Spatial Hamiltonian Identities for Nonlocally Coupled Systems

Abstract

We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler-Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether's theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler-Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.Differential equations of gradient form u t "´∇Epuq or of Hamiltonian form u t " J∇Epuq arise throughout mathematical modeling from maximal energy dissipation first principles, or from least action principles, respectively. Mathematically, the energy E is simply a function defined over a finite or infinite-dimensional space. The associated gradient and Hamiltonian flows are differential equations with equilibria given by the critical points of the energy. Energies over infinite-dimensional spaces are usually defined on function spaces through integrals of nonlinear functions of state variables and their derivatives. Critical points then solve Euler-Lagrange equations, commonly of elliptic type. Dependence of the energy on derivatives of the state variables encodes local interactions, often derived in various types of continuum limits. We are interested here in cases where this continuum limit retains nonlocal interaction terms. More specifically, we are interested in the somewhat specific class of energies E that contain nonlocal interaction term of the formmodeling phenomena such as long-range interactions of agents in social interaction, between particles through nonlocal force fields, or of neurons, labeled in a feature space x through synaptic connections. In all those cases, the convolution structure embodies the modeling assumption of translational invariance of physical space. We are

Global bifurcation of solitary waves for the Whitham equation

“…In this section, we use the integral identity in Proposition 2.2 to exclude the loss of compactness scenario. An alternative route is to employ the Hamiltonian structures for nonlocal problems in [7]. Even though the Whitham kernel does not fit into this framework, a direct differentiation confirms that equation 43 in [7] indeed gives a Hamiltonian for the Whitham equation.…”

“…An alternative route is to employ the Hamiltonian structures for nonlocal problems in [7]. Even though the Whitham kernel does not fit into this framework, a direct differentiation confirms that equation 43 in [7] indeed gives a Hamiltonian for the Whitham equation. We also study how M(s) blows up as s → ∞.…”

Solitary waves in a Whitham equation with small surface tension

“…30 uses the Hamiltonian structure of the full water wave problem. While Equation (1) exhibits a variational formulation in the form investigated by Bakker and Scheel, 33 its smoothing form (26) below does not. As 𝑚 𝜏 does not have an 𝐿 1 Fourier transform, using results in Ref.…”

“…As 𝑚 𝜏 does not have an 𝐿 1 Fourier transform, using results in Ref. 33 would therefore call for a careful examination and adaptation. Thus, it is more appropriate to consider this bifurcation phenomenon in a separate paper and we will not comment further regarding 𝐶 1 .…”

Strain and defects in oblique stripe growth

“…where we used that D + is a symmetric operator with kernel spanned by the constant functions to see that the first summand vanished, and monotonicity of ψ * + ζ to transform the second summand into an integral over ϕ. As a consequence k x = − g is a priori known; see also [22,2,3], where this wavenumber selection mechanism was derived from Hamiltonian identities.…”