Dispersive hydrodynamics in non-Hermitian nonlinear Schrödinger equation with complex external potential

In this paper dispersive hydrodynamics associated with the non-Hermitian nonlinear Schrödinger (NLS) equation with generic complex external potential is studied. In particular, a set of dispersive hydrodynamic equations are obtained. They differ from their classical counterparts (without an external potential), by the presence of additional source terms that alter the density and momentum equations. When restricted to a class of Wadati-type complex potentials, the resulting hydrodynamic system conserves a modified momentum and admits constant intensity/density solutions. This motivates the construction and study of an initial value problem (IVP) comprised of a centred (or non-centred) step-like initial condition that connects two constant intensity/density states. Interestingly, this IVP is shown to be related to a Riemann problem posed for the hydrodynamic system in an appropriate traveling reference frame. The study of such IVPs allows one to interpret the underlying non-Hermitian Riemann problem in terms of an ‘optical flow’ over an obstacle. A broad class of non-Hermitian potentials that lead to modulationally stable constant intensity states are identified. They are subsequently used to numerically solve the associated Riemann problem for various initial conditions. Due to the lack of translation symmetry, the resulting long-time dynamics show a dependence on the location of the step relative to the potential. This is in sharp contrast to the NLS case without potential, where the dynamics are independent of the step location. This fact leads to the formation of diverse nonlinear wave patterns that are otherwise absent. In particular, various gain-loss generated near-field features are present, which in turn drive the optical flow in the far-field which could be comprised of various rich nonlinear wave structures, including DSW-DSW, DSW-rarefaction, and soliton-DSW interactions.

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Introduction
Dispersive evolution equations arise in many areas of the physical sciences [1,84,85].They describe a wide range of phenomena in optics [47,48,55], fluid mechanics [52,72], atomic physics (e.g.Bose-Einstein condensates) [10,29,46,71], and magnetic materials [51,77], to name a few.Universal model equations of physical significance include: (i) the Kortewegde Vries (KdV) (that describes the dynamics of weakly nonlinear and unidirectional longwaves) and (ii) the nonlinear Schrödinger (NLS) that typically arises from a slowly varying envelope approximation.Generally speaking, dispersion/diffraction leads to energy spreading while nonlinearity results in self-steepening or self-defocusing.Due to the interplay between these competing effects, various nonlinear coherent structures can emerge.Important examples include solitons (or solitary waves), breathers and dispersive shock waves (DSW) to name a few [1,5,23,53,84].In this regard, a DSW is a particular type of wave that has attracted much attention in recent years due to its fundamental role in the theory of nonlinear wave breaking.It could be generated, for example, when an initial waveform exhibiting a rapid transition in its intensity is set in motion within a dispersive medium.It manifests itself in the form of a continuously expanding, modulated periodic wave that connects two distinct constant-intensity asymptotic states.Within the framework of Riemann problems, their theoretical characterization was done in [33] for the KdV and in [21,32] the defocusing NLS equation.Another mechanism to generate a DSW (as was done for the NLS) is through the so-called piston problem.Its formation is achieved by compressing a stationary 'optical fluid' through the action of a fast-moving impenetrable, barrier.This idea was first examined within the context of dispersive wave equations in [38].A related problem is the generation of DSW due to optical flows past stationary obstacles of finite size [34,56].DSWs have been experimentally observed and theoretically studied in optical media [7,19,27,69,74,82], quantum condensates (Bose-Einstein, fermionic and non-Hermitian polaritonic type) [6, 11, 12, 25, 35, 37, 39, 42-44, 54, 58], granular crystals [14,15] and fluid dynamic settings [13,16,28,75,80,81] (see the review paper [23] for applications and additional references).
Dispersive hydrodynamics (DH) provides a broad framework for the study of DSW and other nonlinear multiscale wave structures including their interactions.The research efforts in this area have focused on describing wave phenomena both in uniform and inhomogeneous systems (see for e.g.[34,36,37,44,54]).Generally speaking, formulating an associated Riemann problem constitutes a fundamental issue when studying nonlinear wave breaking.However, the absence of constant intensity waveform in inhomogeneous systems hinders the formulation of such problems.Recent works have shown that generalized plane waves (consisting of constant amplitude and nonlinear phase); known as constant intensity waves, (CI waves for short) in non-Hermitian media do exist as was demonstrated, for the NLS equation with an externally imposed Wadati potential [17, 49, 60, 61, 63-65, 70, 73, 88].Motivated by these findings, in this paper we aim to study the dynamical evolution of a step-like initial condition connecting two CI waves, prescribed for the NLS with complex Wadati potential.In an appropriate traveling reference frame, we demonstrate that this class of problems can be formulated as a family of Riemann problems.The starting point for the paper is the defocusing NLS equation subject to a generic class of complex external potentials.A set of dispersive Euler-type equations are derived that describe the evolution of the wave intensity (or density) and its momentum density.When confined to the complex Wadati-type potentials, the resulting non-Hermitian hydrodynamic system admits CI waveforms.Importantly, one can now specify a step-like initial condition that connects two exact constant intensity states.As such this initial value problem (IVP) reduces to a family of Riemann problems.We have identified a broad class of complex-valued potentials leading to 'modulationally' stable states-an essential ingredient in the study of Riemann problems.Such stable hydrodynamic backgrounds are later utilized to define the non-Hermitian Riemann problem.Due to the lack of translational symmetry, a key parameter that plays a role in determining the dynamics is the location of the step-like initial condition.An interesting attribute of the non-Hermitian Riemann problems considered here is the fact that gain and loss (arising from the imaginary part of the potential) balance each other to support a class of stable near-field patterns.They significantly influence the far-field counterflows where multi-scale wave interactions could emerge (that are otherwise absent in their bulk NLS counterparts).As such, the field of non-Hermitian (nonlinear) photonics has recently attracted much attention, due to the possibility of studying rich wave phenomena and pattern formations in physically realizable photonic settings [76].We envision our present work to provide avenues to further explore the phenomenon of wave breaking in such exotic non-Hermitian photonic media.
The outline and organization of the paper is as follows: • In section 2, we derive a general framework for the study of non-Hermitian DH based on the NLS equation with a complex potential.This models wave propagation in optical/photonic media with Kerr nonlinearity and an externally imposed gain-loss distribution.• In section 3, we review the generalized plane wave-type solutions (also known as CI waves) that are supported by non-Hermitian Wadati-type complex potentials.• In section 4, we identify modulationally stable constant intensity waves that residing in non-Hermitian Wadati potentials.In addition, for unstable CI waves, we probe possible pattern formation using direct numerical simulations.• In section 5, we restrict the underlying hydrodynamic formulation of the NLS to a class of complex Wadati potentials.Thereafter, we show that the resulting dispersive hydrodynamic equations subject to a step-like initial condition is related to a non-Hermitian dispersive Riemann problem.Moreover, we solve such Riemann problems using direct numerical simulations.• In section 6, we conclude the paper with some discussions and comments on future work.

