Dark solitons, dispersive shock waves, and transverse instabilities

Dark solitons, dispersive shock waves, and transverse instabilities

M. A. Hoefer222Department of Mathematics, North Carolina State University, Raleigh, NC 27695; mahoefer@ncsu.edu    B. Ilan333School of Natural Sciences, University of California, Merced, CA 95343; bilan@ucmerced.edu

The nature of transverse instabilities to dark solitons and dispersive shock waves for the (2+1)-dimensional defocusing nonlinear Schrödinger / Gross-Pitaevskiĭ (NLS / GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev-Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS / GP equation. The dispersion relation for shallow solitons is obtained asymptotically beyond the KP limit. This yields 1) the maximum growth rate and associated wavenumber of unstable perturbations; and 2) the separatrix between convective and absolute instabilities. The latter result is used to study the transition between convective and absolute instabilities of oblique dispersive shock waves (DSWs). Stationary and nonstationary oblique DSWs are constructed analytically and investigated numerically by direct simulations of the NLS / GP equation. The instability properties of oblique DSWs are found to be directly related to those of the dark soliton. It is found that stationary and nonstationary oblique DSWs have the same jump conditions in the shallow and hypersonic regimes. These results have application to controlling nonlinear waves in dispersive media.

1 Introduction

The instability of one-dimensional structures to weak, long wavelength, transverse perturbations plays an important role in multi-dimensional nonlinear wave propagation. Examples include nonlinear optics [1], Bose-Einstein condensates (BECs) [2], and water waves [3, 4]. Early theoretical work on the transverse instability of solitons for the Kadomtsev-Petviashvili (KP) equation [5, 6] and the nonlinear Schrödinger (NLS) equation [7, 8] focused on its existence and maximum growth rate, both properties of the real portion of the spectrum of unstable modes. Recent numerical simulations of NLS [9] and vector NLS [10] supersonic flow past an obstacle in two-dimensions reveal the excitation of apparently stable, oblique spatial dark solitons for certain flow parameters. The resolution of this inconsistency was explained in [11], where the instability was shown to be of convective type so that transverse perturbations are carried away by the flow parallel to the soliton plane, effectively stabilizing the soliton near the obstacle. The characterization of convective versus absolute instability requires knowledge of the spectrum for a range of wavenumbers in the complex plane [12, 13]. For NLS dark solitons, the criteria can be simplified and involve the imaginary (stable) portion of the spectrum [11].

One of the hallmarks of supersonic flow is the formation of shock waves. In classical, viscous fluids, shock dynamics can be well understood mathematically in the context of a dissipative regularization of conservation laws (cf. [14]). There are, however, a number of fluids with negligible dissipation whose dominant regularizing mechanism is dispersion (see the review [15]). Most notably, superfluidic BECs and optical waves in defocusing nonlinear media fall within this class of dispersive fluids. When a dispersive fluid flows at supersonic speed, it can form a dispersive shock wave (DSW) that possesses an expanding, oscillatory wavetrain with a large amplitude, soliton edge and small amplitude sound wave edge. DSWs appear as special, asymptotic solutions of nonlinear dispersive equations and have been observed in BEC [16, 17, 18] and nonlinear optics [19, 20]. Their theory is much less developed than their classical (dissipative) counterparts. In particular, there has been limited study of DSW stability. Recent works numerically observe transverse instabilities for NLS DSWs resulting from dark pulse propagation on a background in two spatial dimensions [21] and for oblique DSWs in supersonic flow past a corner [22]. In the former case, the transverse instability was mitigated by introducing nonlocal nonlinearity while in the latter case, the convective nature of the instability effectively stabilizes the oblique DSW in certain parameter regimes. In contrast, oblique shock waves in multidimensional, classical gas dynamics are known to be linearly stable when the downstream flow is supersonic [23, 24, 25] (see also the review article [26] for more general results).

