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Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Methods of Solving Flag Partial Differential Equations Xiaoping Xu Institute of Mathematics Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing

  1. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs In representation theory, we are more interested in polynomial solutions of flag partial differential equations. How can we find polynomial solutions of a flag partial differential equation? Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  2. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 . Let B be a commutative associative algebra and let A be a free B -module generated by a filtrated subspace V = � ∞ r =0 V r (i.e., V r ⊂ V r +1 ). Let T 1 be a linear operator on B ⊕ A with a right inverse T − 1 such that T 1 ( B , A ) , T − 1 ( B , A ) ⊂ ( B , A ) , T 1 ( η 1 η 2 ) = T 1 ( η 1 ) η 2 , T − 1 ( η 1 η 2 ) = T − 1 ( η 1 ) η 2 for η 1 ∈ B , η 2 ∈ V . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  3. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 . Let B be a commutative associative algebra and let A be a free B -module generated by a filtrated subspace V = � ∞ r =0 V r (i.e., V r ⊂ V r +1 ). Let T 1 be a linear operator on B ⊕ A with a right inverse T − 1 such that T 1 ( B , A ) , T − 1 ( B , A ) ⊂ ( B , A ) , T 1 ( η 1 η 2 ) = T 1 ( η 1 ) η 2 , T − 1 ( η 1 η 2 ) = T − 1 ( η 1 ) η 2 for η 1 ∈ B , η 2 ∈ V . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  4. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 . Let B be a commutative associative algebra and let A be a free B -module generated by a filtrated subspace V = � ∞ r =0 V r (i.e., V r ⊂ V r +1 ). Let T 1 be a linear operator on B ⊕ A with a right inverse T − 1 such that T 1 ( B , A ) , T − 1 ( B , A ) ⊂ ( B , A ) , T 1 ( η 1 η 2 ) = T 1 ( η 1 ) η 2 , T − 1 ( η 1 η 2 ) = T − 1 ( η 1 ) η 2 for η 1 ∈ B , η 2 ∈ V . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  5. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 . Let B be a commutative associative algebra and let A be a free B -module generated by a filtrated subspace V = � ∞ r =0 V r (i.e., V r ⊂ V r +1 ). Let T 1 be a linear operator on B ⊕ A with a right inverse T − 1 such that T 1 ( B , A ) , T − 1 ( B , A ) ⊂ ( B , A ) , T 1 ( η 1 η 2 ) = T 1 ( η 1 ) η 2 , T − 1 ( η 1 η 2 ) = T − 1 ( η 1 ) η 2 for η 1 ∈ B , η 2 ∈ V . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  6. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 . Let B be a commutative associative algebra and let A be a free B -module generated by a filtrated subspace V = � ∞ r =0 V r (i.e., V r ⊂ V r +1 ). Let T 1 be a linear operator on B ⊕ A with a right inverse T − 1 such that T 1 ( B , A ) , T − 1 ( B , A ) ⊂ ( B , A ) , T 1 ( η 1 η 2 ) = T 1 ( η 1 ) η 2 , T − 1 ( η 1 η 2 ) = T − 1 ( η 1 ) η 2 for η 1 ∈ B , η 2 ∈ V . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  7. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 . Let B be a commutative associative algebra and let A be a free B -module generated by a filtrated subspace V = � ∞ r =0 V r (i.e., V r ⊂ V r +1 ). Let T 1 be a linear operator on B ⊕ A with a right inverse T − 1 such that T 1 ( B , A ) , T − 1 ( B , A ) ⊂ ( B , A ) , T 1 ( η 1 η 2 ) = T 1 ( η 1 ) η 2 , T − 1 ( η 1 η 2 ) = T − 1 ( η 1 ) η 2 for η 1 ∈ B , η 2 ∈ V . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  8. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 . Let B be a commutative associative algebra and let A be a free B -module generated by a filtrated subspace V = � ∞ r =0 V r (i.e., V r ⊂ V r +1 ). Let T 1 be a linear operator on B ⊕ A with a right inverse T − 1 such that T 1 ( B , A ) , T − 1 ( B , A ) ⊂ ( B , A ) , T 1 ( η 1 η 2 ) = T 1 ( η 1 ) η 2 , T − 1 ( η 1 η 2 ) = T − 1 ( η 1 ) η 2 for η 1 ∈ B , η 2 ∈ V . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  9. