On Doubly Periodic Standing Wave Solutions of the Coupled Higgs Field Equation

Based on the Hirota bilinear method and theta function identities, we obtain a new type of doubly periodic standing wave solutions for a coupled Higgs field equation. The Jacobi elliptic function expression and long wave limits of the periodic solutions are also presented. By selecting appropriate parameter values, we analyze the interaction properties of periodic-periodic waves and periodic-solitary waves by some figures.

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Bifurcations of Traveling Wave Solutions for the Coupled Higgs Field Equation

International Journal of Differential Equations ◽

10.1155/2011/547617 ◽

2011 ◽

Vol 2011 ◽

pp. 1-8

Author(s):

Shengqiang Tang ◽

Shu Xia

Keyword(s):

Field Equation ◽

Traveling Wave ◽

Bifurcation Theory ◽

Sufficient Conditions ◽

Higgs Field ◽

Traveling Wave Solutions ◽

Periodic Wave ◽

Solitary Wave Solutions ◽

Periodic Wave Solutions ◽

Wave Solutions

By using the bifurcation theory of dynamical systems, we study the coupled Higgs field equation and the existence of new solitary wave solutions, and uncountably infinite many periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All exact explicit parametric representations of the above waves are determined.

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Abundant soliton and periodic wave solutions for the coupled Higgs field equation, the Maccari system and the Hirota–Maccari system

Physica Scripta ◽

10.1088/0031-8949/85/06/065011 ◽

2012 ◽

Vol 85 (6) ◽

pp. 065011 ◽

Cited By ~ 9

Author(s):

Abdul-Majid Wazwaz

Keyword(s):

Field Equation ◽

Higgs Field ◽

Periodic Wave ◽

Periodic Wave Solutions ◽

Wave Solutions

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Dynamics of localized wave solutions for the coupled Higgs field equation

Nonlinear Dynamics ◽

10.1007/s11071-020-05860-8 ◽

2020 ◽

Vol 101 (2) ◽

pp. 1181-1198

Author(s):

Zhaqilao

Keyword(s):

Field Equation ◽

Higgs Field ◽

Wave Solutions

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Instability of standing waves for non-linear Schrödinger-type equations

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000938x ◽

1988 ◽

Vol 8 (8) ◽

pp. 119-138 ◽

Cited By ~ 46

Keyword(s):

Dynamical Systems ◽

Standing Wave ◽

Optical Waveguides ◽

Standing Waves ◽

Positive Eigenvalue ◽

Shooting Argument ◽

Wave Solutions ◽

Non Linear ◽

Linearized Operator

AbstractA theorem is proved giving a condition under which certain standing wave solutions of non-linear Schrödinger-type equations are linearly unstable. The eigenvalue equations for the linearized operator at the standing wave can be analysed by dynamical systems methods. A positive eigenvalue is then shown to exist by means of a shooting argument in the space of Lagrangian planes. The theorem is applied to a situation arising in optical waveguides.

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