Integrable Dispersive Chains Maxim V. Pavlov Lebedev Institute of Physics 13.02.2014 Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 1 / 19
The General Problem. The Korteweg de Vries equation The Korteweg de Vries equation is associated with the linear Schrödinger equation ψ xx = ( λ + u ) ψ . The function ψ ( x , t , λ ) satisfies the pair of linear equations in partial derivatives ψ t = a ψ x − 1 ψ xx = u ψ , 2 a x ψ . Then the compatibility condition ( ψ xx ) t = ( ψ t ) xx yields the relationship � � − 1 2 ∂ 3 u t = x + 2 u ∂ x + u x a between functions u ( x , t , λ ) and a ( x , t , λ ) . Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 2 / 19
The General Problem. The Korteweg de Vries equation The Korteweg de Vries equation is associated with the linear Schrödinger equation ψ xx = ( λ + u ) ψ . The function ψ ( x , t , λ ) satisfies the pair of linear equations in partial derivatives ψ t = a ψ x − 1 ψ xx = u ψ , 2 a x ψ . Then the compatibility condition ( ψ xx ) t = ( ψ t ) xx yields the relationship � � − 1 2 ∂ 3 u t = x + 2 u ∂ x + u x a between functions u ( x , t , λ ) and a ( x , t , λ ) . If we choose the linear dependences u ( x , t , λ ) = λ + u 1 ( x , t ) and a ( x , t , λ ) = λ + a 1 ( x , t ) , we obtain nothing but the famous Korteweg de Vries equation t = 1 xxx − 3 u 1 4 u 1 2 u 1 u 1 x , Pavlov (FIAN & MSU) where a 1 = − 1 u 1 . Integrable Dispersive Chains 13.02.2014 2 / 19
The General Problem. The Kaup—Boussinesq system Again we consider the relationship � � − 1 2 ∂ 3 u t = x + 2 u ∂ x + u x a between functions u ( x , t , λ ) and a ( x , t , λ ) . Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 3 / 19
The General Problem. The Kaup—Boussinesq system Again we consider the relationship � � − 1 2 ∂ 3 u t = x + 2 u ∂ x + u x a between functions u ( x , t , λ ) and a ( x , t , λ ) . If we choose the quadratic dependence u ( x , t , λ ) = λ 2 + λ u 1 ( x , t ) + u 2 ( x , t ) and again the linear dependence a ( x , t , λ ) = λ + a 1 ( x , t ) , we obtain nothing but the well-known Kaup—Boussinesq system x − 3 t = 1 x − 1 u 1 t = u 2 2 u 1 u 1 u 2 4 u 1 xxx − u 2 u 1 2 u 1 u 2 x , x . Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 3 / 19
The General Problem. The Antonowicz—Fordy Construction Again we consider the relationship � � − 1 2 ∂ 3 u t = x + 2 u ∂ x + u x a between functions u ( x , t , λ ) and a ( x , t , λ ) . Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 4 / 19
The General Problem. The Antonowicz—Fordy Construction Again we consider the relationship � � − 1 2 ∂ 3 u t = x + 2 u ∂ x + u x a between functions u ( x , t , λ ) and a ( x , t , λ ) . Multi-component rational (with respect to the spectral parameter λ ) generalization ( ǫ k are arbitrary parameters) u ( x , t , λ ) = λ M u 0 ( x , t ) + λ M − 1 u 1 ( x , t ) + ... + u M ( x , t ) ǫ M λ M + ǫ M − 1 λ M − 1 + ... + ǫ 0 The authors considered two main subclasses selected by the conditions: ǫ M = 0 and u 0 = 1 (the so called “Generalized KdV type systems”); ǫ M = 0 but u 1 = 1 (the so called “Generalized Harry Dym type systems”). In another paper written together with M. Marvan we found a third narrow subclass determined by a sole restriction u M = 0. Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 4 / 19
