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Because of their exceptional properties the non-Noether symmetries could be effectively used in analysis of Hamiltonian dynamical systems. From the geometric point of view these symmetries are important because of their tight relationship with geometric structures on phase space such as bi-Hamiltonian structures, Frölicher-Nijenhuis operators, Lax pairs and bicomplexes [1]. The correspondence between non-Noether symmetries and conservation laws is also interesting and in regular Hamiltonian systems on 2n dimensional Poisson manifold up to n integrals of motion could be associated with each generator of non-Noether symmetry [1] [3]. As a result non-Noether symmetries could be especially useful in analysis of Hamiltonian systems with many degrees of freedom, as well as infinite dimensional Hamiltonian systems, where large (and even infinite) number of conservation laws could be constructed from the single generator of such a symmetry. Under certain conditions satisfied by the symmetry generator these conservation laws appear to be involutive and ensure integrability of the dynamical system.

Before we consider these models in detail we briefly remind some basic facts concerning symmetries of Hamiltonian systems. Since throughout the article continuous one-parameter groups of symmetries play central role let us remind that each vector field E on the phase space M of the Hamiltonian dynamical system defines continuous one-parameter group of transformations (flow)

g_a = e^aL_E

where L_E denotes Lie derivative along the vector field E. Action of this group on observables (smooth functions on M) is given by expansion

g_a(f) = e^aL_E(f) = f + aL_Ef + ½a²L_E²f + ...

Further it will be assumed that M is 2n dimensional symplectic manifold and the group of transformations g_a will be called symmetry of Hamiltonian system if it preserves manifold of solutions of Hamilton's equation

(1) d dt f = {h , f}

(here { , } denotes Poisson bracket defined in a standard manner by Poisson bivector field {f , g} = W(df ∧ dg) and h is smooth function on M called Hamiltonian) or in other words if for each f satisfying Hamilton's equation g_a(f) also satisfies it. This happens when g_a commutes with time evolution operator

d dt g_a(f) = g_a( d dt f)

If in addition the generator E of the group g_a does not preserve Poisson bracket structure [E , W] ≠ 0 then the g_a is called non-Noether symmetry. Let us briefly recall some basic features of non-Noether symmetries. First of all if E generates non-Noether symmetry then the n functions

(2) Y_k = i_W^k(L_Eω)^k k = 1,2, ... n

(where ω is symplectic form obtained by inverting Poisson bivector W and i denotes contraction) are integrals of motion (see [1] [3]) and if additionally the symmetry generator E satisfies Yang-Baxter equation

(3) [[E[E , W]]W] = 0

these conservation laws Y_k appear to be in involution {Y_k, Y_m} = 0 while the bivector fields W and [E , W] (or in terms of 2-forms ω and L_Eω) form bi-Hamiltonian system (see [1]). Due to this features non-Noether symmetries could be effectively used in construction of conservation laws and bi-Hamiltonian structures.

Now let us focus on non-Noether symmetry of the Toda model – 2n dimensional Hamiltonian system that describes the motion of n particles on the line governed by the exponential interaction. Equations of motion of the non periodic n-particle Toda model are

(4) d dt q_i = p_i
d dt p_i = ε(i − 1)e^{q_{i − 1} − q_i} − ε(n − i)e^{q_i − q_{i + 1}}

(ε(k) = − ε(− k) = 1 for any natural k and ε(0) = 0) and could be rewritten in Hamiltonian form (1) with canonical Poisson bracket defined by

