Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

Yang, Jianke

doi:10.1111/j.1467-9590.2012.00549.x

Cited by 26 publications

(51 citation statements)

References 27 publications

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“…In the model (1), the nonlinearity is only cubic. But extension of our analysis to an arbitrary form of nonlinearity is straightforward without much more effort (see [27]). Solitary waves in Equation (1) are sought of the form…”

Section: Preliminariesmentioning

confidence: 99%

“…Indeed, we have numerically computed the function Q 2 in this condition and plotted it in Figure 2(b); one can see that it is never zero for μ > μ 0 . Because this second condition is not met, this family of asymmetric solitons cannot persist and have to disappear under PT-potential perturbations (27) as well. (14); this function is zero at the red-dot point.…”

Section: Disappearance Of Families Of Asymmetric Solitons Under Weak mentioning

confidence: 99%

“…Because solitons in PT systems could exist for continuous ranges of propagation constants, bifurcations of such solitons become an important issue. In conservative wave systems, various archetypical bifurcations of solitons have been reported, including fold bifurcations (also known as saddle-node or saddle-center bifurcations), symmetry-breaking bifurcations (also known as pitchfork bifurcations), and transcritical bifurcations (see [27] and the references therein). In PT systems, fold bifurcations have been found [12,13,19,20], but the other types of bifurcations are still unknown.…”

Section: Introductionmentioning

confidence: 99%

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Can Parity‐Time‐Symmetric Potentials Support Families of Non‐Parity‐Time‐Symmetric Solitons?

Yang

2013

Stud Appl Math

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For the one-dimensional nonlinear Schrödinger equations with parity-time (PT) symmetric potentials, it is shown that when a real symmetric potential is perturbed by weak PT-symmetric perturbations, continuous families of asymmetric solitary waves in the real potential are destroyed. It is also shown that in the same model with a general PT-symmetric potential, symmetry breaking of PT-symmetric solitary waves does not occur. Based on these findings, it is conjectured that one-dimensional PT-symmetric potentials cannot support continuous families of non-PT-symmetric solitary waves.

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Section: Preliminariesmentioning

confidence: 99%

Section: Disappearance Of Families Of Asymmetric Solitons Under Weak mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Can Parity‐Time‐Symmetric Potentials Support Families of Non‐Parity‐Time‐Symmetric Solitons?

Yang

2013

Stud Appl Math

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“…This is interesting, because Yang, in Ref. [40], only reported transcritical bifurcations by constructing an asymmetric potential, whereas the dumbbell graph is symmetric; see Remark 1.1 for more details. We then show, via a perturbation calculation, and via numerical computation that the same is true for the problem on the quantum graph, which demonstrates the existence of the branch not found in [28].…”

Section: Introductionmentioning

confidence: 96%

NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

Goodman¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We consider the bifurcations of standing wave solutions to the nonlinear Schrödinger equation (NLS) posed on a quantum graph consisting of two loops connected by a single edge, the so-called dumbbell, recently studied in [28]. The authors of that study found the ground state undergoes two bifurcations, first a symmetry-breaking, and the second which they call a symmetry-preserving bifurcation. We clarify the type of the symmetry-preserving bifurcation, showing it to be transcritical. We then reduce the question, and show that the phenomena described in that paper can be reproduced in a simple discrete self-trapping equation on a combinatorial graph of bowtie shape. This allows for complete analysis both by geometric methods and by parameterizing the full solution space. We then expand the question, and describe the bifurcations of all the standing waves of this system, which can be classified into three families, and of which there exists a countably infinite set.

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“…Moreover, an attractive classification is given for solitary wave bifurcations in generalized nonlinear Schrodinger equations in Ref. [17].…”

Section: Introductionmentioning

confidence: 99%

Effects of Quadratic Singular Curves in Integrable Equations

et al. 2014

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In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo-cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first-order derivatives of the new singular solitary wave and periodic waves exist, their second-order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo-cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo-cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.

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Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

Cited by 26 publications

References 27 publications

Can Parity‐Time‐Symmetric Potentials Support Families of Non‐Parity‐Time‐Symmetric Solitons?

Can Parity‐Time‐Symmetric Potentials Support Families of Non‐Parity‐Time‐Symmetric Solitons?

NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

Effects of Quadratic Singular Curves in Integrable Equations

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