The aim of this work is to clarify the role of absolute and convective instabilities as they relate to spatial dark solitons and apply this understanding to DSWs in multiple spatial dimensions. Analytical and computational challenges include:

  • The multi-dimensional nature of the flows.

  • The general criteria for absolute and convective instabilities requires detailed knowledge of the spectrum.

  • Long time integration and large spatial domains are required to properly resolve DSWs numerically.

To address these challenges, we asymptotically determine the spectrum of transverse perturbations to shallow but finite amplitude NLS dark solitons beyond the Kadomtsev-Petviashvili (KP) limit. This enables determination of the maximum growth rate and associated wavenumber of unstable perturbations. Using adjoint methods, we introduce a simple, accurate method for computating the spectrum and its derivatives numerically for arbitrary soliton amplitudes. Simplified criteria for the determination of the separatrix between absolute and convective instabilities are derived. The separatrix is determined in terms of the critical Mach number as it relates to the soliton far field flow. Oblique dark solitons are convectively unstable when and absolutely unstable otherwise. Using our asymptotic and numerical computations of the spectrum, we determine , demonstrating that with a monotonically increasing function of soliton amplitude.

The oblique DSW trailing edge is well-approximated by an oblique dark soliton. In this study, we apply the soliton stability results to the oblique DSW trailing edge in the stationary and nonstationary cases. Stationary oblique DSWs result from the solution of a boundary value problem (supersonic corner flow) while the nonstationary case arises in the solution of a Riemann initial value problem. We find that oblique DSWs with supersonic downstream flows can be absolutely unstable in contrast to classical oblique shocks. We also show that stationary and nonstationary oblique DSWs have the same downstream flow properties in the shallow and hypersonic regimes.

We consider the (2+1)-dimensional defocusing (repulsive) nonlinear Schrödinger /
Gross-Pitaevskiĭ (NLS / GP) equation


along with appropriate initial and/or boundary data. Equation (1) models matter waves in repulsive BECs and intense laser propagation in optically defocusing (i.e., with normal dispersion) media. In the variables


Equation (1) can be recast in terms of the fluid-like variables (density) and (superfluid velocity)


Note that eqs. (3) in the dispersionless regime (neglecting the right hand sides) correspond to the classical shallow water equations (Euler equations of gas dynamics with adiabatic constant ) with the speed of sound [27].

The outline of this paper is as follows. Section 2 discusses the spectrum of unstable transverse perturbations of dark solitons with asymptotic resolution of the maximum growth rate and associated wavenumber in the shallow regime. Using analytic properties of the spectrum, we recap the derivation of the general criteria for absolute and convective instabilities and for oblique solitons, we derive the simplified criteria in Sec. 3. The separatrix is determined. We derive nonstationary oblique DSWs of arbitrary amplitude and stationary oblique DSWs in the shallow regime, showing the connection between their downstream flows in Sec. 4. The stationary case is compared with (2+1)-dimensional numerical simulations. Convective and absolute instability of oblique DSWs is described in terms of the separatrix for the trailing edge dark soliton. Our numerical methods are presented in Sec. 5. Finally, Sec. 6 contains a discussion of the results and the applicability of our methods to other nonlinear dispersive problems.

2 Transverse instability of dark solitons

It is well-known that dark soliton solutions of (1) exhibit an instability to perturbations of sufficiently long wavelength in the transverse direction along the soliton plane [8]. The eigenvalue problem associated with linearizing (1) about the dark soliton leads to the dispersion relation for unstable perturbations. Beyond demonstrating the existence of an instability, knowledge of the dispersion relation for a range of wavenumbers yields important properties of the instability, such as the growth rate , the maximally unstable wavenumber , and whether or not the instability is convective or absolute.

