k/ D 2��.�/ C E�.�/ dy � EC.�/ dy � 2� �1 y 2 0 i.� �2k�/.y�y / C 0 � e .qQ0 � �Q /.�; y ; k/; (2.5) � Z Z � y 1 i � 0 0 �Q .�; y; k/ D 2��.�/ C EC.�/ dy � E�.�/ dy � 2� �1 y 2 0 i.� �2k�/.y�y / � 0 � e .qQ0 � �Q /.�; y ; k/; (2.6) where � is the Dirac function, and E� are defined by � 1; �� > 0, E�.�/ D (2.7) 0; otherwise. � The left scattering data L .k; l/ (k; l 2 R) are defined by 8 Z < i �iTx.k�l/Cy.l2�k2/U � � dx dy e q0.x; y/� .x; y; k/; k 6 l, L .k; l/ D 2� R2 (2.8) : 0; otherwise, 4 LI-YENG SUNG 8 Z < i �iTx.k�l/Cy.l2�k2/U C C dx dy e q0.x; y/� .x; y; k/; k > l, L .k; l/ D 2� R2 (2.9) : 0; otherwise. � Using L we define the time-dependent Jost function from the left �.x; y; t; k/, k 2 R, by the following equation: � Z Z � k 1 � C C � �.x; y; t; k/ D 1 C P dlL .k; l/ C P dlL .k; l/ � �1 k 2 2 3 3 iTx.k�l/Cy.l �k /C4t.k �l /U � e �.x; y; t; l/: (2.10) We can now write down the solution of .1:1/ from the left: Z � Z Z � 1 k 1 2 2 � C iTx.k�l/Cy.l �k /U q.x; y; t/ D dk dlL .k; l/ � dlL .k; l/ e � � R k �1 3 3 4it .k �l / � e Ti.k � l/�.x; y; t; l/ C �x.x; y; t; l/U: (2.11) 2 In summary, given a small q0 2 S.R /, the inverse spectral solution from the left for .1:1/ is obtained through the following steps: (I) Solve .2:5/ and .2:6/ � � � for � .x; y; k/. (II) Define L .k; l/ by .2:8/ and .2:9/ using � .x; y; k/. (III) Solve .2:10/ for �.x; y; t; k/. (IV) The solution q.x; y; t/ is defined by .2:11/ using � L .k; l/ and �.x; y; t; k/. Note that we can also define the solution q.x; y; t/ via the right scattering data: 8 Z < �i �iTx.k�l/Cy.l2�k2/U � � dx dy e q0.x; y/� .x; y; k/; k > l, R .k; l/ D 2� R2 (2.12) : 0; otherwise, 8 Z < �i �iTx.k�l/Cy.l2�k2/U C C dx dy e q0.x; y/� .x; y; k/; k 6 l, R .k; l/ D 2� R2 (2.13) : 0; otherwise. Let the time-dependent Jost function from the right �.x; y; t; k/, k 2 R, be defined by the following equation: � Z Z � 1 k � C C � �.x; y; t; k/ D 1 C P dlR .k; l/ C P dlR .k; l/ � k �1 2 2 3 3 iTx.k�l/Cy.l �k /C4t.k �l /U � e �.x; y; t; l/: (2.14) The solution of (1.1) from the right can then be expressed as Z � Z Z � k 1 1 2 2 � C iTx.k�l/Cy.l �k /U q.x; y; t/ D dk dlR .k; l/ � dlR .k; l/ e � � R �1 k 3 3 4it .k �l / � e Ti.k � l/�.x; y; t; l/ C �x.x; y; t; l/U: (2.15) SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 5 � � The relations among L .k; l/, R .k; l/, �.x; y; k/ and �.x; y; k/, and the equiv- alence of .2:11/ and .2:15/ will be established in Section 3. Finally we recall some relevant results from [6] which are obtained under the smallness assumption Z 2 dy d�.1 C � /jqQ0.�; y/j � 1: (2.16) 2 R (i) The integral equation (2.5) (resp. (2.6)) is uniquely solvable in the Banach 1 space Cb.Ry;L .R�/ � C�.�// for Im k > 0 (resp. Im k 6 0). � C C � (ii) The scattering data L .k; l/ and R .k; l/ (resp. L .k; l/ and R .k; l/) are 1 � � C for k 6 l (resp. k > l). Let F .k; l/ be L .k; l/ or R .k; l/. Then the following estimates hold: � � � � r �C� � @ @ @ � � m 2 2 n � �C� .k � l/ .k � l / C F.k; l/ . .1 C jkj/ � � @k @l @ 6 LI-YENG SUNG �1C�2C�3 @ # �3 3 .