Integrable Dispersive Chains Now we consider ( M = 1 , 2 , ... ) � � 1 + u 1 ( x , t ) + u 2 ( x , t ) + u 3 ( x , t ) u ( x , t , λ ) = λ M + ... , λ 2 λ 3 λ where u k are infinitely many unknown functions. Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 5 / 19
Integrable Dispersive Chains Now we consider ( M = 1 , 2 , ... ) � � 1 + u 1 ( x , t ) + u 2 ( x , t ) + u 3 ( x , t ) u ( x , t , λ ) = λ M + ... , λ 2 λ 3 λ where u k are infinitely many unknown functions. The substitution and the linear dependence a ( 1 ) = λ + a 1 ( x , t ) into � � − 1 2 ∂ 3 u t = x + 2 u ∂ x + u x a yields M th integrable dispersive chain − 1 x + 1 u k t = u k + 1 2 u 1 u k x − u k u 1 4 δ k M u 1 xxx , k = 1 , 2 , ..., x where δ k M is the Kronecker delta and a 1 = − 1 2 u 1 . Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 5 / 19
Higher Commuting Flows Higher commuting flows of the Korteweg de Vries hierarchy are determined by the linear spectral system ψ t k = a ( k ) ψ x − 1 2 a ( k ) ψ xx = ( λ + u 1 ) ψ , x ψ , where k a ( k ) = λ k + a m λ k − m , ∑ m = 1 and functions a m and u 1 depend on the “space” variable x and infinitely many extra “time” variables t k (obviously, t ≡ t 1 ). Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 6 / 19
Higher Commuting Flows Higher commuting flows of the Korteweg de Vries hierarchy are determined by the linear spectral system ψ t k = a ( k ) ψ x − 1 2 a ( k ) ψ xx = ( λ + u 1 ) ψ , x ψ , where k a ( k ) = λ k + a m λ k − m , ∑ m = 1 and functions a m and u 1 depend on the “space” variable x and infinitely many extra “time” variables t k (obviously, t ≡ t 1 ). Substitution � � 1 + u 1 ( x , t ) + u 2 ( x , t ) + u 3 ( x , t ) u ( x , t , λ ) = λ M + ... , λ 2 λ 3 λ into � � − 1 2 ∂ 3 u t = x + 2 u ∂ x + u x a Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 6 / 19
Higher Commuting Flows leads to higher commuting flows (here we define a 0 = 1) � � s u k + m ∂ x + ∂ x u k + m − 1 u k 2 δ k + m ∂ 3 ∑ t s = a s − m , s = 1 , 2 , ..., M x m = 0 Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 7 / 19
Higher Commuting Flows leads to higher commuting flows (here we define a 0 = 1) � � s u k + m ∂ x + ∂ x u k + m − 1 u k 2 δ k + m ∂ 3 ∑ t s = a s − m , s = 1 , 2 , ..., M x m = 0 where all coefficients a m can be found iteratively from the linear system (here we define u 0 = 1 and u − m = 0 for all m = 1 , 2 , ... ) � � s u m − k ∂ x + ∂ x u m − k − 1 2 δ m − k ∂ 3 ∑ a s − m = 0 , k = 0 , 1 , ..., s − 1 . x M m = 0 For instance, a 1 = − 1 a 2 = − 1 2 u 2 + 3 8 ( u 1 ) 2 − 1 a 3 = − 1 2 u 3 + 3 2 u 1 , 8 δ 1 M u 1 4 u 1 u 2 xx , − 5 16 ( u 1 ) 3 + 1 xx ) − 1 x ) 2 − u 1 32 δ 1 M ( 10 u 1 u 1 xx + 5 ( u 1 xxxx − 4 u 2 8 δ 2 M u 1 xx , ... Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 7 / 19