An example numerical computation of the eigenvalues for the spectral problem in eq. (10) is shown in Fig. 1. Since exact expressions are not known, asymptotic approaches leveraging the shallow dark soliton, KP limit [6, 28] and others [29, 11] have been devised. In this section, we complement these results by determining the next order correction to the dispersion relation for shallow dark solitons resulting in accurate approximation across a wider range of soliton amplitudes. We use this to determine and asymptotically. These calculations are verified numerically.

Figure 1: The real (dashes) and imaginary (solid) parts of the discrete eigenvalue of the linearized NLS equation (spectral problem (10)) as functions of for . Delineated on the axes are: i) the cutoff wavenumber [Eq. (14)], where the eigenvalue transitions from purely imaginary to real, ii) the maximal growth wavenumber and growth rate [Eq. (17)], and iii) the critical wavenumber and associated eigenvalue [Eq. (27)] corresponding to the transition between absolute/convective instabilities.

2.1 Dark Soliton

Up to spatio-temporal shifts and an overall phase, the most general line dark soliton solution of (1) is


where is the background density and the phase jump across the soliton determines the depression amplitude as . The soliton is propagating at an angle with respect to the (horizontal) axis, with horizontal and vertical flow velocities and , respectively. Interpreting this solution in the fluid context with density and flow velocity , the soliton is a localized density depression on a uniformly flowing background. The Mach number of the background flow is the total flow velocity divided by the speed of sound


The soliton has the far field behavior

Thus, five parameters determine the soliton uniquely, i.e. .

Using the invariances of Eq. (1) associated with rotation, Galilean transformation, scaling, and phase, we apply the coordinate transformation


leading to the one-parameter family of dark solitons


where and the frame moving with the soliton is

The soliton amplitude is . When the dark soliton is in the shallow amplitude regime. The soliton speed is .

2.2 Linearized eigenvalue problem

To study the transverse instabilities of the dark soliton (7), we consider the ansatz for Eq. (1)

where , are the real and imaginary parts of a small perturbation. Linearizing (1) results in the system


It is expedient to decompose the perturbation as


Substituting (9) into (8) yields the linearized spectral problem






For , and are self-adjoint with respect to the inner product


For small it was shown formally in [8] that: (i) a double eigenvalue bifurcates into two distinct branches with each in ; (ii) there is another zero eigenvalue at the cutoff wavenumber


These calculations were made rigorous in [30] and can be summarized as follows. {theorem}[Rousset, Tzvetkov [30]] For , the system (10) has exactly two purely imaginary eigenvalues which are simple and come in pairs . Therefore, the dark soliton is unstable to sufficiently long wavelength transverse perturbations. Furthermore, for , , the spectrum is real.

For the study of convective/absolute instabilities, knowledge of the stable portion of the spectrum when is required. Based on numerical and asymptotic computations, we conjecture the following.

Conjecture \thetheorem

For , there exist exactly two real, simple eigenvalues .

This conjecture is a natural extension of Thm. 2.2. See Appendix B for further details and comments.

Without loss of generality, we choose such that for and for . Thus, is the dispersion relation for transverse perturbations of the dark soliton (7). By suitable choice of a branch cut, the eigenvalue can be analytically continued for with and square root branch points. We denote the growth rate as


and the eigenfunction associated with as

In Section 5 we discuss our numerical method for computing for . To illustrate the spectrum, Figure 1 presents the dependence of the (real or imaginary) eigenvalue, , on . Figures 2 and 3 present the computed continuous and discrete spectra for particular wavenumbers () and () as well as the associated localized eigenfunctions. Note that the eigenfunctions are neither symmetric nor anti-symmetric.

Figure 2: (a) Numerical approximation of the continuous spectrum () and the two purely imaginary discrete eigenvalues ( +) computed for the linearized system (10) with and [Eq. (14)]. The real (solid) and imaginary (dashed) parts of the corresponding two component localized eigenfunction are shown in (b) and (c) .
Figure 3: Same as for Fig. 2 except and .

