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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: $$m_t\ +\ \underbrace{\ \ um_x\ \ }_{\text{convection}}\ +\ \underbrace{\ \ b\,u_xm\ \ }_{\text{stretching}}\ =\ \underbrace{\ \ \nu\,m_{xx}\ }_{\text{viscosity}} \quad\text{with}\quad u=g*m.$$ Here $u=g*m$ denotes $u(x)=\int_{-\infty}^\infty g(x-y)m(y)\,dy$. This convolution (or filtering) relates velocity $u$ to momentum density $m$ by integration against the kernel $g(x)$. We shall choose $g(x)$ 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 $g(x)$ 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 $-1 < b < 1$ 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 $m=u-\alpha^2u_{xx}$ and $\nu=0$, this profile is given by $u(x)\simeq \text{sech}^2(x/(2\alpha))$ for $b=-2$ and by $u(x)\simeq \text{sech}(x/\alpha)$ 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” $u(x,t)=cg(x-ct)$, which restricts to the “peakon” in the special case $g(x)=e^{-\vert x\vert /\alpha}$ when $m=u-\alpha^2u_{xx}$. Nonlinear interactions among these pulsons or peakons are governed by the superposition of solutions for $b > 1$ and $\nu=0$, $$m(x,t)=\sum_{i=1}^N p_i(t)\,\delta(x-q_i(t)),\quad u(x,t)=\sum_{i=1}^N p_i(t)\,g(x-q_i(t)).$$ These pulson solutions obey a finite-dimensional dynamical system for the time-dependent speeds $p_{i}(t)$ and positions $q_{i}(t)$. 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, $$\frac{\partial}{\partial t}\bold{m} \ + \underbrace{\ \bold{u}\cdot\nabla \bold{m}\ }_{\text{convection}} + \underbrace{\ \nabla \bold{u}^T\cdot\bold{m} + (b-1)\,\bold{m}(\text{div }\bold{u})\ }_{\text{stretching}} =0$$ for a defining relation, $\bold{u}=G*\bold{m}$. 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 $\bold{x}\in{\bbfR^n}$, $s\in{\bbfR^k}$, $n-k > 0$, and $2N$ parameters $\bold{P}_i(s,t),\bold{Q}_i(s,t)\in{\bbfR^n}$, $$\bold{m}(\bold{x},t) = \sum_{i=1}^N\int\bold{P}_i(s,t)\, \delta\bigl(\,\bold{x}-\bold{Q}\,_i(s,t)\,\bigr)ds,\quad \bold{m}\in{\bbfR^n},$$ $$\bold{u}(\bold{x},t) = \sum_{i=1}^N\int\bold{P}_i(s,t)\, G\bigl(\,\bold{x}-\bold{Q}\,_i(s,t)\,\bigr)ds,\quad \bold{u}\in{\bbfR^n}.$$ These are momentum surfaces (or filaments for $k=1$), defined on surfaces (or curves) $\bold{x}=\bold{Q}\,_i(s,t)$, $i=1,2,...,N$. For $b=2$, the $\bold{P}_i(s,t), \bold{Q}_i(s,t)\in{\bbfR^n}$ 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
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