Non-Hermitian optical-hydrodynamic formulation
We begin our study of non-Hermitian DH by considering the one-dimensional, defocusing NLS equation in the presence of an external complex potential with V R , V I being real-valued functions representing the optical waveguide and the gain-loss landscape respectively.In terms of the complex wave function ψ(x, t), the beam intensity and the transverse power flow density are respectively defined by Due to the presence of a complex potential, these quantities are in general not conserved.In fact, their evolution is governed by ) where here, prime denotes derivative with respect to x.An equivalent hydrodynamic formulation of the NLS equation that arises across diverse disciplines (such as super-fluids and 'optical fluids') can be obtained from the application of the Madelung transformation Indeed, substituting equation (2.7) into (2.1), one arrives at the forced dispersive Euler equations ) where ρ ≡ I is the hydrodynamic density, u = ϕ x ≡ S/I is the hydrodynamic velocity, P(ρ) = ρ 2 /2 denotes the so called hydrodynamic pressure law (which defines the speed of 'sound' . Note that the velocity potential ϕ satisfies a Bernoulli-type equation given by ) constitute what we refer to as the non-Hermitian dispersive hydrodynamic system.Note the presence of the 'body' force term proportional to V I ρu that arises as a result of the non-conservation of hydrodynamic 'mass'.For flows that possess vacuum points (where ρ = 0), the momentum is a more appropriate characterizing variable than the hydrodynamic velocity [9,37].Clearly, the phase ϕ and the hydrodynamic velocity u are undefined whenever ρ vanishes.The hydrodynamic formulation of the NLS equation, i.e. equations (2.8) and (2.9)/(2.11)allows one to appreciate it is underlying (and hidden) dual dispersive-hyperbolic flavour.To see this, we consider the scenario of slowly varying hydrodynamic initial conditions ρ(x, 0) = ρ(x), u(x, 0) = ũ(x), (where x = δx, for δ ≪ 1) which together with a slowly varying waveguide (V R (x) = ṼR (x)), weak gain-loss (V I (x) = δ ṼI (x)) and for short times t = δt lead to the quantity Q x in equation (2.9) being negligible (O(δ 2 )).Thus in this slowly varying limit, one can then drop this term, and the hydrodynamic system admits the hyperbolic advection-reaction form: (2.12) where J(u, ρ) is the Jacobian matrix given by As a result of this hyperbolic formulation wave steepening could occur.However, once this becomes significant, the dispersive term Q x is no longer negligible which causes wave regularization.To better understand the short-time behaviour of the waveform, we examine the behaviour of the dispersionless advection-reaction system (equation (2.12)).To this end, the left eigenvectors of the matrix J and their associated eigenvalues are found to be and: The use of the left eigenvectors (and eigenvalues) allows one to recast the system in a diagonal/ Riemann invariant form given by [84]: where r i , c i , i = 1, 2 are the Riemann invariants and the hyperbolic speeds (defined in equation (2.15)) respectively .17) The b i , i = 1, 2 are the reaction terms defined by While the Riemann invariants and the hyperbolic speeds remain identical to those of the corresponding one-dimensional cubic NLS equation without external potential, equation (2.16) acquires new reaction terms b i , which have two contributions.The first is the gradient term, V ′ R /2, while the second term (∓(r 1 − r 2 )V I )), arises from the optical gain and loss distribution.

