[Paper Review] Noether Symmetries and Conservations Laws For Non-critical Kohn-Laplace Equations on Three-Dimensional Heisenberg Group

Igor Leite Freire|ArXiv.org|Jun 12, 2007

Nonlinear Waves and Solitons3 references3 citations

TL;DR

This paper identifies Noether symmetries and derives corresponding conservation laws for non-critical semilinear Kohn-Laplace equations on the three-dimensional Heisenberg group. Using variational structure and Lie point symmetry analysis, it proves that symmetries like translations, rotations, and certain vector fields are Noether symmetries only under specific conditions, with conservation laws derived via the Noether theorem framework on non-Euclidean, sub-Riemannian geometry.

ABSTRACT

We show which Lie point symmetries of non-critical semilinear Kohn-Laplace equations on the Heisenberg group $H^1$ are Noether symmetries and we establish their respectives conservations laws.

Motivation & Objective

• To determine which Lie point symmetries of non-critical semilinear Kohn-Laplace equations on the Heisenberg group $H^1$ are Noether symmetries.
• To derive explicit conservation laws associated with these Noether symmetries using the variational structure of the equation.
• To classify symmetries based on the nonlinearity $f(u)$, distinguishing cases where additional symmetries arise (e.g., $f(u) = 0$, $f(u) = u$, $f(u) = u^p$).
• To establish the conditions under which vector fields such as $Z$, $U$, and $V_1, V_2, V_3$ qualify as Noether symmetries or not.
• To provide a systematic framework for conservation law derivation in sub-Riemannian settings via Noether's theorem on the Heisenberg group.

Proposed method

• The Kohn-Laplace equation on $H^1$ is shown to possess a variational structure with a Lagrangian $\mathcal{L} = \frac{1}{2}u_x^2 + \frac{1}{2}u_y^2 + 2(x^2+y^2)u_t^2 + 2y u_x u_t - 2x u_y u_t - F(u)$, where $F' = f$.
• Lie point symmetries of the equation are classified based on the function $f(u)$, including $T = \partial_t$, $R = y\partial_x - x\partial_y$, $\tilde{X} = \partial_x - 2y\partial_t$, $\tilde{Y} = \partial_y + 2x\partial_t$, and additional symmetries for special $f(u)$.
• Noether symmetries are identified by verifying whether the first-order prolongation of a vector field satisfies the Noether condition: $\mathcal{L}^{(1)}\mathcal{L} + \mathcal{L} \cdot \text{Div}(\xi) = \text{Div}(\mathbf{J})$.
• For each Noether symmetry, the conserved current $\mathbf{J}$ is computed explicitly using the Noether formula, with components derived from the Lagrangian and symmetry vector fields.
• The analysis includes proving that $W_\beta$ is a Noether symmetry when $\Delta_{H^1}\beta = 0$, while $Z$, $U$, and $V_1, V_2, V_3$ are shown to be non-Noether symmetries under certain conditions.
• Computer-assisted verification using Mathematica is employed to confirm the conservation law components, with code available on request.

Experimental results

Research questions

• RQ1 Which Lie point symmetries of the non-critical semilinear Kohn-Laplace equation on $H^1$ are Noether symmetries?
• RQ2 Under what conditions on $f(u)$ do additional symmetries like $V_1, V_2, V_3$ or $D_p$ become Noether symmetries?
• RQ3 How do the conservation laws associated with Noether symmetries differ across cases such as $f(u) = 0$, $f(u) = u$, and $f(u) = u^p$?
• RQ4 Why are certain symmetries like $Z = x\partial_x + y\partial_y + 2t\partial_t$ and $U = u\partial_u$ not Noether symmetries despite being Lie symmetries?
• RQ5 What is the role of the kernel of the Kohn-Laplace operator in generating Noether symmetries via $W_\beta$?

Key findings

• The symmetries $T = \partial_t$, $R = y\partial_x - x\partial_y$, $\tilde{X} = \partial_x - 2y\partial_t$, and $\tilde{Y} = \partial_y + 2x\partial_t$ are Noether symmetries for all $f(u)$, with associated conservation laws derived from the Noether current.
• The symmetry $W_\beta = \beta\partial_u$ is a Noether symmetry if $\Delta_{H^1}\beta = 0$, and its conservation law arises from the divergence of a current involving $\beta$ and $u$.
• The generator $Z = x\partial_x + y\partial_y + 2t\partial_t$ is not a Noether symmetry, as its action does not yield a divergence form matching the Noether condition.
• The symmetry $U = u\partial_u$ is not a Noether symmetry, as shown by the failure of the Noether condition $U^{(1)}\mathcal{L} + \mathcal{L} \cdot \text{Div}(U) = \text{Div}(\mathbf{J})$.
• For $f(u) = 0$, the symmetries $V_1, V_2, V_3$ are Noether symmetries, and their conservation laws are derived analogously to known results for $f(u) = u^3$.
• For $f(u) = u^p$ with $p \neq 0,1,3$, the dilation symmetry $D_p = x\partial_x + y\partial_y + 2t\partial_t + \frac{2}{1-p}u\partial_u$ is a Noether symmetry, and its conservation law is explicitly constructed.