Modulation theory solution for nonlinearly resonant, fifth‐order Korteweg–de Vries, nonclassical, traveling dispersive shock waves

Abstract

A new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equationa Korteweg-de Vries (KdV) type nonlinear wave equation with third-and fifth-order spatial derivatives-in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third-and fifth-order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth-order KdV equation without the third derivative term, hence without any linear resonance. A self-similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV-Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third-order dispersion. The Kawahara-Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third-order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.

Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals

“…which is analogous to the wave conservation equation (35). We then have that the averaged Lagrangian is…”

“…Unfortunately, there are no known general solitary wave or periodic wave solutions of the nematic equations. However, the resonant wavetrain has small amplitude, so that the periodic wave solution will be found as a Stokes' expansion, as was done for the Kahawara equation [29] and the fifth order KdV equation, the Kahawara equation (1) with µ = 0 [35]. We then seek a Stokes' expansion solution of the nematic equations as ρ =ρ + a cos ϕ + a 2 ρ 2 cos 2ϕ + a 3 ρ 3 cos 3ϕ + .…”

“…It can be seen that there is excellent agreement for the lead solitary wave height over both the PDSW and RDSW regimes. When a resonant wavetrain is present, the amplitude of the lead wave of the DSW oscillates as radiation propagates out of the DSW [35]. In this case, the average lead wave amplitude is taken for the numerical comparisons.…”

“…The stability of the Stokes' wave solution of the Kawahara equation (1) has been studied [35]. This work can be used to determine the stability of the trailing edge of the DSW, where the wavenumber is k i given by (53) with m = 0.…”

“…The instability of the resonant wavetrain is now clear, in contrast to the example shown in Figure 1(b), with the breakup of the resonant wavetrain as it propagates in z now prominent. Each wave of the DSW generates a resonant wave as a DSW is a modulated wavetrain [35]. Figure 5 shows that these resonant waves within the DSW are also unstable.…”