W = n Σ i = 1 ∂ ∂p_i ∧ ∂ ∂q_i

corresponding symplectic form

ω = n Σ i = 1 dp_i ∧ dq_i

and Hamiltonian equal to

h = 1 2 n Σ i = 1 p_i² + n − 1 Σ i = 1 e^{q_i − q_{i + 1}}

The group of transformations g_a generated by the vector field E will be symmetry of Toda chain if for each p_i, q_i satisfying Toda equations (4) g_a(p_i), g_a(q_i) also satisfy it. Substituting infinitesimal transformations

g_a(p_i) = p_i + aE(p_i) + O(a²)
g_a(p_i) = q_i + aE(q_i) + O(a²)

into (4) and grouping first order terms gives rise to the conditions

(5) d dt E(q_i) = E(p_i)
d dt E(p_i) = ε(i − 1)e^{q_{i − 1} − q_i} (E(q_{i − 1}) − E(q_i)) − ε(n − i)e^{q_i − q_{i + 1}} (E(q_i) − E(q_{i + 1}))

One can verify that the vector field defined by

(6) E(p_i) = 1 2 p_i² + ε(i − 1)(n − i + 2) e^{q_{i − 1} − q_i} − ε(n − i)(n − i) e^{q_i − q_{i + 1}} +
t 2 (ε(i − 1)(p_{i − 1} + p_i) e^{q_{i − 1} − q_i} − ε(n − i)(p_i + p_{i + 1}) e^{q_i − q_{i + 1}}
E(q_i) = (n − i + 1)p_i − 1 2 i − 1 Σ k = 1 p_k + 1 2 n Σ k = i + 1 p_k +
t 2 (p_i² + ε(i − 1)e^{q_{i − 1} − q_i} + ε(n − i)e^{q_i − q_{i + 1}})

satisfies (5) and generates symmetry of Toda chain. It appears that this symmetry is non-Noether since it does not preserve Poisson bracket structure [E , W] ≠ 0 and additionally one can check that Yang-Baxter equation [[E[E , W]]W] = 0 is satisfied. This symmetry could play important role in analysis of Toda model. First let us note that calculating L_Eω leads to the following 2-form

L_Eω = n Σ i = 1 p_idp_i ∧ dq_i + n − 1 Σ i = 1 e^{q_i − q_{i + 1}} dq_i ∧ q_{i + 1} + Σ i < j dp_i ∧ dp_j

and together ω and L_Eω give rise to bi-Hamiltonian structure of Toda model (compare with [2]). The conservation laws (2) associated with the symmetry reproduce well known set of conservation laws of Toda chain.

I₁ = Y₁ = n Σ i = 1 p₁ + p₂
I₂ = 1 2 Y₁² − Y₂ = 1 2 n Σ i = 1 p_i² + n − 1 Σ i = 1 e^{q_i − q_{i + 1}}
I₃ = 1 3 Y₁³ − Y₁Y₂ + Y₃ = 1 3 n Σ i = 1 p_i³ + n − 1 Σ i = 1 (p_i + p_{i + 1}) e^{q_i − q_{i + 1}}
I₄ = 1 4 Y₁⁴ − Y₁²Y₂ + 1 2 Y₂² + Y₁Y₃ − Y₄ =
1 4 n Σ i = 1 p_i⁴ + n − 1 Σ i = 1 (p_i² + 2p_ip_{i + 1} + p_{i + 1}²) e^{q_i − q_{i + 1}} + 1 2 n − 1 Σ i = 1 e^{2(q_i − q_{i + 1})} + n − 2 Σ i = 1 e^{q_i − q_{i + 2}}
I_m = (− 1)^mY_m + m^{− 1} m − 1 Σ k = 1 (− 1)^kI_{m − k}Y_k

The condition [[E[E , W]]W] = 0 satisfied by generator of the symmetry E ensures that the conservation laws are in involution i. e. {Y_k,Y_m} = 0. Thus the conservation laws as well as the bi-Hamiltonian structure of the non periodic Toda chain appear to be associated with non-Noether symmetry.

