In joint work with Sal, we have finally written down the systematic process for deriving all of Olver’s conservation laws for the water-wave problem from our nonlocal-nonlocal formulation. This post outlines the current state of deriving the conservation laws from this Nonlocal-Nonlocal formulation discussed in the A Weak Formulation of Water Waves in Surface Variables. Click Continue Reading… to learn a little bit more about the process.

Snapshot of fluid domain

Using the general formulation outlined in A Weak Formulation of Water Waves in Surface Variables, we begin by recasting the equations for irrotational, inviscid, fluid flow with a free-boundary given by
&\phi_{xx} + \phi_{zz} =0, & &(x,z)\in\mathscr{D}, \label{eqn:laplace1d} \\
&\phi_t + \frac{1}{2}\vert\nabla\phi\vert^2 + gz + p = 0, & &(x,z)\in\mathscr{D},\label{eqn:bernoulliBulk} \\
&\phi_z =0, & &z = -h, \label{eqn:kinematicBottom1d} \\
&\eta_t + \phi_x\eta_x =\phi_z, &&z = \eta(x,t),\label{eqn:kinematic1d} \\
&\phi_t + \frac{1}{2}\vert\nabla\phi\vert^2 + g\eta =0, && z = \eta(x,t), \label{eqn:dynamic1d} \end{align}
as a system of equations in terms of the boundary variables \(q(x,t) = \phi(x,\eta,t)\), \(Q(x,t) = \phi(x,-h,t)\), and a freely choosen harmonic test function \(\varphi(x,z)\) such that the following integrals make sense:
\displaystyle &\int_\mathscr{S}\left(\frac{d}{dt}\varphi\, -\, q\left(\varphi_{zz}\, – \,\eta_x\varphi_{xz}\right)\right)\,dx = \int_{\mathscr{B}}-Q\varphi_{zz}\,dx\qquad &\textsf{(A)}\\
&\int_\mathscr{S}\left(q_x\eta_t\varphi_{xz} \,- \left(q_t + g\eta\right)\left(\varphi_{zz} \,+ \eta_x\varphi_{xz}\right)\right)\,dx = \int_{\mathscr{B}}\frac{1}{2}Q_x^2\varphi_{zz}\,dx\qquad &\textsf{(B)}\\ \end{align*}

At this point, all that is left is to see what happens when we choose different harmonic test functions \(\varphi\). The simpliest case is to let \(\varphi = x + iz\) which is harmonic everywhere. Substituting \(\varphi = x + iz\) into (A) yields \[\int_{\mathscr{S}}\frac{d}{dt}(x + i\eta)\,dx = 0\qquad \Rightarrow\qquad \frac{d}{dt}\int_{-\infty}^\infty \eta \,dx = 0\]This is precisely \(T_3\) or conservation of mass from Olver’s list of conservation laws! We can proceed to find the remaining conservation laws as outlined in this table:

Conserved DensityRule
\(\displaystyle T_1 = -\eta_x q\)(B) with \(\displaystyle \varphi = \frac{1}{2}(x + iz)^2\)
\(\displaystyle T_2 = \frac{1}{2}q\eta_t + \frac{1}{2}g\eta^2\) (B) with \(\displaystyle \varphi = \frac{1}{6}(x + iz)^3\)
\(\displaystyle T_3 = \eta\)(A) with \(\displaystyle \varphi = x + iz\)
\(\displaystyle T_4 = q + gt\eta\)(B) with \(\displaystyle\varphi = \frac{1}{2}(x + iz)^2\)
\(\displaystyle T_5 = x\eta + t\eta_x q\)(A) with \(\displaystyle\varphi = \frac{1}{2}(x + iz)^2\)
\(\displaystyle T_6 = \frac{1}{2}\eta^2 – \,t\,q – \frac{1}{2}gt^2\eta\)(A) with \(\displaystyle \varphi = \frac{1}{2}(x + iz)^2\)
\(\displaystyle T_7 = (\eta – x \eta_x)q + t\left(-4T_2 + \frac{7}{2}g\eta^2\right)-\frac{7}{2}t^2gq – \frac{7}{6}g^2t^3\eta\)(B) with \(\displaystyle\varphi = \frac{1}{6}(x + iz)^3\)
\(\displaystyle T_8 = (x + \eta\eta_x)q + gtx\eta +\frac{1}{2}t^2g \eta_x q\)(B) with \(\displaystyle\varphi = \frac{1}{6}(x + iz)^3\)
Conserved densities and derivation infromation.

Of course, there are a variety of other settings that we have considered and are in the process of submitting. It’s very exciting as this has lead us towards a few other projects!

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