Non-Noether symmetries in integrable models

Because of their exceptional properties the non-Noether symmetries could beeffectively used in analysis of Hamiltonian dynamical systems.From the geometric point of view these symmetries are importantbecause of their tight relationship with geometric structures on phase spacesuch as bi-Hamiltonian structures, Frölicher-Nijenhuis operators,Lax pairs and bicomplexes [1]. The correspondencebetween non-Noether symmetries and conservation laws is also interesting andin regular Hamiltonian systems on 2n dimensional Poisson manifoldup to n integrals of motion could be associated with each generatorof non-Noether symmetry [1] [3].As a result non-Noether symmetries could be especially useful in analysis ofHamiltonian systems with many degrees of freedom, as well as infinite dimensionalHamiltonian systems, where large (and even infinite) number of conservation lawscould be constructed from thesingle generator of such a symmetry. Under certain conditions satisfied by thesymmetry generator these conservation laws appear to be involutive and ensureintegrability of the dynamical system.

The n-particle non periodic Toda model is one of integrable modelsthat possesses such a nontrivial symmetry. In this model non-Noether symmetry(which is one-parameter group of noncannonical transformations)yields conservation laws that appear to be functionally independent,involutive and ensure the integrability of this dynamical system.Well known bi-Hamiltonian realizationof the Toda model is also related to this symmetry.

Nonlinear Schrödinger equation is another important examplewhere symmetry (again one-parameter group) leads to the infinite sequence ofconservation laws in involution. The KdV and mKdV equations also possessnon-Noether symmetries which are quite nontrivial (but symmetry group isstill one-parameter) and in each model the infinite set of conservation laws isassociated with the single generator of the symmetry.

Before we consider these models in detail we briefly remind some basic factsconcerning symmetries of Hamiltonian systems. Since throughout the articlecontinuous one-parameter groups of symmetries play central role let us remind thateach vector field E on the phase space M of theHamiltonian dynamical system defines continuous one-parameter group oftransformations (flow)g_z = e^zL_Ewhere L_E denotes Lie derivative along thevector field E. Action of this group on observables (smoothfunctions on M) is given by expansiong_z(f) = e^zL_E(f) =f + zL_Ef + ½(zL_E)²f + ⋯Further it will be assumed that M is 2n dimensionalsymplectic manifold and the group of transformations g_zwill be called symmetry of Hamiltonian system if it preserves manifold of solutionsof Hamilton's equationddtf = {h , f}(here { , } denotes Poisson bracket defined in a standard mannerby Poisson bivector field {f , g} = W(df ∧ dg) and his smooth function on M called Hamiltonian) or in other words if foreach f satisfying Hamilton's equation g_z(f)also satisfies it. This happens when g_z commuteswith time evolution operatorddtg_z(f) = g_z(ddtf)If in addition the generator E of the group g_zdoes not preserve Poisson bracket structure[E , W] ≠ 0 then the g_z is callednon-Noether symmetry. Let us briefly recall some basic features of non-Noethersymmetries. First of all if E generates non-Noether symmetrythen the n functionsY_k =i_W^k(L_Eω)^k      k = 1,2, ... n(where ω is symplectic form obtained by inverting Poissonbivector W and s denotes contraction) are integralsof motion (see [1] [3])and if additionally the symmetry generatorE satisfies Yang-Baxter equation[[E[E , W]]W] = 0these conservation laws Y_k appear to be in involution{Y_k, Y_m} = 0while 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 featuresnon-Noether symmetries could be effectively used in construction of conservation lawsand bi-Hamiltonian structures.