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Let T 2 be a linear operator on A such that T 2 ( V r +1 ) ⊂ B V r , T 2 ( f ζ ) = fT 2 ( ζ ) for r ∈ N , f ∈ B , ζ ∈ A . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  10. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Let T 2 be a linear operator on A such that T 2 ( V r +1 ) ⊂ B V r , T 2 ( f ζ ) = fT 2 ( ζ ) for r ∈ N , f ∈ B , ζ ∈ A . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  11. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Then we have { g ∈ A | ( T 1 + T 2 )( g ) = 0 } ∞ � ( − T − 1 T 2 ) i ( hg ) | g ∈ V , h ∈ B ; T 1 ( h ) = 0 } , = Span { i =0 where the summation is finite under our assumption. Moreover, the operator � ∞ i =0 ( − T − 1 T 2 ) i T − 1 is a right inverse of T 1 + T 2 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  12. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Then we have { g ∈ A | ( T 1 + T 2 )( g ) = 0 } ∞ � ( − T − 1 T 2 ) i ( hg ) | g ∈ V , h ∈ B ; T 1 ( h ) = 0 } , = Span { i =0 where the summation is finite under our assumption. Moreover, the operator � ∞ i =0 ( − T − 1 T 2 ) i T − 1 is a right inverse of T 1 + T 2 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  13. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Then we have { g ∈ A | ( T 1 + T 2 )( g ) = 0 } ∞ � ( − T − 1 T 2 ) i ( hg ) | g ∈ V , h ∈ B ; T 1 ( h ) = 0 } , = Span { i =0 where the summation is finite under our assumption. Moreover, the operator � ∞ i =0 ( − T − 1 T 2 ) i T − 1 is a right inverse of T 1 + T 2 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  14. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We remark that the above operator T 1 and T 2 may not commute. Take the notion i , j = { i , i + 1 , ..., j } for two integers i and j such that i ≤ j . Denote by N the additive semigroup of nonnegative integers. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  15. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We remark that the above operator T 1 and T 2 may not commute. Take the notion i , j = { i , i + 1 , ..., j } for two integers i and j such that i ≤ j . Denote by N the additive semigroup of nonnegative integers. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  16. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We remark that the above operator T 1 and T 2 may not commute. Take the notion i , j = { i , i + 1 , ..., j } for two integers i and j such that i ≤ j . Denote by N the additive semigroup of nonnegative integers. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  17. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Define x α = x α 1 1 x α 2 2 · · · x α n for α = ( α 1 , ..., α n ) ∈ N n . n Moreover, we denote i 1 , 0 , ..., 0) ∈ N n . ǫ i = (0 , ..., 0 , � For each i ∈ 1 , n , we define the linear operator ( x i ) on A by: � ( x α ) = x α + ǫ i α i + 1 for α ∈ N n . ( x i ) Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  18. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Define x α = x α 1 1 x α 2 2 · · · x α n for α = ( α 1 , ..., α n ) ∈ N n . n Moreover, we denote i 1 , 0 , ..., 0) ∈ N n . ǫ i = (0 , ..., 0 , � For each i ∈ 1 , n , we define the linear operator ( x i ) on A by: � ( x α ) = x α + ǫ i α i + 1 for α ∈ N n . ( x i ) Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  19. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Define x α = x α 1 1 x α 2 2 · · · x α n for α = ( α 1 , ..., α n ) ∈ N n . n Moreover, we denote i 1 , 0 , ..., 0) ∈ N n . ǫ i = (0 , ..., 0 , � For each i ∈ 1 , n , we define the linear operator ( x i ) on A by: � ( x α ) = x α + ǫ i α i + 1 for α ∈ N n . ( x i ) Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  20. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Define x α = x α 1 1 x α 2 2 · · · x α n for α = ( α 1 , ..., α n ) ∈ N n . n Moreover, we denote i 1 , 0 , ..., 0) ∈ N n . ǫ i = (0 , ..., 0 , � For each i ∈ 1 , n , we define the linear operator ( x i ) on A by: � ( x α ) = x α + ǫ i α i + 1 for α ∈ N n . ( x i ) Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  21. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Furthermore, we let m � (0) � ( m ) � �� � � � = 1 , = · · · ( x i ) ( x i ) ( x i ) ( x i ) and denote � ( α ) � ( α 1 ) � ( α 2 ) � ( α n ) ∂ α = ∂ α 1 x 1 ∂ α 2 x 2 · · · ∂ α n for α ∈ N n . x n , = · · · ( x 1 ) ( x 2 ) ( x n ) � ( α ) is a right inverse of ∂ α for α ∈ N n . We remark Obviously, � ( α ) ∂ α � = 1 if α � = 0 due to ∂ α (1) = 0. that Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  22. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Furthermore, we let m � (0) � ( m ) � �� � � � = 1 , = · · · ( x i ) ( x i ) ( x i ) ( x i ) and denote � ( α ) � ( α 1 ) � ( α 2 ) � ( α n ) ∂ α = ∂ α 1 x 1 ∂ α 2 x 2 · · · ∂ α n for α ∈ N n . x n , = · · · ( x 1 ) ( x 2 ) ( x n ) � ( α ) is a right inverse of ∂ α for α ∈ N n . We remark Obviously, � ( α ) ∂ α � = 1 if α � = 0 due to ∂ α (1) = 0. that Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  23. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Furthermore, we let m � (0) � ( m ) � �� � � � = 1 , = · · · ( x i ) ( x i ) ( x i ) ( x i ) and denote � ( α ) � ( α 1 ) � ( α 2 ) � ( α n ) ∂ α = ∂ α 1 x 1 ∂ α 2 x 2 · · · ∂ α n for α ∈ N n . x n , = · · · ( x 1 ) ( x 2 ) ( x n ) � ( α ) is a right inverse of ∂ α for α ∈ N n . We remark Obviously, � ( α ) ∂ α � = 1 if α � = 0 due to ∂ α (1) = 0. that Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  24. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Furthermore, we let m � (0) � ( m ) � �� � � � = 1 , = · · · ( x i ) ( x i ) ( x i ) ( x i ) and denote � ( α ) � ( α 1 ) � ( α 2 ) � ( α n ) ∂ α = ∂ α 1 x 1 ∂ α 2 x 2 · · · ∂ α n for α ∈ N n . x n , = · · · ( x 1 ) ( x 2 ) ( x n ) � ( α ) is a right inverse of ∂ α for α ∈ N n . We remark Obviously, � ( α ) ∂ α � = 1 if α � = 0 due to ∂ α (1) = 0. that Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  25. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Consider the wave equation in Riemannian space with a nontrivial conformal group: n � u tt − u x 1 x 1 − g i , j ( x 1 − t ) u x i x j = 0 , ( ∗ ) i , j =2 where we assume that g i , j ( z ) are one-variable polynomials. Change variables: z 0 = x 1 + t , z 1 = x 1 − t . Then ∂ 2 t = ( ∂ z 0 − ∂ z 1 ) 2 , ∂ 2 x 1 = ( ∂ z 0 + ∂ z 1 ) 2 . So the equation ( ∗ ) changes to: n � 2 ∂ z 0 ∂ z 1 + g i , j ( z 1 ) u x i x j = 0 . i , j =2 Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  26. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Consider the wave equation in Riemannian space with a nontrivial conformal group: n � u tt − u x 1 x 1 − g i , j ( x 1 − t ) u x i x j = 0 , ( ∗ ) i , j =2 where we assume that g i , j ( z ) are one-variable polynomials. Change variables: z 0 = x 1 + t , z 1 = x 1 − t . Then ∂ 2 t = ( ∂ z 0 − ∂ z 1 ) 2 , ∂ 2 x 1 = ( ∂ z 0 + ∂ z 1 ) 2 . So the equation ( ∗ ) changes to: n � 2 ∂ z 0 ∂ z 1 + g i , j ( z 1 ) u x i x j = 0 . i , j =2 Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  27. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Consider the wave equation in Riemannian space with a nontrivial conformal group: n � u tt − u x 1 x 1 − g i , j ( x 1 − t ) u x i x j = 0 , ( ∗ ) i , j =2 where we assume that g i , j ( z ) are one-variable polynomials. Change variables: z 0 = x 1 + t , z 1 = x 1 − t . Then ∂ 2 t = ( ∂ z 0 − ∂ z 1 ) 2 , ∂ 2 x 1 = ( ∂ z 0 + ∂ z 1 ) 2 . So the equation ( ∗ ) changes to: n � 2 ∂ z 0 ∂ z 1 + g i , j ( z 1 ) u x i x j = 0 . i , j =2 Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  28. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Consider the wave equation in Riemannian space with a nontrivial conformal group: n � u tt − u x 1 x 1 − g i , j ( x 1 − t ) u x i x j = 0 , ( ∗ ) i , j =2 where we assume that g i , j ( z ) are one-variable polynomials. Change variables: z 0 = x 1 + t , z 1 = x 1 − t . Then ∂ 2 t = ( ∂ z 0 − ∂ z 1 ) 2 , ∂ 2 x 1 = ( ∂ z 0 + ∂ z 1 ) 2 . So the equation ( ∗ ) changes to: n � 2 ∂ z 0 ∂ z 1 + g i , j ( z 1 ) u x i x j = 0 . i , j =2 Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  29. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Consider the wave equation in Riemannian space with a nontrivial conformal group: n � u tt − u x 1 x 1 − g i , j ( x 1 − t ) u x i x j = 0 , ( ∗ ) i , j =2 where we assume that g i , j ( z ) are one-variable polynomials. Change variables: z 0 = x 1 + t , z 1 = x 1 − t . Then ∂ 2 t = ( ∂ z 0 − ∂ z 1 ) 2 , ∂ 2 x 1 = ( ∂ z 0 + ∂ z 1 ) 2 . So the equation ( ∗ ) changes to: n � 2 ∂ z 0 ∂ z 1 + g i , j ( z 1 ) u x i x j = 0 . i , j =2 Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  30. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Consider the wave equation in Riemannian space with a nontrivial conformal group: n � u tt − u x 1 x 1 − g i , j ( x 1 − t ) u x i x j = 0 , ( ∗ ) i , j =2 where we assume that g i , j ( z ) are one-variable polynomials. Change variables: z 0 = x 1 + t , z 1 = x 1 − t . Then ∂ 2 t = ( ∂ z 0 − ∂ z 1 ) 2 , ∂ 2 x 1 = ( ∂ z 0 + ∂ z 1 ) 2 . So the equation ( ∗ ) changes to: n � 2 ∂ z 0 ∂ z 1 + g i , j ( z 1 ) u x i x j = 0 . i , j =2 Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  31. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Consider the wave equation in Riemannian space with a nontrivial conformal group: n � u tt − u x 1 x 1 − g i , j ( x 1 − t ) u x i x j = 0 , ( ∗ ) i , j =2 where we assume that g i , j ( z ) are one-variable polynomials. Change variables: z 0 = x 1 + t , z 1 = x 1 − t . Then ∂ 2 t = ( ∂ z 0 − ∂ z 1 ) 2 , ∂ 2 x 1 = ( ∂ z 0 + ∂ z 1 ) 2 . So the equation ( ∗ ) changes to: n � 2 ∂ z 0 ∂ z 1 + g i , j ( z 1 ) u x i x j = 0 . i , j =2 Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  32. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Denote n � T 1 = 2 ∂ z 0 ∂ z 1 , T 2 = g i , j ( z 1 ) ∂ x i ∂ x j . i , j =2 � � Take T − 1 = 1 ( z 1 ) , and 2 ( z 0 ) B = F [ z 0 , z 1 ] , V = F [ x 2 , ..., x n ] , V r = { f ∈ V | deg f ≤ r } . Then the conditions in Lemma 1 hold. Thus we have: Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  33. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Denote n � T 1 = 2 ∂ z 0 ∂ z 1 , T 2 = g i , j ( z 1 ) ∂ x i ∂ x j . i , j =2 � � Take T − 1 = 1 ( z 1 ) , and 2 ( z 0 ) B = F [ z 0 , z 1 ] , V = F [ x 2 , ..., x n ] , V r = { f ∈ V | deg f ≤ r } . Then the conditions in Lemma 1 hold. Thus we have: Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  34. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Denote n � T 1 = 2 ∂ z 0 ∂ z 1 , T 2 = g i , j ( z 1 ) ∂ x i ∂ x j . i , j =2 � � Take T − 1 = 1 ( z 1 ) , and 2 ( z 0 ) B = F [ z 0 , z 1 ] , V = F [ x 2 , ..., x n ] , V r = { f ∈ V | deg f ≤ r } . Then the conditions in Lemma 1 hold. Thus we have: Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  35. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Denote n � T 1 = 2 ∂ z 0 ∂ z 1 , T 2 = g i , j ( z 1 ) ∂ x i ∂ x j . i , j =2 � � Take T − 1 = 1 ( z 1 ) , and 2 ( z 0 ) B = F [ z 0 , z 1 ] , V = F [ x 2 , ..., x n ] , V r = { f ∈ V | deg f ≤ r } . Then the conditions in Lemma 1 hold. Thus we have: Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  36. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Theorem 2 . The space of all polynomial solutions for the equation ( ∗ ) is: ∞ n � � � � ( − 2) − m ( g i , j ( z 1 ) ∂ x i ∂ x j ) m ( f 0 g 0 + f 1 g 1 ) Span { ( z 0 ) ( z 1 ) m =0 i , j =2 | f 0 ∈ F [ z 0 ] , f 1 ∈ F [ z 1 ] , g 0 , g 1 ∈ F [ x 2 , ..., x n ] } Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  37. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Theorem 2 . The space of all polynomial solutions for the equation ( ∗ ) is: ∞ n � � � � ( − 2) − m ( g i , j ( z 1 ) ∂ x i ∂ x j ) m ( f 0 g 0 + f 1 g 1 ) Span { ( z 0 ) ( z 1 ) m =0 i , j =2 | f 0 ∈ F [ z 0 ] , f 1 ∈ F [ z 1 ] , g 0 , g 1 ∈ F [ x 2 , ..., x n ] } Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  38. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Let m 1 , m 2 , ..., m n be positive integers. According to Lemma 1, the set � � (( k 2 + ··· + k n ) m 1 ) ∞ � k 2 + · · · + k k � ( x ℓ 1 ( − 1) k 2 + ··· + k n { 1 ) k 2 , ..., k n ( x 1 ) k 2 ,..., k n =0 × ∂ k 2 m 2 ( x ℓ 2 2 ) · · · ∂ k n m n ( x ℓ n n ) | ℓ 1 ∈ 0 , m 1 − 1 , ℓ 2 , ..., ℓ n ∈ N } x 2 x n forms a basis of the space of polynomial solutions for the equation ( ∂ m 1 x 1 + ∂ m 2 x 2 + · · · + ∂ m n x n )( u ) = 0 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  39. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Let m 1 , m 2 , ..., m n be positive integers. According to Lemma 1, the set � � (( k 2 + ··· + k n ) m 1 ) ∞ � k 2 + · · · + k k � ( x ℓ 1 ( − 1) k 2 + ··· + k n { 1 ) k 2 , ..., k n ( x 1 ) k 2 ,..., k n =0 × ∂ k 2 m 2 ( x ℓ 2 2 ) · · · ∂ k n m n ( x ℓ n n ) | ℓ 1 ∈ 0 , m 1 − 1 , ℓ 2 , ..., ℓ n ∈ N } x 2 x n forms a basis of the space of polynomial solutions for the equation ( ∂ m 1 x 1 + ∂ m 2 x 2 + · · · + ∂ m n x n )( u ) = 0 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  40. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Let m 1 , m 2 , ..., m n be positive integers. According to Lemma 1, the set � � (( k 2 + ··· + k n ) m 1 ) ∞ � k 2 + · · · + k k � ( x ℓ 1 ( − 1) k 2 + ··· + k n { 1 ) k 2 , ..., k n ( x 1 ) k 2 ,..., k n =0 × ∂ k 2 m 2 ( x ℓ 2 2 ) · · · ∂ k n m n ( x ℓ n n ) | ℓ 1 ∈ 0 , m 1 − 1 , ℓ 2 , ..., ℓ n ∈ N } x 2 x n forms a basis of the space of polynomial solutions for the equation ( ∂ m 1 x 1 + ∂ m 2 x 2 + · · · + ∂ m n x n )( u ) = 0 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  41. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Let m 1 , m 2 , ..., m n be positive integers. According to Lemma 1, the set � � (( k 2 + ··· + k n ) m 1 ) ∞ � k 2 + · · · + k k � ( x ℓ 1 ( − 1) k 2 + ··· + k n { 1 ) k 2 , ..., k n ( x 1 ) k 2 ,..., k n =0 × ∂ k 2 m 2 ( x ℓ 2 2 ) · · · ∂ k n m n ( x ℓ n n ) | ℓ 1 ∈ 0 , m 1 − 1 , ℓ 2 , ..., ℓ n ∈ N } x 2 x n forms a basis of the space of polynomial solutions for the equation ( ∂ m 1 x 1 + ∂ m 2 x 2 + · · · + ∂ m n x n )( u ) = 0 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  42. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 indeed gives an algorithm of finding polynomial solutions for more general equations. Let f i ∈ R [ x 1 , ..., x i ] for i ∈ 1 , n − 1 . Consider the equation: ( ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f n − 1 ∂ m n x n )( u ) = 0 . Denote d 1 = ∂ m 1 x 1 , d r = ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f r − 1 ∂ m r for r ∈ 2 , n . x r Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  43. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 indeed gives an algorithm of finding polynomial solutions for more general equations. Let f i ∈ R [ x 1 , ..., x i ] for i ∈ 1 , n − 1 . Consider the equation: ( ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f n − 1 ∂ m n x n )( u ) = 0 . Denote d 1 = ∂ m 1 x 1 , d r = ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f r − 1 ∂ m r for r ∈ 2 , n . x r Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  44. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 indeed gives an algorithm of finding polynomial solutions for more general equations. Let f i ∈ R [ x 1 , ..., x i ] for i ∈ 1 , n − 1 . Consider the equation: ( ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f n − 1 ∂ m n x n )( u ) = 0 . Denote d 1 = ∂ m 1 x 1 , d r = ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f r − 1 ∂ m r for r ∈ 2 , n . x r Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  45. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 indeed gives an algorithm of finding polynomial solutions for more general equations. Let f i ∈ R [ x 1 , ..., x i ] for i ∈ 1 , n − 1 . Consider the equation: ( ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f n − 1 ∂ m n x n )( u ) = 0 . Denote d 1 = ∂ m 1 x 1 , d r = ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f r − 1 ∂ m r for r ∈ 2 , n . x r Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  46. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We will apply Lemma 1 with T 1 = d r , T 2 = � n − 1 i = r f i ∂ m i +1 x i +1 � ( m 1 ) inductively. Take a right inverse d − 1 = ( x 1 ) . Suppose that we have found a right inverse d − s of d s for some s ∈ 1 , n − 1 such that x i d − s = d − s x i , ∂ x i d − s = d − s ∂ x i for i ∈ s + 1 , n ( ∗∗ ) Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  47. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We will apply Lemma 1 with T 1 = d r , T 2 = � n − 1 i = r f i ∂ m i +1 x i +1 � ( m 1 ) inductively. Take a right inverse d − 1 = ( x 1 ) . Suppose that we have found a right inverse d − s of d s for some s ∈ 1 , n − 1 such that x i d − s = d − s x i , ∂ x i d − s = d − s ∂ x i for i ∈ s + 1 , n ( ∗∗ ) Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  48. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We will apply Lemma 1 with T 1 = d r , T 2 = � n − 1 i = r f i ∂ m i +1 x i +1 � ( m 1 ) inductively. Take a right inverse d − 1 = ( x 1 ) . Suppose that we have found a right inverse d − s of d s for some s ∈ 1 , n − 1 such that x i d − s = d − s x i , ∂ x i d − s = d − s ∂ x i for i ∈ s + 1 , n ( ∗∗ ) Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  49. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We will apply Lemma 1 with T 1 = d r , T 2 = � n − 1 i = r f i ∂ m i +1 x i +1 � ( m 1 ) inductively. Take a right inverse d − 1 = ( x 1 ) . Suppose that we have found a right inverse d − s of d s for some s ∈ 1 , n − 1 such that x i d − s = d − s x i , ∂ x i d − s = d − s ∂ x i for i ∈ s + 1 , n ( ∗∗ ) Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  50. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 enable us to take ∞ � d − ( − d − s f s ) i d − s ∂ im s +1 s +1 = x s +1 i =0 as a right inverse of d s +1 . Obviously, x i d − s +1 = d − s +1 x i , ∂ x i d − s +1 = d − for i ∈ s + 2 , n . s +1 ∂ x i By induction, we have found a right inverse d − s of d s such that ( ∗∗ ) holds for each s ∈ 1 , n . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  51. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 enable us to take ∞ � d − ( − d − s f s ) i d − s ∂ im s +1 s +1 = x s +1 i =0 as a right inverse of d s +1 . Obviously, x i d − s +1 = d − s +1 x i , ∂ x i d − s +1 = d − for i ∈ s + 2 , n . s +1 ∂ x i By induction, we have found a right inverse d − s of d s such that ( ∗∗ ) holds for each s ∈ 1 , n . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  52. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Lemma 1 enable us to take ∞ � d − ( − d − s f s ) i d − s ∂ im s +1 s +1 = x s +1 i =0 as a right inverse of d s +1 . Obviously, x i d − s +1 = d − s +1 x i , ∂ x i d − s +1 = d − for i ∈ s + 2 , n . s +1 ∂ x i By induction, we have found a right inverse d − s of d s such that ( ∗∗ ) holds for each s ∈ 1 , n . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  53. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We set S r = { g ∈ R [ x 1 , ..., x r ] | d r ( g ) = 0 } for r ∈ 1 , k . Then m 1 − 1 � R x i S 1 = 1 . i =0 Suppose that we have found S r for some r ∈ 1 , n − 1. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  54. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We set S r = { g ∈ R [ x 1 , ..., x r ] | d r ( g ) = 0 } for r ∈ 1 , k . Then m 1 − 1 � R x i S 1 = 1 . i =0 Suppose that we have found S r for some r ∈ 1 , n − 1. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  55. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs We set S r = { g ∈ R [ x 1 , ..., x r ] | d r ( g ) = 0 } for r ∈ 1 , k . Then m 1 − 1 � R x i S 1 = 1 . i =0 Suppose that we have found S r for some r ∈ 1 , n − 1. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  56. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Given h ∈ S r and ℓ ∈ N , we define ∞ � ( − d − r f r ) i ( h ) ∂ im r +1 x r +1 ( x ℓ σ r +1 ,ℓ ( h ) = r +1 ) , i =0 which is actually a finite summation. Lemma 1 says ∞ � S r +1 = σ r +1 ,ℓ ( S r ) . ℓ =0 By induction, we obtain: Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  57. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Given h ∈ S r and ℓ ∈ N , we define ∞ � ( − d − r f r ) i ( h ) ∂ im r +1 x r +1 ( x ℓ σ r +1 ,ℓ ( h ) = r +1 ) , i =0 which is actually a finite summation. Lemma 1 says ∞ � S r +1 = σ r +1 ,ℓ ( S r ) . ℓ =0 By induction, we obtain: Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  58. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Given h ∈ S r and ℓ ∈ N , we define ∞ � ( − d − r f r ) i ( h ) ∂ im r +1 x r +1 ( x ℓ σ r +1 ,ℓ ( h ) = r +1 ) , i =0 which is actually a finite summation. Lemma 1 says ∞ � S r +1 = σ r +1 ,ℓ ( S r ) . ℓ =0 By induction, we obtain: Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  59. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Given h ∈ S r and ℓ ∈ N , we define ∞ � ( − d − r f r ) i ( h ) ∂ im r +1 x r +1 ( x ℓ σ r +1 ,ℓ ( h ) = r +1 ) , i =0 which is actually a finite summation. Lemma 1 says ∞ � S r +1 = σ r +1 ,ℓ ( S r ) . ℓ =0 By induction, we obtain: Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  60. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Given h ∈ S r and ℓ ∈ N , we define ∞ � ( − d − r f r ) i ( h ) ∂ im r +1 x r +1 ( x ℓ σ r +1 ,ℓ ( h ) = r +1 ) , i =0 which is actually a finite summation. Lemma 1 says ∞ � S r +1 = σ r +1 ,ℓ ( S r ) . ℓ =0 By induction, we obtain: Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  61. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Theorem 3 . The set { σ n ,ℓ n σ n − 1 ,ℓ n − 1 · · · σ 2 ,ℓ 2 ( x ℓ 1 1 ) | ℓ 1 ∈ 0 , m 1 − 1 , ℓ 2 , ..., ℓ n ∈ N } forms a basis of the polynomial solution space S n of the partial differential equation: ( ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f n − 1 ∂ m n x n )( u ) = 0 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  62. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Theorem 3 . The set { σ n ,ℓ n σ n − 1 ,ℓ n − 1 · · · σ 2 ,ℓ 2 ( x ℓ 1 1 ) | ℓ 1 ∈ 0 , m 1 − 1 , ℓ 2 , ..., ℓ n ∈ N } forms a basis of the polynomial solution space S n of the partial differential equation: ( ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f n − 1 ∂ m n x n )( u ) = 0 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  63. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Theorem 3 . The set { σ n ,ℓ n σ n − 1 ,ℓ n − 1 · · · σ 2 ,ℓ 2 ( x ℓ 1 1 ) | ℓ 1 ∈ 0 , m 1 − 1 , ℓ 2 , ..., ℓ n ∈ N } forms a basis of the polynomial solution space S n of the partial differential equation: ( ∂ m 1 x 1 + f 1 ∂ m 2 x 2 + · · · + f n − 1 ∂ m n x n )( u ) = 0 . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  64. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Evolution Equations First we want to solve the following evolution partial differential equation: 2 ∂ x 3 + · · · + x m n − 1 u t = ( ∂ x 1 + x m 1 1 ∂ x 2 + x m 2 n − 1 ∂ x n )( u ) subject to the condition: u (0 , x 1 , ..., x n ) = f ( x 1 , ..., x n ) , where f ( x 1 , x 2 , ..., x n ) is a smooth function. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  65. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Evolution Equations First we want to solve the following evolution partial differential equation: 2 ∂ x 3 + · · · + x m n − 1 u t = ( ∂ x 1 + x m 1 1 ∂ x 2 + x m 2 n − 1 ∂ x n )( u ) subject to the condition: u (0 , x 1 , ..., x n ) = f ( x 1 , ..., x n ) , where f ( x 1 , x 2 , ..., x n ) is a smooth function. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  66. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Evolution Equations First we want to solve the following evolution partial differential equation: 2 ∂ x 3 + · · · + x m n − 1 u t = ( ∂ x 1 + x m 1 1 ∂ x 2 + x m 2 n − 1 ∂ x n )( u ) subject to the condition: u (0 , x 1 , ..., x n ) = f ( x 1 , ..., x n ) , where f ( x 1 , x 2 , ..., x n ) is a smooth function. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  67. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Evolution Equations First we want to solve the following evolution partial differential equation: 2 ∂ x 3 + · · · + x m n − 1 u t = ( ∂ x 1 + x m 1 1 ∂ x 2 + x m 2 n − 1 ∂ x n )( u ) subject to the condition: u (0 , x 1 , ..., x n ) = f ( x 1 , ..., x n ) , where f ( x 1 , x 2 , ..., x n ) is a smooth function. Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  68. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Theoretically, the solution is u = e t ( ∂ x 1 + P n − 1 r =1 x mr ∂ xr +1 ) ( f ) . r Practically, we often need an exact closed formula of the solution! Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  69. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Theoretically, the solution is u = e t ( ∂ x 1 + P n − 1 r =1 x mr ∂ xr +1 ) ( f ) . r Practically, we often need an exact closed formula of the solution! Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  70. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Theoretically, the solution is u = e t ( ∂ x 1 + P n − 1 r =1 x mr ∂ xr +1 ) ( f ) . r Practically, we often need an exact closed formula of the solution! Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  71. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs For convenience, we denote m 0 = 1 and x 0 = 1. Set i − 1 � x m i D i = t for i ∈ 1 , n . ∂ x i +1 i r =0 Denote A = D n , B = − tx m n − 1 n − 1 ∂ x n . Thus D n − 1 = D n + B = A + B . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  72. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs For convenience, we denote m 0 = 1 and x 0 = 1. Set i − 1 � x m i D i = t for i ∈ 1 , n . ∂ x i +1 i r =0 Denote A = D n , B = − tx m n − 1 n − 1 ∂ x n . Thus D n − 1 = D n + B = A + B . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  73. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs For convenience, we denote m 0 = 1 and x 0 = 1. Set i − 1 � x m i D i = t for i ∈ 1 , n . ∂ x i +1 i r =0 Denote A = D n , B = − tx m n − 1 n − 1 ∂ x n . Thus D n − 1 = D n + B = A + B . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  74. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs For convenience, we denote m 0 = 1 and x 0 = 1. Set i − 1 � x m i D i = t for i ∈ 1 , n . ∂ x i +1 i r =0 Denote A = D n , B = − tx m n − 1 n − 1 ∂ x n . Thus D n − 1 = D n + B = A + B . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  75. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs In our special case, the Campbell-Hausdorff formula becomes ∞ � ln e A e B = A + B + a r (ad A ) r ( B ) , a r ∈ R , r =1 equivalently, e A e B = e A + P ∞ i =0 a i ( ad A ) i ( B ) . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  76. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs In our special case, the Campbell-Hausdorff formula becomes ∞ � ln e A e B = A + B + a r (ad A ) r ( B ) , a r ∈ R , r =1 equivalently, e A e B = e A + P ∞ i =0 a i ( ad A ) i ( B ) . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  77. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs In our special case, the Campbell-Hausdorff formula becomes ∞ � ln e A e B = A + B + a r (ad A ) r ( B ) , a r ∈ R , r =1 equivalently, e A e B = e A + P ∞ i =0 a i ( ad A ) i ( B ) . Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  78. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Denote � 0 ∞ ϑ ( x ) = 1 − e − x ( − 1) i − 1 � e yx dy = x i − 1 . = x i ! − 1 i =1 After a long calculation, we obtain mn − 1 e D n = e D n − 1 e t ϑ ( D n − 1 )( x ) ∂ xn . n − 1 Xiaoping Xu Methods of Solving Flag Partial Differential Equations

  79. Polynomial Solutions Evolution Equations Constant-Coefficient PDEs Denote � 0 ∞ ϑ ( x ) = 1 − e − x ( − 1) i − 1 � e yx dy = x i − 1 . = x i ! − 1 i =1 After a long calculation, we obtain mn − 1 e D n = e D n − 1 e t ϑ ( D n − 1 )( x ) ∂ xn . n − 1 Xiaoping Xu Methods of Solving Flag Partial Differential Equations

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