Internal gravitycapillary solitary waves in finite depth
(2017) In Mathematical Methods in the Applied Sciences 40(4). p.10531080 Abstract
We consider a twodimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a timelike coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to... (More)
We consider a twodimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a timelike coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a HamiltonianHopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 0^{2}resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the HamiltonianHopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 0^{2}resonance and recover the results found by Kirrmann.
(Less)
 author
 Nilsson, Dag Viktor ^{LU}
 organization
 publishing date
 20170315
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 Capillarity, Internal waves, Spatial dynamics, Water waves
 in
 Mathematical Methods in the Applied Sciences
 volume
 40
 issue
 4
 pages
 28 pages
 publisher
 John Wiley & Sons
 external identifiers

 scopus:84976904117
 wos:000397302000017
 ISSN
 01704214
 DOI
 10.1002/mma.4036
 language
 English
 LU publication?
 yes
 id
 4ed0882260c045989d62aec0f5866bff
 date added to LUP
 20160726 08:31:36
 date last changed
 20180529 10:24:28
@article{4ed0882260c045989d62aec0f5866bff, abstract = {<p>We consider a twodimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a timelike coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a HamiltonianHopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 0<sup>2</sup>resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the HamiltonianHopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 0<sup>2</sup>resonance and recover the results found by Kirrmann.</p>}, author = {Nilsson, Dag Viktor}, issn = {01704214}, keyword = {Capillarity,Internal waves,Spatial dynamics,Water waves}, language = {eng}, month = {03}, number = {4}, pages = {10531080}, publisher = {John Wiley & Sons}, series = {Mathematical Methods in the Applied Sciences}, title = {Internal gravitycapillary solitary waves in finite depth}, url = {http://dx.doi.org/10.1002/mma.4036}, volume = {40}, year = {2017}, }