Non-Hermitian constant intensity waves
A fundamental ingredient in the study of (classical) NLS DH is the concept of a plane wave.Notably, this leads to the interpretation of a constant 'background' to the underlying regularized Euler equations (obtained by setting V ≡ 0 in equations (2.8) and (2.9)).Several nonlineardispersive excitations such as dark solitons (for the defocusing NLS) reside on such a constant background.Moreover, propagating dispersive shocks and rarefaction waves provide a transition between any such distinct hydrodynamic backgrounds.In Hermitian media (for which V in equation (2.1) is assumed to be real), such elementary constant intensity hydrodynamic backgrounds do not exist.However, for a large class of non-Hermitian potentials, such restrictions can be lifted.In such circumstances, interesting studies including spontaneous pattern formation on constant intensity backgrounds and non-Hermitian Riemann problems reveal rich dispersive hydrodynamic phenomena.Indeed for the complex Wadati potential where w(x) is an arbitrary smooth function, constant intensity states do exist [65,70].They are characterized by a one-parameter family of constant amplitude waveforms: where ρ 0 ⩾ 0 is a constant and θ ′ = w.Interestingly, this non-Hermitian potential can support what we refer to as a topological CI mode [70].In other words, there exists a certain class of w(x) for which the phase difference is not equal to an integer multiple of 2π.Here θ + and θ − denote the right and left asymptotic values of the phase respectively.This is a direct consequence of the underlying non-Hermiticity.In essence, this topological property is counterintuitive since imprinting a phase difference traditionally requires a spatially inhomogeneous wave intensity.An example of such a topological CI wave corresponding to θ = 7π tanh(x)/16 is shown in figure 1, as one can see, they exhibit interesting dynamics when propagating inside the optical medium.One can view such non-Hermitian constant hydrodynamic backgrounds as generalizations to their respective classical plane wave counterparts that exist in bulk medium (V is constant).In this section, we aim to establish regimes in parameter space for which a constant intensity solution given by (3.2) is modulationally stable or unstable.This is particularly important when dealing with non-centred Riemann problems (see section 5) as well as in the study of DSW in non-Hermitian media.To this end, we seek a solution to the NLS equation in the presence of an external potential (equation (2.1)) in the form Keeping linear terms leads to We make the bi-modal ansatz given by where v ≡ [f λ , g λ ] T ; λ is the (in general) complex stability spectra, and the stability operator H is defined by The aim next is to compute the spectra of the matrix operator H corresponding to both localized (discrete eigenvalues) and bounded (continuous spectra) eigenfunction f λ , g λ .Clearly, the existence of purely real spectra implies modulationally stable CI waves.On the other hand, a complex spectrum would lead to absolute instability, with the imaginary part of λ measuring the perturbation growth rate.It is expedient to make some remarks related to the structure of the spectra of H.For localized w(x), the continuous spectra can be found by taking the |x| → ∞ limit, in which case the stability matrix approaches Clearly, the continuous spectra of H ∞ are given by This in turn implies that any instability that could arise would manifest itself only in the near field, i.e. in the region of the optical gain and loss.In other words, any instability has to originate from the localized eigenfunctions associated with the discrete spectrum.Therefore, the non-Hermitian backgrounds are stable in the far field.Further information into the structure of the spectra can be obtained by examining the linear operator H when the hydrodynamic velocity w is assumed to be of even parity.To this end, few remarks are in order: (1) The operator H and the non-Hermitian complex potential V defined in (3.1) are PTsymmetric, i.e.
We next compute the modulation stability spectra corresponding to some typical cases by using the Floquet-Fourier-Hill method.While originally developed to approximate all the spectral elements of operators with periodic coefficients (i.e.bounded domains), the method has been used to compute spectra of non-periodic linear operators defined over the whole real line.With this in mind, we approximate the coefficient w(x) in equation (4.5) with a Fourier series representation on a sufficiently large periodic domain Next, we seek bounded two-mode eigenfunctions given by the Floquet Fourier expansion form where Substituting equations (4.11) and (4.12) into the stability problem (4.4) we obtain a system of infinite dimensional algebraic equations parametrized by the positive integer p ) where and In practice, the above system is replaced by a truncated set of equations that assume the form of a block matrix eigenvalue problem.Subsequently, it is solved using the QR-algorithm.The accuracy of this periodic approximation is determined by the degree to which the periodic bands shrink to isolated discrete eigenvalues as the domain size 2l gets larger.It should be noted that the numerical approximation of the spectra of a non-self-adjoint matrix differential operator defined on the whole real line is a challenging problem that often leads to the appearance of clouds of spurious complex spectra [31].A possible remedy to reduce these artifacts is to choose a larger computational domain together with a greater number of Fourier modes.
Here we consider a PT symmetric potential with a balanced gain and loss configuration which stands out for its fundamental importance in non-Hermitian photonics [26,62,67].A typical single waveguide considered in this paper corresponds to a potential of the form On the other hand, ψ CI becomes modulationally unstable whenever β > β c with the instability spectra appearing in complex quartets {λ, −λ * , −λ, λ * }.For example, when β = 2, the instability eigenfunctions associated with the complex eigenvalue λ = −0.206− 0.052i is shown in figure 4.Moreover, we have verified that the computed two-component instability eigenmodes associated with λ, λ * , −λ * , and −λ indeed satisfy the symmetry relations mentioned above.Interestingly, for the β values used in figure 3, the spectrum of the linear operator We have also computed the spectrum of the linear operator H corresponding to CI waves with different intensities ρ 0 while keeping the non-Hermitian potential V fixed.Our computations indicate that for ρ 0 > 1, the phase transition occurs at higher values of β c .We present numerical findings shown in figure 5, where a linear relationship between the stability threshold β c and amplitude √ ρ 0 is observed.This is evident from the agreement with a best-fit (in the least squares sense) line that nearly interpolates all the relevant data-points.To understand the source of this relationship, we turn to the symmetry exhibited by the stability eigenvalue problem detailed in remark 7.This property suggests that if a CI wave of density ρ 0 is characterized by a bifurcation point β c , then the threshold for the stability of a CI wave  of density α 2 ρ 0 is marked by αβ c .This suggests a linear relationship between β c and the wave amplitude √ ρ 0 .

