In reversible (Hamiltonian) physical systems, the total energy of the system is constant in time. A local change of the energy density W at a point x in time t is balanced by the energy flux S from/to the point x according to the energy balance equation:
Wt + Sx = 0.In irreversible (active and/or dissipative) systems, the total energy changes in time due to the energy sinks/sources of density ρ as in the extended energy balance equation:
Wt + Sx = - ρ.In mathematical physics, energy balance is used for analysis of well-posedness of partial differential equations. Solutions of well-posed differential equations remain bounded in a suitable function space, starting with a bounded initial data. For illustration, we consider the energy balance for the heat equation ut = uxx that takes the form:
(u2)t - 2 (u ux)x = - 2 (ux)2.Suppose the initial data u(x,0) belong to space of real-valued square-integrable functions such that the initial energy E(0) = ∫ u2(x,0) dx is bounded from above. The energy E(t) decreases at later times such that 0 < E(t) = ∫ u2(x,t) dx < E(0). These simple estimates of energy analysis immediately imply the following properties of the heat equation:
In modeling of various physical phenomena, momentum balance equations are useful for analysis of integral properties of solutions of underlying equations at infinite or finite intervals. For instance, adiabatic dynamics of localized pulses, envelope wave packets, and radiative wave trains can be studied with the momentum balance equations, when a solitary wave changes under the action of external perturbations, variations of physical parameters, internal instabilities, various resonances, and interactions with other wave structures. In the simplest version of the soliton perturbation theory, effects of external perturbations to a physical system are captured by slow variations of soliton parameters. The dynamical rate of change of soliton parameters is found by substituting the soliton solutions in the momentum balance equations. For illustration, we consider the perturbed sine-Gordon equation with dissipative and external harmonical forces:
utt - uxx + sin(u) = ε R(u) = ε [-α ut + Γ sin(ωt) (1 - cos(u))],where ε is small parameter, α is parameter of damping and Γ is amplitude of the external force such that ω < 1 (no resonance occurs). With the account of the perturbation, the balance equation for momentum is
(ut ux)t - 0.5 (ut2 + ux2 + 2 cos(u))x = εR(u)ux.As a result, the rate of change of momentum P(t) = ∫ ut ux dx is given by P'(t) = ε ∫ ux R[u] dx, provided that (ut2 + ux2)->0 and cos(u)->1 as x goes to infinity. The unperturbed sine-Gordon equation at ε = 0 has the kink solution:
uk(x,t) = 4 arctan exp(x - vt - x_0)(1 - v2)-1/2,where |v| < 1 is the kink's velocity and x0 is its position. The momentum of the unperturbed kink is a function of its velocity: Pk(v) = - 8 v (1-v2)-1/2. The kink solution satisfies conditions at infinity. Assuming that the velocity of the kink v = v(t) changes due to the external perturbation, we use the momentum balance equation and find a particle-like equation of motion for the kink'a adiabatic dynamics:
Pk'(t) + ε α Pk = 2 π ε Γ sin(ωt).The equation has a simple solution:
Pk(t) = c exp(-ε α t) - 2 π ε ω Γ(ω2 + ε2 α2)-1 cos(ωt) + 2 π ε2 α Γ(ω2 + ε2 α2)-1 sin(ωt),where c is found from initial condition Pk(0) = P0.
Adibatic dynamics of kinks and solitary waves often generate strong radiation field that takes away parts of momentum of the localized solution. The radiation field can be taken into account from other balance equations, such as the mass balance. Radiative effects usually occur in the second order of the perturbation theory, leading to radiative decay of solitary waves or their perturbations. Thus, the balance equations for energy, momentum, and mass can be used for both qualitative and quantitative estimates of the interaction between localized and radiative components of the nonlinear wave in solutions of nonlinear wave equations.