Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs

By Darryl D. Holm, Martin F. Staley. SIAM Journal on Applied Dynamical Systems, 2002

302 citations recorded by the source corpus.

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 Burg...

Subfields: Computer Networks and Communications, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d05658560929ba25c1a7ab1b66875361c8440e58

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Citations in the corpus

Global weak solutions for the Novikov equation (2011, background): “3) The Degasperis–Procesi equation can be regarded as a model for nonlinear shallow water dynamics [18].”
Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation (2006, background): “Holm and Staley [ 28 ] studied stability of solitons and peakons numerically to Eq. (1.2).”
ON THE UNIQUENESS OF DISCONTINUOUS SOLUTIONS TO THE DEGASPERIS–PROCESI EQUATION (2007, methodology): “We refer to [10] for a discussion of Camassa–Holm, Degasperis–Procesi, and other related equations, along with many numerical examples.”
On the Blow-up Phenomena for the Degasperis–Procesi Equation (2010, methodology): “The DP equation is also in dimensionless space-time variables (x, t) an approximation to the incompressible Euler equations for shallow water under the Kodama transformation [25, 26] and its asymptotic accuracy is the same as that of the Camassa-Holm shallow water equation, where u(t, x) is consider”
A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation (2006, background): “For a discussion of Camassa–Holm and related equations as well as their analytical/numerical properties, see the recent article Holm and Staley (2003).”
Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary pde (2002, background): “(B1) In the stable peakon region b > 1, the Steepening Lemma for the b−equation (10) proven in [5] for b = 2 and in [10] for 1 < b ≤ 3 allows inflection points with negative slopes to escape verticality by producing a jump in spatial derivative at the peak of a traveling wave that eliminates the inf”
Inverse scattering transform for the Degasperis–Procesi equation (2010, background): “We conclude by pointing out that while there are interesting multidimensional versions of the DP equation, they are in general nonintegrable, but admit singular (peakon-type) solutions [32].”
Stability of peakons for the Degasperis‐Procesi equation (2007, methodology): “Holm and Staley [20] studied stability of solitons and peakons numerically to equation (1.5).”
Global weak solutions and blow-up structure for the Degasperis–Procesi equation (2006, background): “2) and Holm and Staley [27] studied stability of solitons and peakons numerically.”