Now let us focus on non-Noether symmetry of the Toda model –2n dimensional Hamiltonian system that describes the motionof n particles on the line governed by the exponential interaction.Equations of motion of the non periodic n-particle Toda model areddtq_s = p_sddtp_s =ε(s − 1)e^{q_{s − 1} − q_s} −ε(n − s)e^{q_s − q_{s + 1}}(ε(k) = − ε(− k) = 1 for any natural k and ε(0) = 0) and could be rewritten in Hamiltonian form(3) with canonical Poisson bracket derived from symplectic formω = n∑s = 1dp_s ∧ dq_sand Hamiltonian equal toh = ½n∑s=1p_s² +n−1∑s=1e^{q_s − q_{s + 1}}The group of transformations g_z generated by the vector fieldE will be symmetry of Toda chain if for eachp_s, q_s satisfying Toda equations(7)g_z(p_s), g_z(q_s)also satisfy it.Substituting infinitesimal transformationsg_z(p_s) = p_s + zE(p_s) + O(z²)g_z(p_s) = q_s + zE(q_s) + O(z²)into (7) and grouping first order terms gives rise to theconditionsddtE(q_s) = E(p_s)ddtE(p_s) = ε(s − 1)e^{q_{s − 1} − q_s}(E(q_{s − 1}) − E(q_s))− ε(n − s)e^{q_s − q_{s + 1}}(E(q_s) − E(q_{s + 1}))One can verify that the vector field defined byE(p_s) = ½p_s² +ε(s − 1)(n − s + 2)e^{q_{s − 1} − q_s} −ε(n − s)(n − s) e^{q_s − q_{s + 1}} + t2(ε(s − 1)(p_{s − 1} + p_s)e^{q_{s − 1} − q_s} −ε(n − s)(p_s + p_{s + 1})e^{q_s − q_{s + 1}}E(q_s) = (n − s + 1)p_s −½s−1∑k=1p_k +½n∑k=s+1p_k+ t2(p_s² +ε(s − 1)e^{q_{s − 1} − q_s} +ε(n − s)e^{q_s − q_{s + 1}})satisfies (11) and generates symmetry of Toda chain. It appears that this symmetry is non-Noether since it does notpreserve Poisson bracket structure [E , W] ≠ 0and additionally one can check that Yang-Baxter equation[[E[E , W]]W] = 0 is satisfied.This symmetry could play important role inanalysis of Toda model. First let us note that calculating L_Eωleads to the following 2-formL_Eω = n∑s=1p_sdp_s ∧ dq_s +n−1∑s=1e^{q_s − q_{s + 1}}dq_s ∧ q_{s + 1} +  ∑r < sdp_r ∧ dp_sand together ω and L_Eω give rise tobi-Hamiltonian structure of Toda model (compare with [2]).The conservation laws (5)associated with the symmetry reproduce well knownset of conservation laws of Toda chain.I₁ = Y₁ = n∑s=1p_sI₂ = ½Y₁² − Y₂ =½n∑s=1p_s² +n−1∑s=1e^{q_s − q_{s + 1}}I₃ = 13Y₁³ − Y₁Y₂ + Y₃ =13n∑s=1p_s³ +n−1∑s=1(p_s + p_{s + 1})e^{q_s − q_{s + 1}}I₄ = ¼Y₁⁴ −Y₁²Y₂ + ½Y₂² +Y₁Y₃ − Y₄= ¼n∑s=1p_s⁴ +n−1∑s=1(p_s² + 2p_sp_{s + 1} +p_{s + 1}²) e^{q_s − q_{s + 1}}+ ½n−1∑s=1e^{2(q_s − q_{s + 1})} +n−2∑s=1e^{q_s − q_{s + 2}} I_m = (− 1)^mY_m + m^{− 1}m−1∑k=1(− 1)^kI_{m − k}Y_kThe condition [[E[E , W]]W] = 0 satisfied by generator of thesymmetry E ensures that the conservation laws are in involutioni. e. {Y_k,Y_m} = 0.Thus the conservation laws as well as the bi-Hamiltonian structureof the non periodic Toda chain appear to be associated with non-Noether symmetry.