2.3 Asymptotic eigenvalue

It follows from (14) that for shallow amplitude solitons, , the cutoff wavenumber is small, i.e., . In Appendix A we prove: {proposition} For shallow amplitude, , and either or , the eigenvalue for (10) satisfies


where the first (leading order) term is the dispersion relation for the KP equation and the second term is the correction arising from the NLS equation.

Equation (16) gives an asymptotic approximation to the eigenvalue for long wave perturbations of shallow dark solitons. The dispersion relation for the KP equation is well known (cf. [6, 28]). The new correction term enables us to accomplish the following.

  • Implement an accurate, explicit calculation of the maximum growth rate and associated wavenumber of unstable perturbations (Sec. 2.4).

  • Show that the separatrix between absolute and convective instabilities is supersonic (Sec. 3.3).

  • Validate the numerical computations of , which are sensitive and computationally demanding, especially in the shallow regime.

2.4 Calculation of the maximum growth rate

The maximal growth wavenumber and the maximum growth rate are defined by


Since is real for it follows that (see Fig. 1). Using Proposition 2.3 we find

Corollary \thetheorem

The proof follows by expanding and for small and solving eq. (17) with the approximation (16).

A comparison of these results with numerical computations (discussed in Sec. 5) is shown in Fig. 4. The computations exhibit excellent agreement with the asymptotics as well as the expected scaling of the errors with .

Figure 4: Numerically computed maximum growth rate (a) and maximally unstable wavenumber (c) as functions of for dark solitons of the NLS equation (1). The KP limit and its first order correction are presented for comparison. Plots (b) and (d) are the corresponding differences between the highly-accurate computed values and asymptotic approximations (18)–(19), exhibiting the expected scaling with .

3 Convective and absolute instabilities of dark solitons

We begin by reviewing the notions of absolute/convective instabilities and the general criteria for distinguishing between them. For more detailed discussions see [12, 13, 31, 32, 33]. Qualitatively, absolute and convective instabilities can be defined as follows (see illustration in Fig. 5).

Definition \thetheorem

A solution is said to be absolutely unstable if generic, small, localized perturbations grow arbitrarily large in time at each fixed point in space. A solution is said to be convectively unstable if small, localized perturbations grow arbitrarily large in time but decay to zero at any fixed point in space.

It is important to note that Definition 3 depends implicitly on the reference frame as can be gleaned from Fig. 5 where panel (b) is a rotation in the - plane of panel (a). Such a rotation implies that the observer in (b) is moving faster to the left than the observer in (a). Thus, if the observer “outruns” the growing perturbation, then the instability is convective. Equivalently, if the background flow speed is faster than the expanding, unstable perturbation, and after sufficient time passes the solution returns to is unperturbed state, the instability is convective.

Figure 5: Illustration of (a) absolutely and (b) convectively unstable waves.

3.1 Review of the general criteria for distinguishing between instabilities

Absolute and convective instabilities can be distinguished analytically. Consider an initial value problem on the entire line, i.e. a (1+1)-dimensional linearized system on . The usual approach for studying instabilities is to consider a small, spatially extended plane wave perturbation of some wavenumber and corresponding frequency determined by a zero of the dispersion function . The zero state is stable if and only if for all zeros of the dispersion function. However, the evolution of a particular, localized perturbation involves a Fourier integral over all real wavenumbers so that treating a single wavenumber is insufficient to fully describe any instabilities observed (or not observed) in a physical system [12].

The resolution calls for a different approach to instability analysis. Instead of a plane wave perturbation one assumes that the system is perturbed by a localized impulse, i.e. a Dirac delta function at position and . In this case the solution is the Green’s function


where the Fourier integral is carried out over real wavenumbers and the Bromwich frequency contour lies above all zeros of . In connection with the plane-wave analysis, the system is unstable if and only if the solution grows without bound along some reference frame, i.e. there is a velocity such that for fixed ,

However, when considering a particular reference frame, say for fixed , if the solution grows without bound (resp. decays to zero) at a certain fixed point in space, , then the system is absolutely (resp. convectively) unstable in this reference frame, i.e.