Direct numerical simulations and pattern formation
So far, the above results were obtained from a linear stability eigenvalue problem (4.4).To further supplement these findings, we perform direct numerical simulations using equation (2.1) subject to the perturbed initial condition given by where ρ 1 and ϕ 1 are the perturbations to the density and phase respectively.Here ρ 1 and ϕ 1 are generated by a superposition of Gaussians with random centres (c n , e n ), random amplitudes (a n , b n ) and width d.Specifically, they are given by The amplitudes and centres are sampled from a uniform distribution on the intervals [0, 1] and [−500, 500] respectively.Here, D(x) is a window function used to localize the otherwise extended random Gaussians.It is given by: The parameters δ and ν control the amplitude and window width of this 'filter' function.Since any instabilities that could arise are associated with the discrete eigenvalues (and the associated localized eigenfunctions), it is sufficient to consider such localized disturbances.Two numerical experiments were performed, one for the modulationally stable case and the other for an unstable waveform.In the first scenario, corresponding to w = πsech 2 (x)/2 with β = (π/2), we did not observe any growth in the density for 0 ⩽ t ⩽ 300, confirming the findings obtained from the linear stability analysis.The second situation involves CI waves having hydrodynamic velocity given by w = 2.5sech 2 (x) (β c < 2.5), that are predicted to be modulationally unstable according to the linear stability analysis.In this case, the direct numerical simulation shows a transient growth in amplitude that seems to agree with the spectral linear stability analysis.As a result of this absolute instability, several dark solitons are seen to form in the lossy optical region.Upon nonlinear saturation, there is a formation of a 'quasistationary' localized mode, residing on a constant background, whose peak density is located within the region of optical gain (see figure 8(b)).It should be noted that this quasi-steady pattern is formed in the transverse near-field, and is a direct consequence of the external Wadati potential.Accompanying this 'state' are two far-field counter-propagating DSWs each of which resembles a classical NLS DSW (see figures 8(a),(c) and (d)).The right propagating one, in particular, interacts with the dark solitons that are generated inside the lossy optical region (figure 8(d)).Our findings are summarized in figure 6 together with the time snapshots in figures 7 and 8. Interestingly enough, this quasi-stationary mode can be approximated by seeking a solution to the governing hydrodynamic equations in the form subject to an external complex potential of the Wadati-type (3.1) with w ≡ w(x) and β = 2.5.Substituting this ansatz in (2.8) and (2.10) leads to Note that the end density ϱ and the Wadati mode density distribution ϱ are part of the same expression and need to be recovered simultaneously.This intricacy is mainly due to the nontrivial gain-loss distribution w ′ .We suspect that for a further analytical description of the wavepattern, it might be useful to resort to asymptotic arguments (in particular restricting w to the delta-function limit), which we leave for a future work.