� .x; y; k/ � 1/ . .1 C jyj / for .x; y; k/ 2 R : (2.23) @x� SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 7 � Z jf .l/j . dl C 2 lk>0 .1 C jk � ljjkj/ Z � jf .l/j C dl : (3.2) 2 lk<0 .1 C jk C ljjkj/ Using (3.2) we obtain for any K > 0 that, for 1 6 p 6 1, �Z Z � 1 0 � jf .l/j jf .l/j � kLCf kLp.k>K/ . � � dl 2 C dl 2 � 0 .1 C jk � ljK/ �1 .1 C jk C ljK/ Lp.k>K/ � Z Z � 1 0 � jf .l/j jf .l/j � � � . dl C dl � 2 2 � 0 .1 C jk � ljK/ �1 .1 C jk C ljK/ Lp.Rk/ 1 . kf kLp.R/; (3.3) K and similarly 1 C kL f kLp.k<�K/ . kf kLp.R/: (3.4) K C It follows from (3.3), (3.4), the smoothness of L .k; l/ and the Fréchet–Kolmo- C gorov theorem (respectively the Ascoli–Arzelà theorem) (cf. [19]) that L is a p compact operator on L .R/ for 1 6 p < 1 (respectively Cb.R/). C p We note that I � L has the same kernel on L .R/ and Cb.R/. Let f 2 Cb.R/ C belong to Ker.I � L /. We derive from (2.17) that for any K > 0 Z k jf .l/j C kL f kL1.k<�K/ . dl 2 2 2 �1 .1 C jk � l j/ Z k jf .l/j 1 . dl . kf kL1.k<�K/: (3.5) 2 �1 .1 C jk � ljK/ K C Since f .k/ D .L f /.k/, it follows from (3.5) that f .k/ D 0 for k sufficiently negative, and then Gronwall’s inequality (cf. [18]) implies f .k/ D 0 for all k 2 R. C The invertibility of L follows from the Fredholm alternative (cf. [19]). 2 � t � t � � Let .L / and .R / be the transposes of L and R respectively, i.e., Z l � t � T.L / f U.l/ D � dkL .k; l/f .k/ and �1 Z l � t � T.R / f U.l/ D � dkR .k; l/f .k/: (3.6) �1 We have the following analog of Lemma 3.1. � t � t LEMMA 3.2. The operators I � .L / and I � .R / are bounded and invertible p on L .R/ .1 6 p < 1/ and Cb.R/. 8 LI-YENG SUNG We define 2 # �i.xk�yk / # � .x; y; k/ D e � .x; y; k/; where # D C;�; l; or r: (3.7) Then (2.19) and (2.20) can be rewritten as � � l � � r � D .I � L /� and � D .I � R /� : (3.8) From .3:8/ and Lemma 3.1 we derive the following equations in Cb.R/: r C �1 C l r � �1 � l � D .I � R / .I � L /� and � D .I � R / .I � L /� : (3.9) l For k 2 R, let the functions � .x; y; k/ be defined by the following analog of (2.21): Z y i 2 0 � � l 0 i.� �2k�/.y�y / e l 0 �Q .�; y; k/ D 2��.�/ C dy e q0 � �Q .�; y ; k/; (3.10) 2� �1 1 which is uniquely solvable in Cb.Ry;L .R�/�C�.�// under the assumption (2.16). l l Of course � enjoys the same properties as � . l Let � .x; y; k/ be defined by 2 l �i.xk�yk / l � .x; y; k/ D e � .x; y; k/; (3.11) then we have (cf. (2.1)–(2.4)) l l l l l l � y D �i�xx C iq0� and �y D �i�xx C iq0� : (3.12) LEMMA 3.3. Let �.�; y; k/ .resp. � .�; y; k// be the Fourier transform of l .� .x; y; k/�1/ .resp. the complex conjugate of .�.x; y; k/ � 1// in the x variable. Then we have Z Z lim d� sup j�.�; y; k/j D lim d� sup j�.�; y; k/j D 0: (3.13) y!�1 y!�1 R k2R R s2R Proof. It suffices to discuss the case of �.�; y; k/. From (2.21) we obtain Z y 2 0 0 i.� �2k�/.y�y / 0 �.�; y; k/ D i dy e qQ0.�; y / C �1 Z y i 2 0 0 i.� �2k�/.y�y / 0 C dy e .qQ0 � � /.�; y ; k/: (3.14) 2� �1 It follows immediately from (3.14) that �.�; y; k/ is continuous in all the variables and hence sup j�.�; y; k/j is a measurable function in � . Moreover, we y;k2R find from the Neumann series solution of (3.14) that 1 X sup j�.