On the Model AC=BD and Trigram" Structures of the Soliton - PowerPoint PPT Presentation

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship On the Model AC=BD and “Trigram" Structures of the Soliton Theory Shou-Fu Tian joint work with Prof. Hong-Qing Zhang (My PhD Supervisor) School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P . R. China. Workshop on the tenth ASCM in Beijing, 2012

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Outline The problem of soliton theory 1 The model AC=BD and its applications 2 The model AC=BD Algebraic-geometry solution Sato theory The Trigram structures and Trigram identities 3 The Trigram structures and Trigram identities Exterior decomposition Trigram identities Interior decomposition Trigram identities The relationship between Tau function and Theta 4 function

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship It is well known that searching for soliton solutions of the nonlinear differential equations is one of the most important topics in soliton theory; therefore, various methods of finding soliton solutions have been developed. Some of the most important methods are

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship It is well known that searching for soliton solutions of the nonlinear differential equations is one of the most important topics in soliton theory; therefore, various methods of finding soliton solutions have been developed. Some of the most important methods are the inverse scattering transformation (IST), Algebraic geometry solution, Lie group method, Sato theory and Hirota’s bilinear method, Determinant technique, and so on.

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Question What is the unified and fundamental structure of all soliton equations?

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Question What is the unified and fundamental structure of all soliton equations? Recently, Hirota (2004) indicates that soliton solutions expressed by the Pfaffans and soliton equations (or the bilinear equations) are just equivalent to the Pfaffan identities.

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship In 1978, one of authors Zhang proposed his “AC=BD" theory to get general solutions of linear partial differential equations. After that, he extend the “AC=BD" theory to solve some nonlinear partial differential equations.

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship In 1978, one of authors Zhang proposed his “AC=BD" theory to get general solutions of linear partial differential equations. After that, he extend the “AC=BD" theory to solve some nonlinear partial differential equations. In this talk further investigate AC=BD theory. present compositively our “Trigram" theory. present a new method to construct the relationship between Tau- and Theta- function.

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD In this section, by virtue of Zhang (1978) and (2008), we recall the following results.

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD In this section, by virtue of Zhang (1978) and (2008), we recall the following results. Definition 2.1 Suppose X is a linear space and assume that A , B , C and D are some operators from X to X . For an arbitrary v ∈ X satisfying ACv = B · Dv (2.1) where AC ( v ) = A ( Cv ) , BD ( v ) = B · ( Dv ) , then ACv = B · Dv is called AC=B · D model .

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD Definition 2.2 For an arbitrary operator A , if there exist operators B , C , D , satisfying AC = B · D , C ker D =Ker A , where Ker A = { u | Au = 0 } , Ker D = { v | Dv = 0 } , then Au = 0 is called C-D integrable system . If C ker D � Ker A , then Au = 0 is called local integrable system .

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD Definition 2.2 For an arbitrary operator A , if there exist operators B , C , D , satisfying AC = B · D , C ker D =Ker A , where Ker A = { u | Au = 0 } , Ker D = { v | Dv = 0 } , then Au = 0 is called C-D integrable system . If C ker D � Ker A , then Au = 0 is called local integrable system . Theorem 2.3 Suppose X is a linear space and assume that A , B , C and D are some operators from X to X . If AC = B · D , B 0 = 0, C Ker D ⊇ Ker A , then u = Cv is a general solution of Au = 0 where v satisfies Dv = 0.

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD Theorem 2.4 Suppose X is a Banach space, AC : X → R is a function, and D is a monodromy operator. If ACD − 1 is Gateaux differential, and ACv = 0 is derived by Dv = 0, then there exist an operator B satisfying ACv = BvDv .

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution Trace formulas and Dubrovin equations Step 1. Au=0: Here we take KdV hierarchy for example, whose Lax equation and associate zero-curvature equation , respectively, given by ˙ L = [ L , M ] , U t n − V n + 1 , x + [ U , V n + 1 ] = 0 , (2.2) � � 0 1 where [ L , M ] = LM − ML , and U = U ( z ) = , − z + u 0 � � G n − 1 ( z ) F n ( z ) V n + 1 = V n + 1 ( z ) = . − H n + 1 ( z ) − G n − 1 ( z )

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution Step 2. u=Cv: Trace formula u = � 2 n m = 0 E m − 2 � n j = 1 µ j , where µ j are zeros of F n ( z ) , and E m are zeros of R 2 n + 1 ( z ) satisfying the hyperelliptic curve � 1 � F r ( z ) − 1 F r , x ( z ) 2 − ( u − z ) � F r , x , x ( z ) � � � K r : y 2 − R 2 r + 1 ( z ) = y 2 − F r ( z ) 2 2 4 of the KdV hierarchy. It is worth emphasizing that � 2 n m = 0 E m is an arbitrary constant by comparison of z 2 n for R 2 n + 1 ( z ) . From z = − 2 � n j = 1 µ j , we obtain n n 2 n � � � F r ( µ j ) + 1 µ j , x � � E m � ACv = − 2 µ j , t r + 2 F r , x , x , x ( µ j ) − 2 F r , x ( µ j ) . 2 j = 1 j = 1 m = 0

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution Step 3. Dv=0: Dubrovin equations � D x j = 2 � ( µ j − µ k ) − 1 , � B x j v = 0 : µ j , x = − 2 iy ( � µ j ) F r ( µ j ) , k = 1 k � = j � D t µ j , t n = − 2 i � ( µ j − µ k ) − 1 , � B t j v = 0 : F r ( µ j ) y ( � µ j ) j = − 2 , k = 1 k � = j where ( � µ j , � ν j ) is a new point in hyperelliptic curve K n , that is µ j ( x ) = ( µ j , iG n − 1 , x ( µ j ( x ) , x )) , j = 1 , . . . , n , � � ν j ( x ) = ( ν l , − iG n − 1 , x ( ν l ( x ) , x )) , l = 0 , . . . , n . (2.3) Now we obtain ACv = � n j v + � n j = 1 B x j vD x j = 1 B t j vD t j v , i.e., AC = � 2 n j = 1 B j D j .

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution Its-Matveev formula Step 1. Au=0: Here we take a generalized Lax equation for example given by ˙ L = [ L , M ] with its general form : N ( u , u t , u x 1 , . . . , u x N , . . . ) = 0 , (2.4) where X = ( x 1 , . . . , x N ) , N is a differential polynomial.

The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution Its-Matveev formula Step 1. Au=0: Here we take a generalized Lax equation for example given by ˙ L = [ L , M ] with its general form : N ( u , u t , u x 1 , . . . , u x N , . . . ) = 0 , (2.4) where X = ( x 1 , . . . , x N ) , N is a differential polynomial. Step 2. u=Cv: Its-Matveev formula: u = u 0 + a ∂ m Λ ln ϑ ( ξ ) , where u 0 is an initial solution, a is a constant, Λ = x m 1 1 x m 2 . . . x m N N , 2 m = m 1 + m 2 + · · · + m N , ϑ ( ξ ) = � ∞ n = −∞ e π i � τ n , n � + 2 π i � ξ, n � with ξ = ( ξ 1 , . . . , ξ n ) , ξ i = k i x 1 + l i x 2 + . . . + ρ i x N + ω i t + ε i , i = 1 , 2 , . . . , n , is a Riemann theta function.