Non-Hermitian dispersive Riemann problem
In this section, we aim to formulate and study a class of Riemann problems associated with a non-Hermitian variant of the NLS equation (2.1) in the presence of complex external potentials defined in (3.1).To begin our study we consider the following NLS with arbitrary w(x).Interestingly, the loss-gain driven model (5.1) possesses a local conservation law, even in the absence of any PT -symmetry, given by (see [68]) where Q and F are the density and flux respectively.The hydrodynamic equations associated with equation (5.1) are obtained via the use of the Madelung transformation (ψ = √ ρ exp(iϕ), u = ϕ x ), which yields where P ≡ ρ 2 /2 and R ≡ − 1 4 ρ(lnρ) xx are the hydrodynamic and quantum pressures respectively.We have a local conservation law for a modified hydrodynamic momentum density that results from (5.2).The hydrodynamic quantity ρ(u − w) is thus globally conserved for rapidly decaying intensities. (5.5) In the dispersionless hydrodynamic limit, for which the quantum pressure R is absent, system (5.3) and (5.4) can be put in a Riemann invariant form similar to (2.16), with the exception that the reaction terms b 1,2 now read ) . (5.6) As mentioned before, equation (5.1) supports a broad class of CI waves characterized by their phase θ, (with θ ′ = w) and density ρ 0 (see equation (3.2)).Their modulation stability properties were established in section 4. With this key ingredient (i.e.modulationally stable CI states), we next proceed to study the time-evolution of a three-parameter family of step initial conditions (in density).These three parameters are: step location x 0 that arises due to the absence of spacetranslational symmetry; ρ (1) 0 , which correspond to the constant densities when x < x 0 and x > x 0 , respectively due to lack of reflection and scaling symmetries in equation (5.1).Thus, the dispersive hydrodynamic system in equations (5.3) and (5.4) is subject to the step initial conditions 0 , as x < x 0 , ρ (2) 0 , as x > x 0 . (5.7) In terms of the underlying NLS equation (5.1), this family of initial conditions (equation (5.7)) translates to 0 exp (iθ (x)) , as x > x 0 . (5.8) The initial condition (5.7) exhibits a jump discontinuity only in density, while the hydrodynamic velocity remains continuous.As such the dispersive hydrodynamic system (5.3) and (5.4) subject to (5.7) resembles a Riemann problem (that requires all initial conditions to be piecewise constant).
However with the use of certain change of variables, one can actually formulate a special class of Riemann problems that correspond to piecewise constant densities and zero velocities.Indeed, substituting ψ(x, t) = Ψ(x, t) exp(iθ(x)) into equation (5.1) yields (5.9) In terms of the Madelung ansatz Ψ(x, t) = √ ρ(x, t) exp(iΦ(x, t)), where Φ = ϕ − θ and U = Φ x , the hydrodynamic formulation of equation (5.9) reads (note that (5.10) (5.11) For the system, one can now define a class of Riemann problems with zero initial hydrodynamic velocity given by (5.12) In terms of the wavefunction Ψ(x, t), this family of Riemann initial conditions corresponds to 0 , as x > x 0 .
(5.13) Some remarks are in order: (1) The linear stability properties of the hydrodynamic backgrounds Ψ 0 = √ ρ 0 exp(−iρ 0 t) (a solution to equation (5.9)) are given by the same stability eigenvalue problem in equation (4.4).(2) If the wavefunction Ψ(x, t) is a solution to equation (5.9), with Wadati velocity w(x), then αΨ(αx, α 2 t) is also a solution with hydrodynamic velocity αw(αx) for α ̸ = 0. (3) Throughout the rest of this section, the Wadati potential is fixed in such a way that w resembles a localized function with localization length L. (4) The characteristic length scale associated with the non-Hermitian Riemann problem (equations (5.1) and (5.14)) is set by the localization scale L of w.This in turn defines the near-field wave dynamics.This is in contrast to the bulk NLS case (w ≡ 0), where selfsimilar theory plays a crucial role in the complete description of the Riemann problem dynamics (see for e.g.[20]).( 5) As a result of item (4), the family of Riemann problems considered here can be interpreted in terms of a dispersive hydrodynamic flow over a localized obstacle.The related equations (5.3) and (5.4) supplemented with (5.7) provides such a description for rapidly decaying w.For ease of computations, we replace the step initial condition (5.13) with a smoothened one given by (5.14) 0 .The parameter α ≫ 1 controls the transition width between two CI states of densities ρ Remark: To test the accuracy of the numerical simulations, we obtain a rate equation for the hydrodynamic momentum below.We shall restrict the derivation to a rapidly decaying w and a hydrodynamic velocity U(x, t).As a consequence of (5.11), the identity is readily found.To this end, throughout the rest of this paper, we report on extensive numerical experiments performed on the non-Hermitian Riemann problems equation (5.9) supplemented with various initial conditions given by (5.14).The numerical tests presented in this paper are performed for w = πsech 2 (x)/2, and system parameters given by The time-stepping method used to solve the Riemann problem is based on a fourth-order Cauchy-type exponential time differencing (ETD) scheme (see [45] for further details).The spatial discretization is performed via a Fourier integral representation.With this in mind, the non-localized initial data given in equation (5.14) is altered (to justify the use of Fourier-based methods) by multiplying it with a rapidly decaying window of size 2ν (ν ≫ 1) and amplitude  4.20).As such, the numerical simulations are performed on equation (5.9) subject to the modified initial condition posed on a large computational domain (typically on the order of 10 4 ).This is necessary to ensure that the density has enough space to expand while significantly staying away from the computational domain boundaries.For all the direct numerical simulations presented below, we have monitored the linear growth rate for the hydrodynamic momentum defined in equation (5.15) to make sure it is indeed satisfied up to an error of the order of 10 −9 .

Centred Riemann problem with ρ
(2) In this section, we simulate centred Riemann problems (x 0 = 0) for ρ (2) 0 = 0 and various ρ (1) 0 .For all the numerical experiments, we observe the appearance of a quasistationary kink-type mode residing in the transverse near field.On both sides of this stationary Wadati-kink structure, counterpropagating nonlinear waves are observed.As expected, the width of these kinks is on the order of magnitude of the localization length scale L. To fix ideas, we discuss the results for the case when ρ (1) 0 = 2 first (see figure 10).The Wadati-kink mode observed in the transverse near field agrees with the one obtained from the solution to equations (4.23) and (4.24) (after substituting φ = ϑ + θ) for w = πsech 2 (x)/2 subject to the boundary conditions (5.17) (5.18) The solution to this boundary value problem is shown in figure 10(b) (red solid line) along side the mode observed using the direct numerical simulation.As one can see, they both display excellent agreement.Far away from the potential, the left propagating DSW resembles the NLS DSW (figure 10(c)).The DSW 'jump condition' (constancy of the Riemann invariant r 2 ) [23] was seen to hold across its structure.Moreover, a right propagating rarefaction wave (figure 10(d)) provides a transition to the vacuum state in the transverse far-field.Across this rarefaction wave, the Riemann invariant r 1 was seen to be constant.We have also performed long-time direct numerical simulations (up to t = 300) corresponding to various end state densities ρ (1) 0 .The near-field behaviour remains qualitatively similar for all cases, in the sense that a Wadati kink-mode is formed.However, with increasing ρ (1) 0 , the left propagating nonlinear wave changes its pattern from a dispersive shock to a rarefaction wave as is shown in figures 11(a) and (c).Interestingly, there is a transition in the density ρ (1) 0 = 3.6, for which the DSW disappears.This behaviour can be seen in figure 11(b).