  • absolutely unstable.

  • convectively unstable.

Exponential integrals of the type in (20) have two competing effects. Zeros of the dispersion function can lead to exponential growth when or cancellation and decay due to rapid oscillation when for large . To ascertain whether the system is absolutely or convectively unstable one needs to discover which of these opposite tendencies dominates. A number of methods for distinguishing between convective and absolute instabilities have been suggested, dating back to the work of Sturrock [12] andBriggs [13]. See also [34, 35, 36]. For completeness we outline the general criteria below.

Here we assume that is known explicitly. The -integral in (20) is along a contour that lies above all the zeros of for each fixed, real and we further assume that is entire in above this contour. Hence, for the -integral may be carried out by closing the contour in the lower half-plane and summing over the residues of the dispersion function expanded at each of its roots. Assuming the roots of are simple (multiple roots do note pose a serious difficulty [32]), the resulting integral can be written as


where the sum is over the zeros of the dispersion function

The problem is to determine the long time behavior of (21) for which the method of steepest descent is applicable (cf. [37]). For this, we restrict ourselves to the point moving with speed , . Then, by suitable deformation of the real line to the steepest descent contour, the dominant contributions arise from the saddle points of the exponent satisfying

allowing for multiple saddle points along each branch of the dispersion relation. Note that the zeros of , double roots of the dispersion function, do not contribute appreciably to the integral because they cancel in the sum (21). Using the method of steepest descent, one recovers the dominant long time behavior


Thus, if , then an impulse perturbation at , grows without bound along the line and the instability is absolute. Otherwise, if , the perturbation decays along the line and so the instability is convective. These have been referred to as the Bers-Briggs criteria [13, 32].

3.2 Simplified criteria for the separatrix of soliton instabilities

Many previous studies applied the general criteria for classifying instabilities to dissipative systems (plasma physics, viscous fluids, etc.) where the dispersion relation was known explicitly. Given explicit (and sufficiently simple) dispersion relations, the analysis of the stationary points can be carried out directly. However, the dispersion relation is unknown for dark solitons of the NLS equation. It can be computed numerically, but this makes the analysis of saddle points in the complex- and / or complex- planes quite challenging. Fortunately, as derived below, there are simplified analytic criteria for the transition point between absolute and convective instabilities of NLS solitons that rely solely on computations of the dispersion relation for real .

Using the Laplace transform in eq. (8), the linearized evolution of an initial perturbation to the dark soliton satisfies

where the Bromwich contour lies above all eigenvalues of . In order to investigate the unstable transverse dynamics in , we project onto the eigenfunction and perform the contour integration over resulting in the following representation of the dynamics

The integral is taken over by use of the invariance of the eigenpair

By performing a Galilean shift in the NLS equation (1) as


the dispersion relation for transverse perturbations becomes


where is the flow speed parallel to the plane of the dark soliton (7). This is equivalent to investigating the behavior of the perturbation in eq. (22) along the line . With this substitution, we consider eq. (22) whose long time asymptotic behavior requires the evaluation of


where the dependence on is suppressed. The integral over is negligible because the dispersion relation is purely real (the stationary phase method yields algebraic decay in , cf. [37]). Introducing the change to a complex variable , eq. (24) becomes


where and the contour is . Two distinct possibilities arise.

Figure 6: Integration contours (solid curves) in the complex plane for eq. (25) and the real interval (dashed lines). The filled circles correspond to poles of the integrand where , , which in (a) prevent the smooth deformation of to giving rise to an absolute instability. Parameter values are , . (a) . (b) . See also Fig. 7.
  1. A zero of gives a residue contribution to Cauchy’s theorem when deforming to the real interval as in Fig.6(a). In this case the integral diverges exponentially as and the instability is absolute.