Unlike the Toda model the dynamical systems in our next examples are infinite dimensional and in order to ensure integrability one should construct infinite number of conservation laws. Fortunately in several integrable models this task could be effectively done by identifying appropriate non-Noether symmetry. First let us consider well known infinite dimensional integrable Hamiltonian system – nonlinear Schrödinger equation (NSE)

ψ_t = i(ψ_xx + 2ψ²ψ)

where ψ is a smooth complex function of (t, x) ∈ R². On this stage we will not specify any boundary conditions and will just focus on symmetries of NSE. Supposing that the vector field E generates the symmetry of NSE one gets the following restriction

(7) E(ψ)_t = i[E(ψ)_xx + 2ψ²E(ψ) + 4ψψE(ψ)]

(obtained by substituting infinitesimal transformation ψ → ψ + aE(ψ) + O(a²) generated by E into NSE). It appears that NSE possesses nontrivial symmetry that is generated by the vector field

E(ψ) = i(ψ_x + x 2 ψ_xx + ψφ + xψ²ψ) − t(ψ_xxx + 6ψψψ_x)

(here φ is defined by φ_x = ψψ). In order to construct conservation laws we also need to know Poisson bracket structure and it appears that invariant Poisson bivector field could be defined if ψ is subjected to either periodic ψ(t, − ∞) = ψ(t, + ∞) or zero ψ(t, − ∞) = ψ(t, + ∞) = 0 boundary conditions. In terms of variational derivatives the explicit form of the Poisson bivector field is

W = i + ∞ ∫ − ∞ dx δ δψ ∧ δ δψ

while corresponding symplectic form obtained by inverting W is

ω = i + ∞ ∫ − ∞ dx δψ ∧ δψ

Now one can check that NSE could be rewritten in Hamiltonian form

ψ_t = {h , ψ}

with Poisson bracket { , } defined by W and

h = + ∞ ∫ − ∞ dx (ψ²ψ² − ψ_xψ_x)

Knowing the symmetry of NSE that appears to be non-Noether ([E, W] ≠ 0) one can construct bi-Hamiltonian structure and conservation laws. First let us calculate Lie derivative of symplectic form along the symmetry generator

L_Eω = + ∞ ∫ − ∞ [δψ_x ∧ δψ + ψδφ ∧ δψ + ψδφ ∧ δψ]dx

The couple of 2-forms ω and L_Eω exactly reproduces the bi-Hamiltonian structure of NSE proposed by Magri [4] while the conservation laws associated with this symmetry are well known conservation laws of NSE

I₁ = Y₁ = 2 + ∞ ∫ − ∞ ψψ dx
I₂ = Y₁² − 2Y₂ = i + ∞ ∫ − ∞ (ψ_xψ − ψ_xψ) dx
I₃ = Y₁³ − 3Y₁Y₂ + 3Y₃ = 2 + ∞ ∫ − ∞ (ψ²ψ² − ψ_xψ_x) dx
I₄ = Y₁⁴ − 4Y₁²Y₂ + 2Y₂² + 4Y₁Y₃ − 4Y₄ =
+ ∞ ∫ − ∞ [i(ψ_xψ_xx − ψ_xψ_xx) + 3i(ψψ²ψ_x − ψψ²ψ_x)] dx
I_m = (− 1)^mmY_m + m − 1 Σ k = 1 (− 1)^kI_{m − k}Y_k

The involutivity of the conservation laws of NSE {Y_k, Y_m} = 0 is related to the fact that E satisfies Yang-Baxter equation [[E[E , W]]W] = 0.

Now let us consider other important integrable models – Korteweg-de Vries equation (KdV) and modified Korteweg-de Vries equation (mKdV). Here symmetries are more complicated but generator of the symmetry still can be identified and used in construction of conservation laws. The KdV and mKdV equations have the following form

u_t + u_xxx + uu_x = 0 [KdV]

and

u_t + u_xxx − 6u²u_x = 0 [mKdV]

(here u is smooth function of (t, x) ∈ R²). The generators of symmetries of KdV and mKdV should satisfy conditions

E(u)_t + E(u)_xxx + u_xE(u) + uE(u)_x = 0 [KdV]

and

E(u)_t + E(u)_xxx − 12uu_xE(u) − 6u²E(u)_x = 0 [mKdV]

(again this conditions are obtained by substituting infinitesimal transformation u → u + aE(u) + O(a²) into KdV and mKdV, respectively). Further we will focus on the symmetries generated by the following vector fields