ρ
(1) 0 = 0. We now examine a centred non-Hermitian Riemann problem for which the left end state ρ (1) 0 vanishes.This helps us shed more light on the rich dynamics exhibited by the hydrodynamic equations emanating from the loss of reflection symmetry.For example, when ρ (2) 0 = 2, the long-time dynamics of the density are shown in figure 12.While the profile in the transverse far-field (red solid) approximately follows that of the bulk NLS case (blue dashed) (figure 12(a)), nonetheless, the waveform in the near-field is significantly modified due to the formation of a breathing wave structure (see figure 12(b)).This pulsating behaviour leads to the generation of right and left propagating dark soliton-like trains within the rarefaction fan structure (figures 12(c) and (d)).

Non-centred Riemann problem with vanishing right end state
In this section we study non-centred Riemann problems corresponding to initial states centred (i) away from the transverse near field, (ii) inside the region of optical gain (where w ′ > 0), and (iii) inside the loss region (where w ′ < 0).These three distinct circumstances are depicted in figure 9(a), (b) and (c) respectively.For each case, the right amplitude ρ (2) 0 is chosen to be zero.In what follows, we report on numerical results for each individual situation.

5.2.1.
x 0 = −200.Intuitively, we expect the short-time dynamics to follow that of the bulk NLS.Indeed, our numerical simulation corroborates this, as can be seen in figure 13(a).In other words, we observe a rarefaction wave.However, as the propagating optical field hits the non-Hermitian 'obstacle', a small amplitude defect 'state' emerges on the fan profile (figure 13(b)), whose amplitude starts growing as the fan continues to expand.For intermediate times (t = 300), the optical flow develops into a large amplitude and highly oscillatory DSWtype train upstream of the obstacle, while gradually relaxing (connecting) to the rarefaction wave (figures 13(c) and (d)).Tracking the development of the optical flow for even larger times (figures 13(e)-(h)), we note the appearance of the Wadati kink mode observed for the centred Riemann counterpart (see figure 10(b)) at the location in the near-field.Moreover, as a result of nonlinear wave interaction, the landscape to the left of the kink mode leads to the incipient formation of a soliton train.To the right of the kink mode, the wave profile begins to relax (barring the transient oscillations) to a rarefaction wave profile providing a transition to the vacuum state.

x
We next examine the dynamics of a Riemann problem for which the initial step is located inside the region of optical gain.Remarkably, in this case the stationary Wadati kink mode observed for the centred Riemann counterpart (see figure 10(b)) appears at the same location as before, as seen in figure 14(b).However, an upstream propagating dark soliton is observed to form, which eventually gets trapped inside the left propagating dispersive shock wave [79].Two additional left propagating trailing dark solitons are also observed, which alter this landscape.However, the properties of the far-field DSW and rarefaction wave are otherwise similar in character (identical end states) to those observed for the centred case.

x
Lastly, when the initial step is now centred in the loss region, a slightly different pattern is found.That is to say, in the near-field the quasi-stationary Wadati mode is formed (at the same location), accompanied by a far-field left propagating DSW and right traveling rarefaction wave.This behaviour is summarized in figure 14(c).The counterflows in the far-field are modified by small amplitude dispersive radiation originating at the near field.

Centred Riemann problem with non-zero densities
Next, we discuss the centred Riemann problem constrained to non-zero densities with ρ 0 .We fix the left density ρ In this case, the temporal evolution of the density and hydrodynamic velocity are quite different compared to their zero background counterparts as well as the classical NLS without potential [21].First, we examine the case when ρ   (2) 0 [21].On the other hand, the left asymptotic state of the Wadati kink results in a  (2) 0 > 0.1 is more intricate, as the near-field (quasi)-stationary mode no longer exists.Instead, it is replaced by a breathing wave pattern.In turn, this changes the landscape of the Riemann problem dramatically, leading to the creation of dark soliton trains and soliton-DSW interactions.We present the resulting dynamics for ρ (2) 0 = 0.5, 1 in figures 15(c) and (d).In each of these cases, the inability of the breathing wave structures to relax to a stationary mode is reminiscent of well-known scenarios in transcritical conservative flows over obstacles [22,30,56], wherein closer to the edges of the transcritical regime, such unsteady behaviour is expected.Remark: For the case w = πsech 2 (x)/2, our modulation stability results indicate that CI waves (and consequently, their plane wave counterparts to equation (5.9)) are linearly stable whenever ρ 0 > 0.65.However, as mentioned in section 4.1, any instability that could arise would originate from the transverse near field alone.For the Riemann problems defined by parameters ρ (2) 0 = 0.1, 0.5, we did not observe any transient growth associated with such nearfield instability.

Non-centred Riemann problems with non-vanishing densities
To this end, we report on the dynamics of non-centred Riemann problems corresponding to non-zero hydrodynamic states.We exemplify this case by choosing parameters ρ (1) 0 = 2, ρ (2) 0 = 1 and x 0 = −200.Given that the non-Hermitian potential is localized, the evolution at an intermediate time scale should be governed predominantly by the bulk NLS.We have verified this in figure 17(a), wherein we obtain the usual right propagating DSW which connects to a left propagating rarefaction wave via an expanding hydrodynamic background.Eventually, the DSW would encounter the optical barrier, that leads to interesting wave dynamics.In such circumstances, the pulsating waveform observed in section 5.3 (see figure 15(e)) reappears, thus setting a similar complex pattern.Furthermore, as before, these pulsations drive the DSW-soliton interactions in the transverse far-field.These dynamics are depicted in figure 15.