  2. The zeros of do not lie between and the real line as in Fig.6(b) (they may lie on the real axis) so that there is a smooth deformation of the contour to the real interval . In this case the integral decays to zero as and the instability is convective.

Figure 7: Plot of for real and . The minimum of this curve corresponds to the coalescence of the poles in Fig. 6 and the critical transverse flow speed at which the instability changes from absolute to convective. The dashed lines correspond to the values of used to compute Fig. 6(a) (lower, absolute instability) and Fig. 6(b) (upper, convective instability).

As discussed in the previous section, the saddle points give the long time asymptotic behavior , . As the transverse flow speed is varied, the type of instability changes from absolute to convective. The transition from absolute to convective instability occurs at when two zeros of merge on the real line. That is, they form a double zero so that and is minimum. This behavior is depicted in Figs. 6 and 7 with 6(a) showing two complex conjugate zeros for while 6(b) reveals their splitting into two real zeros for . These real zeros are depicted in Fig. 7 for . By an appropriate choice of the branch cut, one can show that so that complex zeros of come in conjugate pairs. This proves {proposition} The critical wavenumber and critical transverse velocity for the transition between absolute and convective instability are real. They satisfy the simplified criteria


These conditions were first proposed in [11].


The proof follows from (23) and (26).

When the transverse flow speed is subcritical, , the dark soliton is absolutely unstable and when the dark soliton is convectively unstable. The soliton family is parametrized by its amplitude , thus forms a separatrix between absolute and convective instabilities. The separatrix can also be interpreted as the speed at which an initially localized perturbation spreads in time. Thus a convective instability occurs when the background flow speed, carrying the perturbation’s center of mass, exceeds the speed at which the perturbation spreads out.

In general, the determination of via (27) requires numerical computation. Even so, Eqs. (27) are much easier to use than the general criteria because the general criteria depend on over the complex- plane whereas (27) only depends on for real .

3.3 The separatrix in the shallow amplitude regime

The shallow-amplitude asymptotics of the dispersion relation (16) enable us to explicitly compute and , determine the critical wavenumber and find the separatrix between absolute and convective instabilities. Here it is convenient to use the wavenumber scaling (see Appendix A)

The asymptotic dispersion relation (23) becomes

The simplified criteria (27) give

Equating like coefficients of and using (27), yields {proposition} The first order asymptotic approximation of the critical velocity and wavenumber are


For comparison, Fig. 8 shows the numerical solution of the system (27) and the asymptotics in (28). The numerical details are presented in Sec. 5.1.

Figure 8: (a) The separatrix between absolute and convective instabilities. For speeds (), the dark soliton is convectively (absolutely) unstable. (c) The critical wavenumber satisfying the condition (26a). (b) and (d) are the differences in the numerically computed values of , and the first order approximations in (28a) and (28b), respectively, showing the expected error scalings.

3.4 Convective/absolute instabilities of spatial dark solitons

The natural reference frame for studying the convective or absolute nature of soliton instabilities is the one moving with the soliton. In this reference frame, both the soliton density and velocity are independent of time. The dark soliton is referred to as a spatial dark soliton. Such structures arise, for example, in the context of flow past an impurity [9, 38], flow over extended obstacles, and dispersive shock waves [39, 40].

The spatial dark soliton in (4) satisfies , which determines the phase jump


This soliton is uniquely determined by four parameters rather than five. We use the background density and background velocity as three of these parameters along with either the normalized soliton amplitude or the soliton angle , the two being related via (29) through


The spatial dark soliton exhibits either an absolute or convective instability depending on the Mach number of the background flow (5) and either the amplitude or, equivalently, the soliton angle . By moving in the reference frame of the normalized dark soliton (7), the background flow has velocity normal to the soliton and velocity parallel to the soliton. The critical Mach number of the background flow and its first order asymptotic approximation are


Transverse perturbations are absolutely unstable for and convectively unstable for .

