TY - JOUR
T1 - Existence of perturbed solitary wave solutions to a model equation for water waves
AU - Hunter, John K.
AU - Scheurle, Jurgen
N1 - Funding Information:
Unlike the KdV solitary wave, the travelling wave solutions we construct do not decay to zero as Ix\[ ~ + oo. Instead, when x is large, they approach small amplitude oscillations (cf. remark 3.2(a)). Our motivation for studying (1.1) is to understand the effect of surface tension on solitary water waves. (By a solitary water wave, we mean a travelling wave solution of the water wave equations for which the free surface approaches a constant height as Ix\[ ~ + oo.) The KdV equation (1.2) is obtained by a formal asymptotic expansion for small amplitude, long water waves, and it has a solitary wave solution. This *Partially supported by the NSF under grant DMS-8601879. **Partially supported by the NSF under grant DMS-8404506.
PY - 1988/9
Y1 - 1988/9
N2 - We prove the existence of travelling wave solutions to a fifth order partial differential equation, which is a formal asymptotic approximation for water waves with surface tension. These travelling waves are arbitrarily small perturbations of solitary waves, but are not solitary waves themselves, because they approach small amplitude oscillations for large values of the independent variable. This result suggests that for Bond numbers less than one third, there are branches of travelling wave solutions to the water wave equations, which are perturbations of supercritical elevation solitary waves, and which bifurcate from Froude number one and Bond number one third.
AB - We prove the existence of travelling wave solutions to a fifth order partial differential equation, which is a formal asymptotic approximation for water waves with surface tension. These travelling waves are arbitrarily small perturbations of solitary waves, but are not solitary waves themselves, because they approach small amplitude oscillations for large values of the independent variable. This result suggests that for Bond numbers less than one third, there are branches of travelling wave solutions to the water wave equations, which are perturbations of supercritical elevation solitary waves, and which bifurcate from Froude number one and Bond number one third.
UR - http://www.scopus.com/inward/record.url?scp=0000219244&partnerID=8YFLogxK
U2 - 10.1016/0167-2789(88)90054-1
DO - 10.1016/0167-2789(88)90054-1
M3 - Article
AN - SCOPUS:0000219244
SN - 0167-2789
VL - 32
SP - 253
EP - 268
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
IS - 2
ER -