Discussions and conclusions
The last decade or so has witnessed an intense interest in the general area of DH and its associated Riemann problems.Central to this effort is the investigation into the structure and formation of dispersive shock waves.Much of the research done along these lines has revolved around nonlinear wave propagation inside a homogeneous and conservative medium, where the total power or integrated density is time-independent.Extensions to scenarios involving weak dissipation have however been proposed and studied, with the KdV-Burger's equation being a prototypical test bed model [8,66,84].On the other hand, certain nonlinear wave phenomena (such as DSW) can occur in a spatially non-uniform media, as is the case, for example, in Bose-Einstein condensate flows [12], optical waveguide arrays [40] and resonant fluid flows over variable topography [30], to name a few.Often, the interplay between dissipation and inhomogeneity lead to the formation of novel multi-scale wave excitations (for example, in polariton condensate flows [6,43]).Generally speaking, in inhomogeneous media (conservative or dissipative) plane waves of constant amplitude are no longer viable.In this regard, given the existence of non-Hermitian constant intensity waves in Wadati-type potentials [65], it is thus intriguing to study nonlinear wave phenomena in the presence of inhomogeneities.As such, the main focus of this paper was to carry out a thorough investigation into such patternforming systems using the NLS equation in the presence of a Wadati-type complex external potential as an example of non-Hermitian system.The associated dispersive hydrodynamic equations were derived for a general complex potential and seen to possess source terms that are proportional to the real and imaginary parts of the external potential.When restricted to the class of complex Wadati potentials, this hydrodynamic system (besides admitting CI waves) conserves a 'modified' hydrodynamic momentum density (see equation (5.5)).The existence of such uniform-intensity waveforms motivated the study of the dynamics of a step-like initial condition connecting two CI backgrounds.For localized potentials, such IVPs could be interpreted as interactions between nonlinear waves (DSW/rarefaction waves) and obstacles comprised of gain and loss.Moreover, with an appropriate transformation of variables, the above problem was shown to be related to a three-parameter family of Riemann problems.Riemann problems are at the heart of the study of wave breaking in nonlinear media [18,21,24,33,57] and have been primarily posed for homogeneous systems.In particular, the long-time dynamics are described by self-similar theory.On the other hand, for the class of Riemann problems examined in the present work, the dynamics are significantly influenced by the interplay between inhomogeneity and non-Hermiticity.The long-time evolutions of these non-Hermitian Riemann problems reveal rich dynamics, particularly due to the loss of space-translational and reflection symmetries.Central to the description of the emerging wave patterns from step-like initial data is the presence (or absence) of the quasi-stationary nearfield Wadati modes.Their creation was seen to lead to hydrodynamic singularities on either side of the non-Hermitian 'obstacle', causing counterpropagating flows emanating from the near-field.In this regard, two distinct scenarios were studied, corresponding to ρ 0 , quasi-stationary Wadati modes with an even parity in the density and velocity distributions connecting two identical hydrodynamic backgrounds were obtained.As such, this Wadati end state and the nonlinear waves in the far-field were seen to coincide with the equivalent bulk NLS Riemann problem.On the other hand, when ρ (1) 0 > ρ (2) 0 , we observed wave dynamics reminiscent of the classical transcritical fluid flow problem.This was first studied in 78], and later utilized to characterize the (transcritical) Bose-Einstein condensate flow past a broad obstacle [56] and the bi-directional, transcritical shallow water flow problem [22].To explain the differences between the two cases of ρ 0 , whenever quasi-stationary modes were observed, their end states (ϱ 1,2 , ζ 1,2 ) were thus seen to be determined by the gain-loss distribution w ′ .This is in sharp contrast to ρ (1) 0 , where the Wadati mode end states were observed to be independent of the gain-loss distribution (equation (5.19)).For the case ρ (1) 0 , the emergence of unsteady near-field features could additionally act as a continuous source of counter-propagating wavetrains leading to rich nonlinear wave interactions in the far-field.Both the steady and unsteady near-field features were observed within the same family of non-Hermitian Riemann problems (section 5.3) for distinct parameter ranges.A theoretical framework to understand this change in the near-field behaviour can be developed within a weak gain-loss and large space (large time) limit.In this case, one would examine slowly varying (albeit O(1) in magnitude) w(X) with X = δx and T = δt where δ ≪ 1.With this scaling, the quantum pressure R drops out from equations (5.10) and (5.11), thus yielding to leading order an approximate hydrodynamic system, for which one can probe the existence of stationary solutions.
Some other intriguing features of the non-Hermitian Riemann problems considered here include 'near-field universality' and 'reciprocity'.By universality, we mean the independence of the near field dynamics (in the long time limit) with respect to the (initial) step location x 0 .As a result, the wave patterns over a long time are similar up to dark solitons/dispersive radiation trapped/transmitted through the dispersive hydrodynamic excitations.Another curious feature, arising from the aspect of universality was reciprocity.Here, it was observed that identical near-field dynamics are attained by the non-Hermitian Riemann problems either via a 1-wave or 2-wave interaction, (each arising from the same bulk NLS Riemann problem) with the non-Hermitian obstacle.This points to yet another interesting direction for future research when the initial step is located in the far field (away from the region of gain and loss).Over intermediate time scales, the optical flow developed into large-scale DSW/rarefaction waves as governed by the bulk NLS.Therefore, the approach to describe the subsequent dynamics reduces to studying optical flow past a localized non-Hermitian barrier.The significant scale separation between these dispersive hydrodynamic excitations and a sufficiently welllocalized complex potential could be leveraged to simplify the analysis.A pertinent example in this regard corresponds to the characterization of wave patterns that arise in the context of condensate flows past delta-impurities [34,54].
In the present work, we have focused our study exclusively on localized non-Hermitian external potentials.The reason for this being three-fold: (i) it facilitates the connection to the bulk NLS in the far-field limit, as the observed CI waves could be viewed as possessing a localized velocity distribution on a hydrodynamic background (ii) it simplifies the study of modulational stability of non-Hermitian backgrounds and (iii) it allows one to conceptualize the Riemann problems as a flow over a 'non-Hermitian obstacle'.However, for an extended complex potential (such as periodic), these uniform intensity waves possess an extended (spatial) velocity distribution.As a consequence, the resulting problems cannot be viewed through lens of dispersive hydrodynamic flows over an obstacle.Nonetheless, the notion of Riemann problems in a relevant traveling reference frame is still available through equations (5.10)-(5.12),providing an avenue to explore rich dispersive hydrodynamic phenomena.In this regard, one can utilize the machinery of Whitham modulation theory developed for the integrable models (including the NLS) [2-4, 59, 79, 83, 84] and its perturbed variants [41] to investigate such non-Hermitian periodic Riemann problems.

