zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 +um x convection +bu x m stretching =νm xx viscosity withu=g*m·

Here u=g*m denotes u(x)= - 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 ν=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 ν=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-α 2 u xx and ν=0, this profile is given by u(x)sech 2 (x/(2α)) for b=-2 and by u(x)sech(x/α) for b=-3.

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

m(x,t)= i=1 N p i (t)δ(x-q i (t)),u(x,t)= 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,

t𝐦+𝐮·𝐦 convection +𝐮 T ·𝐦+(b-1)𝐦(div𝐮) stretching =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 n , s k , n-k>0, and 2N parameters 𝐏 i (s,t),𝐐 i (s,t) n ,

𝐦(𝐱,t)= i=1 N 𝐏 i (s,t)δ 𝐱 - 𝐐 i (s,t) ds,𝐦 n ,
𝐮(𝐱,t)= i=1 N 𝐏 i (s,t)G 𝐱 - 𝐐 i (s,t) ds,𝐮 n ·

These are momentum surfaces (or filaments for k=1), defined on surfaces (or curves) 𝐱=𝐐 i (s,t), i=1,2,···,N. For b=2, the 𝐏 i (s,t),𝐐 i (s,t) n satisfy canonical Hamiltonian equations for geodesic motion on the space of n-vector valued k-surfaces with cometric G.


MSC:
76D33Waves in incompressible viscous fluids
37N10Dynamical systems in fluid mechanics, oceanography and meteorology