Bifurcation diagrams of coupled Schrödinger equations

Essman, Michael Christopher; Shi, Junping

doi:10.1016/j.amc.2012.09.061

Cited by 4 publications

(2 citation statements)

References 34 publications

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“…We also mention that result like Theorem 1.1 can be proved using other method even for a general bounded domain, see [9], but our main emphasis here is to present a systematic approach for radially symmetric solutions and obtain a complete structure of solutions to the shooting problem (1.3), that is information not provided in [9]. We also mention that the shooting problem like (1.3) can be solved numerically, and bifurcation diagrams showing the regions of different types of solutions as in Theorems 1.3 and 1.4 can be numerically obtained [17].…”

Section: ) Does Not Possess a Solution Of Type Gmentioning

confidence: 97%

On the uniqueness and structure of solutions to a coupled elliptic system

Chen

Chern

Shi

et al. 2010

Journal of Differential Equations

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In this paper, we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear Schrödinger equation. We establish the existence and uniqueness of solutions to the Dirichlet problem on the ball. In addition, the nonexistence of the ground state solutions under certain conditions on the nonlinearities and the complete structure of different types of solutions to the shooting problem are proved.

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Section: ) Does Not Possess a Solution Of Type Gmentioning

confidence: 97%

On the uniqueness and structure of solutions to a coupled elliptic system

Chen

Chern

Shi

et al. 2010

Journal of Differential Equations

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“…The bifurcation theory analysis is one of the most common methods in dealing with a system of differential equations; see e.g. [28][29][30][31][32][33][34][35][36][37][38][39]. Recent methods dealing with NLEEs are generalized exponential rational function method [40], ameliorated form of a modified F-expansion method [41], Φ 6 − model expansion method [42] and the super thermal index method [43].…”

Section: Introductionmentioning

confidence: 99%

Qualitative analysis and new soliton solutions for the coupled nonlinear Schrödinger type equations

Elbrolosy

2021

Phys. Scr.

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This work is interested in constructing new traveling wave solutions for the coupled nonlinear Schrödinger type equations. It is shown that the equations can be converted to a conservative Hamiltonian traveling wave system. By using the bifurcation theory and Qualitative analysis, we assign the permitted intervals of real propagation. The conserved quantity is utilized to construct sixteen traveling wave solutions; four periodic, two kink, and ten singular solutions. The periodic and kink solutions are analyzed numerically considering the affect of varying each parameter keeping the others fixed. The degeneracy of the solutions discussed through the transmission of the orbits illustrates the consistency of the solutions. The 3D and 2D graphical representations for solutions are presented. Finally, we investigate numerically the quasi-periodic behavior for the perturbed system after inserting a periodic term.

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New traveling wave solutions for Gilson–Pickering equation in plasma via bifurcation analysis and direct method

Elmandouh

Elbrolosy

2022

Math Methods in App Sciences

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This work aims to study the wave propagation for the Gilson–Pickering equation appearing in plasma. By using an appropriate wave transformation, this equation is converted into a singular dynamical system that can then be converted into a regular dynamical system through a suitable point transformation for the independent variable. We demonstrate that both systems have the same first integral. In light of the topological equivalence between the phase orbits of both systems, we give a brief description of each system's phase plane. On the basis of the bifurcation analysis, we introduce two theorems summarizing the conditions on the parameters giving rise to periodic and solitary solutions, besides the conditions of unbounded wave solutions. Consequently, we construct some parametric wave solutions which are periodic, solitary, and unbounded wave solutions. We perform a numerical study to clarify these solutions graphically and to confirm the efficacy of the analytical study. We also analyze the influence of the parameters on some of the obtained solutions numerically.

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Bifurcation diagrams of coupled Schrödinger equations

Cited by 4 publications

References 34 publications

On the uniqueness and structure of solutions to a coupled elliptic system

On the uniqueness and structure of solutions to a coupled elliptic system

Qualitative analysis and new soliton solutions for the coupled nonlinear Schrödinger type equations

New traveling wave solutions for Gilson–Pickering equation in plasma via bifurcation analysis and direct method

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