Multi-hump Collapsing Solutions in the Nonlinear Schr{ö}dinger Problem: Existence, Stability and Dynamics
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2025
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| author | Chapman, Jon S. Kavousanakis, Mihalis Charalampidis, Efstathios G. Kevrekidis, Ioannis G. Kevrekidis, Panayotis G. |
| author_facet | Chapman, Jon S. Kavousanakis, Mihalis Charalampidis, Efstathios G. Kevrekidis, Ioannis G. Kevrekidis, Panayotis G. |
| contents | In the present work we examine multi-hump solutions of the nonlinear Schr{ö}dinger equation in the blowup regime of the one-dimensional model with power law nonlinearity, bearing a suitable exponent of $σ>2$. We find that families of such solutions exist for arbitrary pulse numbers, with all of them bifurcating from the critical case of $σ=2$. Remarkably, all of them involve ``bifurcations from infinity'', i.e., the pulses come inward from an infinite distance as the exponent $σ$ increases past the critical point. The position of the pulses is quantified and the stability of the waveforms is also systematically examined in the so-called ``co-exploding frame''. Both the equilibrium distance between the pulse peaks and the point spectrum eigenvalues associated with the multi-hump configurations are obtained as a function of the blowup rate $G$ theoretically, and these findings are supported by detailed numerical computations. Finally, some prototypical dynamical scenarios are explored, and an outlook towards such multi-hump solutions in higher dimensions is provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09746 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-hump Collapsing Solutions in the Nonlinear Schr{ö}dinger Problem: Existence, Stability and Dynamics Chapman, Jon S. Kavousanakis, Mihalis Charalampidis, Efstathios G. Kevrekidis, Ioannis G. Kevrekidis, Panayotis G. Pattern Formation and Solitons In the present work we examine multi-hump solutions of the nonlinear Schr{ö}dinger equation in the blowup regime of the one-dimensional model with power law nonlinearity, bearing a suitable exponent of $σ>2$. We find that families of such solutions exist for arbitrary pulse numbers, with all of them bifurcating from the critical case of $σ=2$. Remarkably, all of them involve ``bifurcations from infinity'', i.e., the pulses come inward from an infinite distance as the exponent $σ$ increases past the critical point. The position of the pulses is quantified and the stability of the waveforms is also systematically examined in the so-called ``co-exploding frame''. Both the equilibrium distance between the pulse peaks and the point spectrum eigenvalues associated with the multi-hump configurations are obtained as a function of the blowup rate $G$ theoretically, and these findings are supported by detailed numerical computations. Finally, some prototypical dynamical scenarios are explored, and an outlook towards such multi-hump solutions in higher dimensions is provided. |
| title | Multi-hump Collapsing Solutions in the Nonlinear Schr{ö}dinger Problem: Existence, Stability and Dynamics |
| topic | Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2504.09746 |