TY - JOUR
T1 - A method for studying waves with spatially localized envelopes in a class of nonlinear partial differential equations
AU - King, Melvin E.
AU - Vakakis, Alexander F.
N1 - Funding Information:
This work was supportedb y an NSF GraduateS tudent Fellowship, and partially by NSF 0731 8. Dr. DevendraG arg is the Grant monitor.
PY - 1994/6
Y1 - 1994/6
N2 - A methodology for investigating stationary and travelling waves with spatially localized envelopes is presented. The nonlinear governing partial differential equations considered possess a constant first integral of motion, and are separable in space and time when the small parameter of the problem is set to zero. To study stationary waves, a coordinate transformation on the governing nonlinear partial differential equation is imposed which eliminates the time dependence from the problem. An amplitude modulation function is then introduced to express the response of the system at an arbitrary point as a nonlinear function of a reference response. Analytic approximations to the amplitude modulation function are developed by expressing it in power series, and asymptotically solving sets of singular functional equations at the various orders of approximation. Travelling solutions may be computed from stationary ones, by imposing appropriate Lorentz transformations. As an application of the methodology, stationary and travelling breathers of a nonlinear partial differential equation are analytically computed.
AB - A methodology for investigating stationary and travelling waves with spatially localized envelopes is presented. The nonlinear governing partial differential equations considered possess a constant first integral of motion, and are separable in space and time when the small parameter of the problem is set to zero. To study stationary waves, a coordinate transformation on the governing nonlinear partial differential equation is imposed which eliminates the time dependence from the problem. An amplitude modulation function is then introduced to express the response of the system at an arbitrary point as a nonlinear function of a reference response. Analytic approximations to the amplitude modulation function are developed by expressing it in power series, and asymptotically solving sets of singular functional equations at the various orders of approximation. Travelling solutions may be computed from stationary ones, by imposing appropriate Lorentz transformations. As an application of the methodology, stationary and travelling breathers of a nonlinear partial differential equation are analytically computed.
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U2 - 10.1016/0165-2125(94)90004-3
DO - 10.1016/0165-2125(94)90004-3
M3 - Article
AN - SCOPUS:0028443683
SN - 0165-2125
VL - 19
SP - 391
EP - 405
JO - Wave Motion
JF - Wave Motion
IS - 4
ER -