Unlike the Toda model the dynamical systems in our next examples areinfinite dimensional and in order to ensure integrability one should constructinfinite number of conservation laws. Fortunately in several integrable modelsthis 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)u_t = i(u_xx + 2u²ū)where u is a smooth complex function of(t, x) ∈ ℝ2. On this stage we will not specify anyboundary conditions and will just focus on symmetries of NSE. Supposing that thevector field E generates the symmetry of NSE one gets the followingrestrictionE(u)_t = i[E(u)_xx + 2u²E(ū)+ 4uūE(u)](obtained by substituting infinitesimal transformationu → u + zE(u) + O(z2) generated by Einto NSE). It appears that NSE possesses nontrivial symmetry that is generated by thevector fieldE(u) = i(u_x + x2u_xx + uv + xu²ū) −t(u_xxx + 6uūu_x)(here v is defined by v_x = uū).In order to construct conservation laws we also need to know Poisson bracketstructure and it appears that invariant Poisson bivector field could be definedif u is subjected to either periodicu(t, − ∞) = u(t, + ∞) or zerou(t, − ∞) = u(t, + ∞) = 0 boundaryconditions. In terms of variational derivatives the explicit form of the Poisson bivector field isW = i+ ∞∫− ∞dx δδu ∧ δδūwhile corresponding symplectic form obtained by inverting W isω = i+ ∞∫− ∞dx δu ∧ δūNow one can check that NSE could be rewritten in Hamiltonian formu_t = {h , u}with Poisson bracket { , } defined by W andh = + ∞∫− ∞dx (u²ū² − u_xū_x)Knowing the symmetry of NSE that appears to be non-Noether([E, W] ≠ 0) one can construct bi-Hamiltonian structure andconservation laws. First let us calculate Lie derivative of symplectic form along the symmetrygeneratorL_Eω = + ∞∫− ∞[δu_x ∧ δū + uδv ∧ δū + ūδv ∧ δu]dxThe couple of 2-forms ω and L_Eωexactly reproduces the bi-Hamiltonian structure of NSE proposed by Magri[4] while the conservation laws associated with this symmetryare well known conservation laws of NSEI₁ = Y₁ = 2+ ∞∫− ∞uū dxI₂ = Y₁² − 2Y₂ = i+ ∞∫− ∞(ū_xu − u_xū) dxI₃ = Y₁³ − 3Y₁Y₂+ 3Y₃ = 2+ ∞∫− ∞ (u²ū² − u_xū_x) dxI₄ = Y₁⁴ − 4Y₁²Y₂ +2Y₂² + 4Y₁Y₃ − 4Y₄ = + ∞∫− ∞[i(ū_xu_xx − u_xū_xx)+ 3i(ūu²ū_x − uū²u_x)] dx I_m = (− 1)^mmY_m +m − 1∑k = 1(− 1)^kI_{m − k}Y_kThe involutivity of the conservation laws of NSE{Y_k, Y_m} = 0 is related to the fact thatE 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 beidentified and used in construction of conservation laws. The KdV and mKdV equationshave the following formu_t + u_xxx + uu_x = 0 [KdV]andu_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 conditionsE(u)_t + E(u)_xxx + u_xE(u) + uE(u)_x = 0 [KdV]andE(u)_t + E(u)_xxx − 12uu_xE(u) − 6u²E(u)_x = 0 [mKdV](again this conditions are obtained by substituting infinitesimal transformationu → u + zE(u) + O(z2) into KdV and mKdV, respectively).Further we will focus on the symmetries generated by the following vector fieldsE(u) = 12u_xx + 16u² +124u_xv + x8(u_xxx + uu_x)− t16(6u_xxxxx + 20u_xu_xx +10 uu_xxx + 5u²u_x) [KdV]andE(u) = − 32u_xx + 2u³+ u_xw − x2(u_xxx − 6u²u_x) − 3t2(u_xxxxx − 10u²u_xxx− 40uu_xu_xx − 10u_x³+ 30u⁴u_x) [mKdV](here v and w are defined by v_x = uand w_x = u²)To construct conservation laws we need to know Poisson bracket structureand again like in the case of NSE the Poisson bivector field is well definedwhen u is subjected to either periodicu(t, − ∞) = u(t, + ∞) or zerou(t, − ∞) = u(t, + ∞) = 0 boundaryconditions. For both KdV and mKdV the Poisson bivector field isW = + ∞∫− ∞dx δδu ∧ δδvwith corresponding symplectic formω = + ∞∫− ∞dx δu ∧ δvleading to Hamiltonian realization of KdV and mKdV equationsu_t = {h , u}with Hamiltoniansh = + ∞∫− ∞(u_x² − u³3) dx [KdV]andh = + ∞∫− ∞(u_x² + u⁴) dx [mKdV]By taking Lie derivative of thesymplectic form along the generators of the symmetries one getsanother couple of symplectic formsL_Eω = + ∞∫− ∞dx (δu ∧ δu_x + 23uδu ∧ δv) [KdV]L_Eω = + ∞∫− ∞dx (δu ∧ δu_x − 2uδu ∧ δw) [mKdV]involved in bi-Hamiltonian realization of KdV/mKdV hierarchies andproposed by Magri [4]. The conservation laws associated withthe symmetries reproduce infinite sequence of conservation laws of KdV equationI₁ = Y₁ =23+ ∞∫− ∞u dx I₂ = Y₁ − 2Y₂ =49+ ∞∫− ∞u² dx I₃ = Y₁³ − 3Y₁Y₂ + 3Y₃ =89+ ∞∫− ∞(u³3 − u_x²) dx I₄ = Y₁⁴ − 4Y₁²Y₂ +2Y₂² + 4Y₁Y₃ − 4Y₄ = 6445+ ∞∫− ∞(536u⁴ −53uu_x² + u_xx²) dx I_m = (− 1)^mmY_m +m − 1∑k = 1(− 1)^kI_{m − k}Y_kand mKdV equationI₁ = 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 = 2565+ ∞∫− ∞(5 u⁸+ 70u⁴u_x² − 7u_x⁴+ 14u²u_xx² + u_xxx²) dx I_m = (− 1)^mmY_m +m − 1∑k = 1(− 1)^kI_{m − k}Y_kThe involutivity of these conservation laws is well known and in terms of the symmetrygenerators it is ensured by conditions [[E[E , W]]W] = 0.Thus the conservation laws and bi-Hamiltonian structures of KdV and mKdVhierarchies are related to the non-Noether symmetries of KdV and mKdV equations.