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-symmetric KdV solutions and their algebraic extension with zero-width resonances.

Kumar Abhinav1, Aradhya Shukla2, Prasanta K Panigrahi3,4

  • 1Centre for Theoretical Physics and Natural Philosophy, Nakhonsawan Studiorum for Advanced Studies, Mahidol University, Nakhonsawan, 60130, Thailand. kumar.abh@mahidol.ac.th.

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Researchers identified complex breather and soliton solutions for KdV and mKdV equations using a Pöschl-Teller potential. Further extensions are needed to achieve the broken-phase, enabling non-trivial zero-width resonances.

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics

Background:

  • The Korteweg-de Vries (KdV) and modified KdV (mKdV) equations model various nonlinear phenomena.
  • Pöschl-Teller potentials are often used in quantum mechanics and nonlinear systems.

Purpose of the Study:

  • To identify complex breather and soliton solutions for KdV and mKdV equations.
  • To investigate the conditions for achieving the broken-phase solutions.

Main Methods:

  • Utilizing a Pöschl-Teller type potential with specific symmetry properties.
  • Analyzing the spectral properties of the potential in the complex plane.

Main Results:

  • Complex breather and soliton solutions were identified for KdV and mKdV equations under a Pöschl-Teller potential.
  • These solutions initially represented the unbroken-phase due to isospectrality with an infinite potential well.
  • Achieving the broken-phase requires an extended potential satisfying the potential algebra and supporting zero-width resonances.

Conclusions:

  • The study identifies a class of solutions but highlights the need for potential extension to access the broken-phase.
  • The findings pave the way for exploring more complex nonlinear phenomena and resonances.