. 16 )
with β being a positive constant.Its index guiding as well as the external gain-loss landscape are depicted in figure2(a).This class of potentials are obtained from equation (3.1) by choosing w = βsech 2 (x) or θ = βtanh(x).For the non-Hermitian potentials given in (4.16), we examine the stability of CI modes for different wave intensities ρ 0 and potential depth β.In figure 3, we show an example of the spectra of H with ρ 0 = 1 and w = βsech 2 (x), for various values of β.Through extensive numerical experiments, we have identified a critical value of β = β c ≈ 1.8 as the phase transition point, below which the spectrum is entirely real.As a consequence, for 0 < β < β c the constant intensity mode ψ CI = exp(i(βtanh(x) − t)) is modulationally stable.

Figure 2 .
Figure 2. A PT -symmetric waveguide whose real part (black dashed line) represents the refractive index of the optical medium, while the imaginary part (red solid line) represents the gain-loss distribution.The non-Hermitian potential is given by equation (3.1) with w (a) defined in (4.16) and β = π/2.

Figure 5 .
Figure 5.The critical stability threshold βc for the potential V as a function of the wave amplitude √ ρ 0 .It is determined via direct computations of the operator (equation (4.5)) spectra (in red dashed).Also shown in black solid line is the least squares fit given by βc ≈ 1.262 √ ρ 0 + 0.425.

. 24 )Figure 6 .
Figure 6.Top view of the density ρ(x, t), illustrating the long-time features of the CI wave instability in the potential V with β = 2.5.It is obtained from direct numerical simulations of the NLS (5.1) subject to the external potential specified by w = 5sech 2 (x)/2 and initial condition given in equation (4.17) for ρ 0 = 1.The perturbations to the density and phase take the form given in (4.18) and (4.19).A window function (4.20) is used to scale (δ ≪ 1) and localize the perturbations.

Figure 7 .Figure 8 .
Figure 7. Time snapshots of the hydrodynamic density ρ taken from the space-time evolution shown in figure 6: (a) t = 0. (b) t = 60, which marks the end of the transient stage of density growth.(c) t = 150, two counter-propagating flows that are initiated across the Wadati mode.(d) t = 300, the fully developed counterpropagating flows.

Figure 9 .
Figure 9. Examples of non-centred Riemann problems where the initial density ρ(x, 0) distribution is shown in black-solid line.Also shown is the gain-loss distribution (i.e.w ′ ) in red-solid line.The three scenarios correspond to the step being (a) far away from the gain-loss region, (b) located in the region of gain, and (c) situated in the lossy arm.
two while varying the right background density, ρ

Figure 13 .
Figure 13.Time snapshots of the hydrodynamic density ρ corresponding to: ρ (1) 0 = 2, ρ (2) 0 = 0 and x 0 = −200.They are shown for (a) t = 25, (b) t = 125, depicting a small amplitude defect on the rarefaction profile, and t = 300 : (c) Far view: the overall wave pattern (in red) overlaid on the bulk NLS solution (blue dashed), (d) Zoomed-in view depicting the oscillatory wave train, (e), (f) a quasi-stationary Wadati mode emerges in the near-field, (g) the Wadati mode alongside the incipient soliton train formation (to the left) and the emergent rarefaction wave to the right and (h) Zoomed-in view of (g) at t = 600 showing the Wadati mode (direct numerics in red solid line and solution obtained from a boundary value problem in blue dashed line).

Figure 16 .0
Figure 16.Snaphots of the hydrodynamic density pattern corresponding to system parameters ρ (1) 0 = 2, ρ (2) 0 = 3 and x 0 = 0. (a) At t = 300, depicting the quasi-stationary Wadati mode counterpropagating flows and (b) Zoomed view showing the pattern (in red solid), and compared to a steady Wadati pattern in blue dashed, obtained from a boundary value problem solver.

0
> 2. A representative case is shown in figure16for ρ (2) 0 = 3.Here, a snapshot of the density profile is shown at t = 300, depicting different qualitative behaviour than when ρ Wadati mode possessing end hydrodynamic states lim |x|→∞ ϱ(x) = ϱ and lim |x|→∞ ϑ ′ (x) = ζ together with an even parity is formed in the near-field.Across this mode, a left-propagating NLS DSW and a right-propagating rarefaction wave (with a trailing dark soliton) are observed.Curiously, the set of Wadati hydrodynamic end states were empirically (numerically) observed to fit the formula ϱ = 1 4