 Award ID(s):
 2009647
 Publication Date:
 NSFPAR ID:
 10332337
 Journal Name:
 Nonlinearity
 Volume:
 34
 Issue:
 10
 Page Range or eLocationID:
 7227 to 7254
 ISSN:
 09517715
 Sponsoring Org:
 National Science Foundation
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We construct a broad class of bounded potentials of the onedimensional Schroedinger operator that have the same spectral structure as periodic finitegap potentials, but that are neither periodic nor quasiperiodic. Such potentials, which we call primitive, are nonuniquely parametrized by a pair of positive Hoelder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded nonvanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.

A new type of wave–mean flow interaction is identified and studied in which a smallamplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, largescale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket–mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wavepacket by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit socalled hydrodynamic reciprocity recently described in Maiden et al. ( Phys. Rev. Lett. , vol. 120, 2018, 144101) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models.

In this paper, we are concerned with a semidiscrete complex shortpulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultrashortpulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semidiscrete equation are studied with both zero and nonzero boundary conditions. To be specific, for the focusing sdCSP equation, the multibright solution (zero boundary conditions), multibreather and highorder rogue wave solutions (nonzero boundary conditions) are derived, while for the defocusing sdCSP equation with nonzero boundary conditions, the multidark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Ling et al . 2016 Physica D 327 , 13–29 ( doi:10.1016/j.physd.2016.03.012 ); Feng et al . 2016 Phys. Rev. E 93 , 052227 ( doi:10.1103/PhysRevE.93.052227 )).

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