E(u) = 1 2 u_xx + 1 6 u² + 1 24 u_xv + x 8 (u_xxx + uu_x) −
t 16 (6u_xxxxx + 20u_xu_xx + 10 uu_xxx + 5u²u_x) [KdV]

and

E(u) = − 3 2 u_xx + 2u³ + u_xw − x 2 (u_xxx − 6u²u_x) −
3t 2 (u_xxxxx − 10u²u_xxx − 40uu_xu_xx − 10u_x³ + 30u⁴u_x) [mKdV]

(here v and w are defined by v_x = u and w_x = u²) To construct conservation laws we need to know Poisson bracket structure and again like in the case of NSE the Poisson bivector field is well defined when u is subjected to either periodic u(t, − ∞) = u(t, + ∞) or zero u(t, − ∞) = u(t, + ∞) = 0 boundary conditions. For both KdV and mKdV the Poisson bivector field is

W = + ∞ ∫ − ∞ dx δ δu ∧ δ δv

with corresponding symplectic form

ω = + ∞ ∫ − ∞ dx δu ∧ δv

leading to Hamiltonian realization of KdV and mKdV equations

u_t = {h , u}

with Hamiltonians

h = + ∞ ∫ − ∞ (u_x² − u³ 3 ) dx [KdV]

and

h = + ∞ ∫ − ∞ (u_x² + u⁴) dx [mKdV]

By taking Lie derivative of the symplectic form along the generators of the symmetries one gets another couple of symplectic forms

L_Eω = + ∞ ∫ − ∞ dx (δu ∧ δu_x + 2 3 uδu ∧ δv) [KdV]

L_Eω = + ∞ ∫ − ∞ dx (δu ∧ δu_x − 2uδu ∧ δw) [mKdV]

involved in bi-Hamiltonian realization of KdV/mKdV hierarchies and proposed by Magri [4]. The conservation laws associated with the symmetries reproduce infinite sequence of conservation laws of KdV equation

I₁ = Y₁ = 2 3 + ∞ ∫ − ∞ u dx
I₂ = Y₁ − 2Y₂ = 4 9 + ∞ ∫ − ∞ u² dx
I₃ = Y₁³ − 3Y₁Y₂ + 3Y₃ = 8 9 + ∞ ∫ − ∞ ( u³ 3 − u_x²) dx
I₄ = Y₁⁴ − 4Y₁²Y₂ + 2Y₂² + 4Y₁Y₃ − 4Y₄ =
64 45 + ∞ ∫ − ∞ ( 5 36 u⁴ − 5 3 uu_x² + u_xx²) dx
I_m = (− 1)^mmY_m + m − 1 Σ k = 1 (− 1)^kI_{m − k}Y_k

and mKdV equation

I₁ = Y₁ = − 4 + ∞ ∫ − ∞ u² dx
I₂ = Y₁ − 2Y₂ = 16 + ∞ ∫ − ∞ (u⁴ + u_x²) dx
I₃ = Y₁³ − 3Y₁Y₂ + 3Y₃ = − 32 + ∞ ∫ − ∞ (2u⁶ + 10 u²u_x² + u_xx²) dx
I₄ = Y₁⁴ − 4Y₁²Y₂ + 2Y₂² + 4Y₁Y₃ − 4Y₄ =
256 5 + ∞ ∫ − ∞ (5 u⁸ + 70u⁴u_x² − 7u_x⁴ + 14u²u_xx² + u_xxx²) dx
I_m = (− 1)^mmY_m + m − 1 Σ k = 1 (− 1)^kI_{m − k}Y_k

The involutivity of these conservation laws is well known and in terms of the symmetry generators it is ensured by conditions [[E[E , W]]W] = 0. Thus the conservation laws and bi-Hamiltonian structures of KdV and mKdV hierarchies are related to the non-Noether symmetries of KdV and mKdV equations.

Non-Noether symmetries in integrable models

References