Evolution of initial discontinuities in the DNLS equation theory

Abstract

We present the full classification of wave patterns evolving from an initial step-like discontinuity for arbitrary choice of boundary conditions at the discontinuity location in the DNLS equation theory. In this non-convex dispersive hydrodynamics problem, solutions of the Whitham modulation equations are mapped to parameters of a modulated wave by two-valued functions what makes situation much richer than that for a convex case of the NLS equation type. In particular, new types of simple-wave-like structures, such as trigonometric shocks and combined shocks, appear as building elements of the whole wave pattern observable in the long-time evolution of pulses after the wave-breaking point. The developed theory can find applications to propagation of light pulses in fibers and to the theory of Alfvén dispersive shock waves.

Gurevich–Pitaevskii problem and its development

“…A classification of such structures evolving from the initial discontinuity in accordance with the Gardner equation u t +6(u±αu 2 )u x +u xxx = 0 that occurs in the theory of internal water waves was given in [88]. In the theory of the modified NLS equation iψ t + 1 2 ψ xx − i(|ψ| 2 ψ) x = 0, which has applications in nonlinear optics and magnetohydrodynamic waves, the use of all three sets of parameters becomes necessary: periodic solutions and Whitham's equations were obtained in [89], and the evolution of the initial discontinuity was analyzed in [90][91][92]. Finally, the most complicated case of this type, a ferromagnet with 'easy plane' anisotropy and the equivalent limit for two-component Gross-Pitaevskii equations, was studied in [93,94].…”

Riemann problem for the light pulses in optical fibers for the generalized Chen-Lee-Liu equation

“…At the same time, the generalized Chen-Lee-Liu theory is also used in the investigation of modulated wave dynamics of propagating through a single nonlinear transmission network which presents some practical interest [49]. The method presented here is quite flexible and was also applied to other systems with non-convex hydrodynamics [50][51][52].…”

“…Motivated by applications of the generalized CLL equation (2), we consider the method which permits one to predict a wave pattern arising from any given data for an initial discontinuity. The method is quite general and it was applied to the generalized NLS equations with self-steepening nonlinearity [50,51] and to the Landau-Lifshitz equation for magnetics with easyplane anisotropy [52]. Here we extend the theory to notgenuinely case of the generalized CLL equation (2).…”

Dynamic soliton-mean flow interaction with nonconvex flux

“…The GP problem for these equations includes double waves involving contact DSWs (Landau-Lifshitz equation) and undercompressive DSWs (Miyata-Camassa-Choi equation). Nonconvex mean flows have also been studied for variants of the nonlinear Schrödinger (NLS) equation including the defocusing complex mKdV equation [42], the derivative NLS equation [36], the discrete NLS equation from which mKdV can be derived [39], and NLS with self-steepening terms [33,32], all of which include contact DSW solutions.…”

“…Equation ( 36) is a nonlinear oscillator equation in the potential −Q(u). Figure 3 shows representative potential curves Q(u) for both signs of the dispersion coefficient µ. Travelling wave solutions exist in the regions where Q(u) > 0 (shaded regions) and can be obtained by integrating (36) in terms of Jacobi elliptic functions. The cases µ < 0 and µ > 0 are treated separately.…”