Higher Commuting Flows Thus all higher commuting flows are written also in an evolution form. For instance, the first commuting flow to − 1 x + 1 t = u k + 1 4 δ k u k 2 u 1 u k x − u k u 1 M u 1 xxx , k = 1 , 2 , ..., x is (here we identify y ≡ t 2 ) � � − 1 − 1 2 u 2 + 3 8 ( u 1 ) 2 − 1 u k y = u k + 2 2 u 1 u k + 1 8 δ 1 M u 1 u k x − u k + 1 u 1 + x x xx x � � x + 3 x − 1 + 1 + u k − u 2 2 u 1 u 1 4 δ 1 M u 1 4 δ k + 1 u 1 xxx xxx M � � + 1 xxx − 3 4 [( u 1 ) 2 ] xxx + 1 4 δ k u 2 4 δ 1 M u 1 . M xxxxx Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 8 / 19
Local Hamiltonian Structures A hierarchy of integrable dispersive chains − 1 x + 1 u k t = u k + 1 2 u 1 u k x − u k u 1 4 δ k M u 1 k = 1 , 2 , ..., xxx , x possesses infinitely many local Hamiltonian structures: � � δ H s + 1 s + 1 u k + m − 1 ∂ x + ∂ x u k + m − 1 − 1 u k 2 δ k + m − 1 ∂ 3 ∑ t s = δ u m ; M x m = 1 δ H s + 2 u 1 t s = − 2 ∂ x δ u 1 , � � δ H s + 2 s + 2 u k + m − 2 ∂ x + ∂ x u k + m − 2 − 1 u k 2 δ k + m − 2 ∂ 3 ∑ t s = δ u m ; x M m = 2 � � δ H s + 3 δ H s + 3 δ H s + 3 u 1 ∂ x + ∂ x u 1 -1 2 δ 1 u 1 δ u 2 , u 2 M ∂ 3 t s = -2 ∂ x t s = -2 ∂ x δ u 1 - δ u 2 , x � � δ H s + 3 s + 3 u k + m − 3 ∂ x + ∂ x u k + m − 3 − 1 2 δ k + m − 3 u k ∑ ∂ 3 t s = δ u m . M x m = 3 Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 9 / 19
Conservation Laws All higher local conservation laws can be found from the observation a m = δ H m + s , m = 0 , 1 , ...; s = 1 , 2 , ... δ u s In such a case all Hamiltonians can be found from above variation derivatives, for instance � � � � u 2 − 1 u 1 dx , 4 ( u 1 ) 2 H 1 = H 2 = dx , � � � u 3 − 1 2 u 1 u 2 + 1 8 ( u 1 ) 3 + 1 16 δ 1 M ( u 1 x ) 2 H 3 = dx , � � u 4 − 1 2 u 1 u 3 − 1 4 ( u 2 ) 2 + 3 8 ( u 1 ) 2 u 2 − 5 64 ( u 1 ) 4 H 4 = � � � + 1 x ) 2 − 1 + 1 xx ) 2 + 4 u 1 32 δ 1 − 5 u 1 ( u 1 2 ( u 1 x u 2 16 δ 2 M ( u 1 x ) 2 dx , ... M x Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 10 / 19
Elementary Reductions Obviously for any natural number N � M the reduction u N + 1 = 0 of M th dispersive chain − 1 x + 1 u k t = u k + 1 2 u 1 u k x − u k u 1 4 δ k M u 1 xxx , k = 1 , 2 , ..., x leads to N component integrable dispersive systems: Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 11 / 19
Elementary Reductions Obviously for any natural number N � M the reduction u N + 1 = 0 of M th dispersive chain − 1 x + 1 u k t = u k + 1 2 u 1 u k x − u k u 1 4 δ k M u 1 xxx , k = 1 , 2 , ..., x leads to N component integrable dispersive systems: 1 . N = M = 1, the Korteweg de Vries equation; Pavlov (FIAN & MSU) Integrable Dispersive Chains 13.02.2014 11 / 19
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