Figure 9: Critical Mach number (Eq. (31a)) and its asymptotic approximations ( for KP, Eq. (31b) for the first order correction) as functions of the soliton amplitude (a) and the spatial soliton angle (b). For , the spatial dark soliton is convectively unstable, otherwise it is absolutely unstable.

We also compute the critical Mach number’s dependence on the soliton angle by use of eq. (30) leading to a transformation between and

Therefore, shallow spatial dark solitons at critical velocity have a small , so that


Figure 9 shows the numerically calculated dependence of on and and comparisons with the asymptotic results (31b) and (32). Combining the asymptotic result (31b) with these computations leads to

Conclusion \thetheorem

The transition between convective and absolute instability for spatial dark solitons always occurs at supersonic speeds . A sufficient condition for a spatial dark soliton with background Mach number to be absolutely unstable is

A sufficient condition for a spatial dark soliton with background Mach number to be convectively unstable is

Additionally, is monotonically increasing. In sum,

Remark \thetheorem

In [11], the bounds were obtained. The leading order term in eq. (16) was used to show that in the shallow regime. Equation (31b) improves the lower bound on and demonstrates that is strictly supersonic for all finite soliton amplitudes. The upper bound in [11] was calculated from a rational approximation of the spectrum for large soliton amplitudes [29]. Equation (33) gives the accurate upper bound.

4 Oblique dispersive shock waves

In a dispersive fluid where dissipation is negligible, a jump in the density/velocity may be resolved by an expanding oscillatory region called a dispersive shock wave. The Whitham averaging technique [41] has been successfully used to describe a DSW’s long time asymptotic behavior in a number of physical systems, for example [42, 43, 17, 44, 45]. We briefly recap the rudiments of DSW theory. A DSW is a modulated wavetrain composed of a large amplitude, soliton edge and a small amplitude, oscillatory edge, each moving with different speeds. In the relatively simple case where a DSW connects two constant states, the speeds associated with each edge are determined by jump conditions [46], in analogy with the Rankine-Hugoniot jump conditions of classical, viscous gas dynamics. The jump conditions result from a simple wave solution of the Whitham modulation equations connecting the zero wavenumber, soliton edge to the zero amplitude, oscillatory edge. The existence of a DSW for a particular jump in the fluid variables is guaranteed when an appropriate entropy condition is satisfied. For a left-going DSW, we define the leading (trailing) edge to be the leftmost (rightmost) edge – and vice versa for a right-going DSW. The sign of the dispersion determines the locations of the soliton and small amplitude edges. For systems with positive dispersion such as the NLS eq. (1), the soliton is a depression wave that resides at the trailing edge of the DSW.

While DSWs in (1+1)-dimensions have been well-studied, the theory of supersonic dispersive fluid dynamics in multiple spatial dimensions is in its infancy. Perhaps the simplest DSW in multiple dimensions is an oblique DSW, which has been studied in the stationary [47, 48, 49, 39] and non-stationary [40] regimes (see Figs. 10 and 13). In this section, the analysis from the previous section is applied to the stationary and nonstationary oblique DSW soliton trailing edge to determine the separatrix between convective and absolute instabilities. In addition, in the weak shock and hypersonic regimes, we find that the jump conditions for stationary and nonstationary oblique DSWs are the same. As in classical gas dynamics, oblique DSWs can serve as building blocks for more complicated boundary value problems. Therefore, understanding the instability properties of oblique DSWs is important and relevant to supersonic dispersive flows. This has been further demonstrated by recent numerical simulations of NLS supersonic flow past a corner [39, 40].

In Sec. 4.1, the jump conditions and instability properties of nonstationary oblique DSWs are presented. The following Sec. 4.2 contains a derivation of a stationary oblique DSW in the shallow regime, its stability, and comparisons with numerical simulation. Finally, Sec. 4.3 demonstrates the connections between stationary and nonstationary oblique DSWs.

