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Van der Waals Equation01:10

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The ideal gas law is an approximation that works well at high temperatures and low pressures. The van der Waals equation of state (named after the Dutch physicist Johannes van der Waals, 1837−1923) improves it by considering two factors.
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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Negative-order Korteweg-de Vries equations.

Zhijun Qiao1, Engui Fan

  • 1Department of Mathematics, The University of Texas-Pan American, 1201 W. University Drive, Edinburg, TX 78539, USA. qiao@utpa.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
PubMed
Summary

This study explores negative-order Korteweg-de Vries (NKdV) equations, detailing their Hamiltonian structures and deriving explicit multisoliton and multikink wave solutions. The research uncovers unique singular interactions and quasiperiodic wave solutions, distinct from regular KdV behavior.

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Area of Science:

  • Nonlinear Partial Differential Equations
  • Mathematical Physics
  • Soliton Theory

Background:

  • The study builds upon the established Korteweg-de Vries (KdV) system.
  • Negative-order KdV (NKdV) equations are investigated, derived from the first member of the negative-order KdV hierarchy.
  • NKdV equations exhibit connections to Camassa-Holm, Ermakov-Pinney, and Kupershmidt systems.

Purpose of the Study:

  • To analyze the Hamiltonian structures, Lax pairs, and conservation laws of NKdV equations.
  • To derive explicit multisoliton and multikink wave solutions for NKdV equations.
  • To investigate the properties and relationships of quasiperiodic wave solutions.

Main Methods:

  • Bilinear Bäcklund transformations are employed to find explicit multisoliton and multikink wave solutions.
  • Trace identity and Lax pairs are used to construct bi-Hamiltonian structures and Darboux transformations.
  • Multidimensional binary Bell polynomials facilitate the derivation of N-soliton solutions.

Main Results:

  • Explicit formulas for single and double kink wave and bell soliton solutions are obtained.
  • The collisions of two-kink and two-bell soliton solutions exhibit singular interactions, differing from regular KdV behavior.
  • A direct scheme for constructing quasiperiodic wave solutions is proposed, with their convergence to soliton solutions demonstrated.

Conclusions:

  • The study provides a comprehensive analysis of NKdV equations, including their fundamental properties and wave solutions.
  • Novel insights into the behavior of solitons and quasiperiodic waves in NKdV systems are presented.
  • The findings contribute to the understanding of integrable systems and nonlinear wave phenomena.