Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs

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 $u(x)\simeq{\rm sech}^2(x/(2\alpha))$ for b=-2 and by $u(x)\simeq{\rm sech}(x/\alpha)$ 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 $g(x)=e^{-|x|/\alpha}$ 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 pi(t) and positions qi(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}\mathbf{m} \ + \underbrace{\ \mathbf{u}\cdot\nabla \mathbf{m}\ }_{\text{convection}} + \underbrace{\ \nabla \mathbf{u}^T\cdot\mathbf{m} + (b-1)\,\mathbf{m}({\rm div\,}\mathbf{u})\ }_{\text{s