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Wave structure and nonlinear balances in a family of evolutionary PDEs. (English) Zbl 1088.76531

Summary: We investigate the following family of evolutionary $1+1$ PDEs that describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids:

${m}_{t}\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}\underset{\text{convection}}{\underbrace{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}u{m}_{x}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}}}\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}\underset{\text{stretching}}{\underbrace{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}b\phantom{\rule{0.166667em}{0ex}}{u}_{x}m\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}}}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}\underset{\text{viscosity}}{\underbrace{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\nu \phantom{\rule{0.166667em}{0ex}}{m}_{xx}\phantom{\rule{4pt}{0ex}}}}\phantom{\rule{1.em}{0ex}}\text{with}\phantom{\rule{1.em}{0ex}}u=g*m·$

Here $u=g*m$ denotes $u\left(x\right)={\int }_{-\infty }^{\infty }g\left(x-y\right)m\left(y\right)\phantom{\rule{0.166667em}{0ex}}dy$. This convolution (or filtering) relates velocity $u$ to momentum density $m$ by integration against the kernel $g\left(x\right)$. We shall choose $g\left(x\right)$ to be an even function so that $u$ and $m$ have the same parity under spatial reflection. When $\nu =0$, this equation is both reversible in time and parity invariant. We shall study the effects of the balance parameter $b$ and the kernel $g\left(x\right)$ on the solitary wave structures and investigate their interactions analytically for $\nu =0$ and numerically for small or zero viscosity.

This family of equations admits the classic Burgers ”ramps and cliffs” solutions, which are stable for $-1 with small viscosity.

For $b<-1$, the Burgers ramps and cliffs are unstable. The stable solution for $b<-1$ moves leftward instead of rightward and tends to a stationary profile. When $m=u-{\alpha }^{2}{u}_{xx}$ and $\nu =0$, this profile is given by $u\left(x\right)\simeq {\text{sech}}^{2}\left(x/\left(2\alpha \right)\right)$ for $b=-2$ and by $u\left(x\right)\simeq \text{sech}\left(x/\alpha \right)$ for $b=-3$.

For $b>1$, the Burgers ramps and cliffs are again unstable. The stable solitary traveling wave for $b>1$ and $\nu =0$ is the “pulson” $u\left(x,t\right)=cg\left(x-ct\right)$, which restricts to the “peakon” in the special case $g\left(x\right)={e}^{-|x|/\alpha }$ when $m=u-{\alpha }^{2}{u}_{xx}$. Nonlinear interactions among these pulsons or peakons are governed by the superposition of solutions for $b>1$ and $\nu =0$,

$m\left(x,t\right)=\sum _{i=1}^{N}{p}_{i}\left(t\right)\phantom{\rule{0.166667em}{0ex}}\delta \left(x-{q}_{i}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}u\left(x,t\right)=\sum _{i=1}^{N}{p}_{i}\left(t\right)\phantom{\rule{0.166667em}{0ex}}g\left(x-{q}_{i}\left(t\right)\right)·$

These pulson solutions obey a finite-dimensional dynamical system for the time-dependent speeds ${p}_{i}\left(t\right)$ and positions ${q}_{i}\left(t\right)$. We study the pulson and peakon interactions analytically, and we determine their fate numerically under adding viscosity.

Finally, as outlook, we propose an $n$-dimensional vector version of this evolutionary equation with convection and stretching, namely,

$\frac{\partial }{\partial t}𝐦\phantom{\rule{4pt}{0ex}}+\underset{\text{convection}}{\underbrace{\phantom{\rule{4pt}{0ex}}𝐮·\nabla 𝐦\phantom{\rule{4pt}{0ex}}}}+\underset{\text{stretching}}{\underbrace{\phantom{\rule{4pt}{0ex}}\nabla {𝐮}^{T}·𝐦+\left(b-1\right)\phantom{\rule{0.166667em}{0ex}}𝐦\left(\text{div}\phantom{\rule{0.166667em}{0ex}}𝐮\right)\phantom{\rule{4pt}{0ex}}}}=0$

for a defining relation, $𝐮=G*𝐦$. These solutions show quasi-one-dimensional behavior for $n,k=2,1$ that we find numerically to be stable for $b=2$. The corresponding superposed solutions of the vector $b$-equation in $n$ dimensions exist, with coordinates $𝐱\in {ℝ}^{n}$, $s\in {ℝ}^{k}$, $n-k>0$, and $2N$ parameters ${𝐏}_{i}\left(s,t\right),{𝐐}_{i}\left(s,t\right)\in {ℝ}^{n}$,

$𝐦\left(𝐱,t\right)=\sum _{i=1}^{N}\int {𝐏}_{i}\left(s,t\right)\phantom{\rule{0.166667em}{0ex}}\delta \left(\phantom{\rule{0.166667em}{0ex}}𝐱-𝐐{\phantom{\rule{0.166667em}{0ex}}}_{i}\left(s,t\right)\phantom{\rule{0.166667em}{0ex}}\right)ds,\phantom{\rule{1.em}{0ex}}𝐦\in {ℝ}^{n},$
$𝐮\left(𝐱,t\right)=\sum _{i=1}^{N}\int {𝐏}_{i}\left(s,t\right)\phantom{\rule{0.166667em}{0ex}}G\left(\phantom{\rule{0.166667em}{0ex}}𝐱-𝐐{\phantom{\rule{0.166667em}{0ex}}}_{i}\left(s,t\right)\phantom{\rule{0.166667em}{0ex}}\right)ds,\phantom{\rule{1.em}{0ex}}𝐮\in {ℝ}^{n}·$

These are momentum surfaces (or filaments for $k=1$), defined on surfaces (or curves) $𝐱=𝐐{\phantom{\rule{0.166667em}{0ex}}}_{i}\left(s,t\right)$, $i=1,2,···,N$. For $b=2$, the ${𝐏}_{i}\left(s,t\right),{𝐐}_{i}\left(s,t\right)\in {ℝ}^{n}$ satisfy canonical Hamiltonian equations for geodesic motion on the space of $n$-vector valued $k$-surfaces with cometric $G$.

##### MSC:
 76D33 Waves in incompressible viscous fluids 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
##### Keywords:
invariant manifolds; Hamilton’s principle