�; y; k/j 6 �j .�/; (3.15) y;k2R jD0 where SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 9 � � j� Z �jC1 1 k�jkL1.R/ 6 d� dyjqQ0.�; y/j : (3.16) 2� R2 Combining (2.16), (3.15) and (3.16) we have Z d� sup j�.�; y; k/j < 1: (3.17) R y;k2R The first limit in (3.13) follows from (3.14) and (3.17). 2 LEMMA 3.4. Let f .k/ 2 S.R/, then Z Z 1 l l dx� .x; y; k/ ds� .x; y; s/f .s/ D f .k/ 8y 2 R: (3.18) 2� R R Proof. It follows from (2.23) and (3.12) that the iterated integral on the left-hand side of .3:18/ is independent of y. Using the notation in Lemma 3.3 we can also rewrite the left-hand side of (3.18) by the Fourier inversion formula as Z Z 1 l l dx� .x; y; k/ ds� .x; y; s/f .s/ 2� R R Z Z 1 2 2 �i.xk�yk / l i.xs�ys / l D dx e .� .x; y; k/ � 1/ ds e .� .x; y; s/ � 1/f .s/C 2� R R Z Z 1 2 2 �i.xk�yk / l i.xs�ys / C dx e .� .x; y; k/ � 1/ ds e f .s/C 2� R R Z Z 1 2 2 �i.xk�yk / i.xs�ys / l C dx e ds e .� .x; y; s/ � 1/f .s/ C f .k/ 2� R R 2 Z Z iyk e 2 �iys D ds d��.k � s � �; y; k/�.�; y; s/e f .s/C 2 .2�/ R R 2 Z iyk e 2 �iys C dsT�.k � s; y; k/ C �.k � s; y; s/Ue f .s/ C f .k/: (3.19) 2� R The lemma follows from (3.13) and (3.19). 2 2 LEMMA 3.5. The following identity holds as operators on L .R/: � �1 � C �1 C .I � R / .I � L / D .I � R / .I � L /: (3.20) � �1 � l C �1 C l Proof. From .3:9/ we have .I�R / .I�L /� D .I�R / .I�L /� . Let 1 g 2 L .R/. Then we have Z Z l l dk � .x; y; k/G�.k/ D dk � .x; y; k/GC.k/; (3.21) R R where 10 LI-YENG SUNG � t � t �1 G� D TI � .L / UTI � .R / U g: (3.22) 1 l Since G� 2 L .R/ by Lemma 3.2 and � .x; y; k/ is bounded and continuous in all the variables, we find by (3.18), (3.21) and Fubini’s theorem that Z Z dkf .k/G�.k/ D dkf .k/GC.k/ 8f 2 S.R/: (3.23) R R It follows from (3.23) that Z � �1 � dkT.I � R / .I � L /f U.k/g.k/ R Z C �1 C D dkT.I � R / .I � L /f U.k/g.k/: (3.24) R 1 2 Since g 2 L .R/ is arbitrary and S.R/ is dense in L .R/, the identity (3.20) follows from (3.24). 2 We now show that (2.11) and (2.15) define the same function q.x; y; t/. We can rewrite (2.10) and (2.14) as � � � C �1 C � �1 P E.x;y;t/.I � L /E .x;y;t/ C P E.x;y;t/.I � L /E.x;y;t/ � D 1; (3.25) � � � C �1 C � �1 P E.x;y;t/.I � R /E .x;y;t/ C P E.x;y;t/.I � R /E.x;y;t/ � D 1; (3.26) 2 3 i.xk�yk C4tk / where TE.x;y;t/f U.k/ D e f .k/. Combining (3.20), (3.25) and (3.26) we obtain the following lemma. LEMMA 3.6. The functions �.x; y; t; k/ and �.x; y; t; k/ are related by C �1 C �1 � D E.x;y;t/.I � R / .I � L /E .x;y;t/� � �1 � �1 D E.x;y;t/.I � R / .I � L /E .x;y;t/�: (3.27) PROPOSITION 3.7. The formulas .2:11/ and .2:15/ define the same function q.x; y; t/. Proof. The case where t D 0 follows from (2.24). For t D6 0, in view of (2.25) and (2.26), we can rewrite (2.11) and (2.15) as q.x; y; t/ � Z � 1 @ � � C � �1 D dk E.x;y;t/..I � L / � .I � L //E .x;y;t/ �.x; y; t; l/ ; (3.28) � @x R q.x; y; t/ � Z � 1 @ � � C � �1 D dk E.x;y;t/..I � R / � .I � R //E .x;y;t/ �.x; y; t; l/ : (3.29) � @x R The proposition follows from (3.27)–(3.29). 2

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