4.1 Nonstationary Oblique DSWs

Figure 10: Schematic of an oblique DSW.

In this section, we first recap the derivation of a nonstationary oblique DSW [40] and then discuss its instability properties.

A schematic of a non-stationary oblique DSW at a specific time in its evolution is depicted in Fig. 10. An incoming upstream, supersonic flow is turned through the oblique DSW by the deflection angle . To accommodate the deflection, the oblique DSW expands along the wave angle . The leading edge consists of small amplitude waves propagating into the upstream flow while the trailing edge is composed of a dark soliton whose amplitude and speed are asymptotically calculated from the oblique DSW jump conditions.

The nonstationary oblique DSW results from the long time evolution of an initial jump in the density and velocity component normal to the DSW wave angle , in the direction , and continuity of the velocity parallel to , in the direction . We consider the upstream state

and the downstream state

The normal 1-DSW associated with the dispersionless characteristic (left-going wave) satisfies the simple wave condition [43]


A NLS governed fluid experiences potential flow (see eq. (2)). By restricting the spatial variation of the solution to the direction and integrating the irrotationality constraint along the direction , we obtain the continuity of the parallel velocity component


Choosing the reference frame in which the soliton trailing edge is fixed, the speed of the soliton edge satisfies [43]


The jump conditions (34), (35), and (36) for the oblique DSW relate the upstream quantities , and one of the angles or to the downstream quantities , and the other angle. Introducing the Mach numbers , along with some manipulation, the jump conditions become [40]


Further manipulations lead to the equivalent relations

The associated entropy condition is , which, when incorporated into the jump conditions, gives

These state that the upstream flow must be supersonic, the flow always turns into the DSW, and the wave angle is larger than the Mach angle . The Mach angle is half the opening angle of the Mach cone inside of which infinitesimally small disturbances are confined to propagate in dispersionless supersonic flow. A convenient way to visualize these results is by the -- diagram in Fig. 11 that relates the deflection and wave angles for a given upstream Mach number . Figure 11 includes the sonic curve (to the right/left the flow is sub/supersonic).

A natural question is whether oblique DSWs with supersonic downstream flow conditions are convectively or absolutely unstable. To address this question, we use:

Definition \thetheorem

Transverse perturbations to the nonstationary and stationary oblique DSW are convectively (absolutely) unstable whenever the trailing, dark soliton edge is convectively (absolutely) unstable.

Figure 11: The -- diagram for non-stationary oblique DSWs of the NLS equation (1). Each upstream Mach number leads to a relationship between and . The separatrix curve (solid) between convectively and absolutely unstable solitons is supersonic, i.e. in the region (left of the sonic line, dashes). The separatrix curve asymptotes to the sonic line as .

See further discussion in Sec. 6.

Spatial dark solitons exhibit the constraint (29). When applied to the oblique DSW trailing edge in Fig. 10 with background flow parameters , we find

Using the jump conditions in eqs. (37), we determine the normalized soliton amplitude


where is related to by (37a). The Mach number of the downstream flow adjacent to the soliton is so the absolute/convective stability criterion (31a) determines the separatrix


with given in (38). Conclusion 3.4 implies

Corollary \thetheorem

Nonstationary oblique DSWs with subsonic downstream flow are absolutely unstable. Supersonic downstream flow can be either convectively or absolutely unstable.

This conclusion can also be gleaned from Fig. 11. To the right of the separatrix, the trailing edge oblique soliton is absolutely unstable because while to its left, the soliton is convectively unstable. The region to the right of the separatrix and to the left of the sonic line represents absolutely unstable oblique DSWs with supersonic downstream flow conditions. Below we derive additional properties of the separatrix.

From Fig. 11, we observe a minimum wave angle , below which the oblique DSW is convectively unstable. Setting in eq. (37b) and solving for we find

which has a minimum for , . We therefore have a sufficient condition for the oblique DSW trailing edge to be convectively unstable