Abstract
We study a coupled dispersionless system in 
dimensions with a time-dependent coefficient in the nonlinear coupling. The model consists of a real field u(x, t) and a complex field 
, driven by a prescribed modulation function 
. A 
Lax pair of Zakharov–Shabat type is constructed, showing that the system remains integrable with the temporal dependence confined to diagonal terms of the Lax matrices. Using a Darboux transformation adapted to this Lax pair, we derive Wronskian-type formulas for multi-soliton solutions and obtain explicit families of bright and dark traveling waves for several representative choices of 
. In addition, a time reparametrization transforms the model to an autonomous form whose traveling-wave reduction yields a planar Hamiltonian system with equilibria and phase portraits identical to the constant-coefficient coupled dispersionless equations. The modulation 
therefore leaves the intrinsic traveling-wave dynamics unchanged but reshapes how these waves are embedded in the (x, t)-plane, enabling management of soliton amplitudes, velocities and curved or oscillatory trajectories.
Keywords: Coupled dispersionless system, Soliton management, Variable-coefficient system, Darboux transformation, Wronskian solutions.
Subject terms: Mathematics and computing, Physics
Introduction
Coupled dispersionless (CD) systems form a widely studied class of integrable models for nonlinear waves in regimes where dispersive effects are either negligible or can be removed via dispersionless limits1. Canonical examples were introduced and analyzed by Kakuhata and Konno, who established Lagrangian and Hamiltonian formulations, identified conserved quantities, and derived loop–soliton solutions describing current–fed strings in external fields2–4. Later developments extended these ideas to semi–discrete, multicomponent, and short–pulse–related settings, demonstrating that CD–type equations appear naturally as reductions or limits of broader integrable hierarchies5–10.
Variable–coefficient integrable equations provide an important framework for modeling wave propagation in inhomogeneous or actively managed media. In such systems, spatial or temporal modulation of coefficients can be used to control soliton velocity, amplitude, and width, leading to phenomena including curved trajectories, nonuniform backgrounds, and the formation of bound states11–13. Related soliton–management mechanisms have been explored in many variable–coefficient models (KdV, NLS, Boussinesq and others), where exact or semi–analytic localized solutions reveal how time-dependent parameters influence the evolution and robustness of nonlinear excitations14–17. From the methodological viewpoint, Darboux transformations and dressing methods remain central tools for constructing explicit solutions; they have been systematically adapted to higher–order, matrix, and variable–coefficient integrable systems, and more recently combined with data–driven approaches such as Lax–pair–informed neural networks9,18–20.
Konno and Oono originally proposed a coupled integrable dispersionless model, which is now widely known as the Konno–Oono system. Their formulation can be written in the general form21 as
 |
1 |
Here, 
, 
, and 
represent arbitrary constants that significantly influence the qualitative behavior of the system. Considerable attention has been devoted to understanding the contribution of these parameters in shaping the dynamics of the model.
The Konno–Oono framework has appeared in several physical contexts, including geometric transport phenomena along curves under external magnetic effects and interactions involving current-driven strings4,6. Earlier studies mainly concentrated on the construction of traveling wave profiles for the coupled Konno–Oono system using different analytical approaches22–25.
An interesting feature of system 1 is that, under a suitable transformation, it reduces to another coupled dispersionless integrable structure. There is a direct algebraic reduction connecting system 1 to system 2. The substitution
 |
into 1, we obtain
 |
2 |
which is the coupled dispersionless system originally obtained in3.
The coupled dispersionless system 2, and its generalization 3 studied in the present work, arise directly from the physical model of a current-fed elastic string moving in an external magnetic field. The absence of dispersion reflects the limit in which string tension dominates bending stiffness, a regime relevant to several magnetically driven filament dynamics problems2,4. The time-dependent coefficient 
models temporal variations in the applied magnetic field strength or in the current carried by the string. Such temporal inhomogeneity is encountered, for instance, in pulsed-field experiments and in optical-fibre analogues where the nonlinear coupling strength is modulated by an external pump11,12. From a mathematical viewpoint, the variable-coefficient system 3 provides the minimal dispersionless setting in which to study soliton management: because dispersion is absent, all amplitude and width information resides in the nonlinear coupling, and 
directly controls the coupling strength without coupling to dispersive modes.
To obtain the variable-coefficient model, we identify
 |
so that 
. Since 
, the equations for r and s in 2 are complex conjugates of each other and reduce to the single equation 
. Replacing the constant coupling factor 
with a prescribed real function 
, we have
 |
3 |
where 
is a real field. A nontrivial 
allows one to vary the effective nonlinear coupling in time and thus to implement temporal soliton management within a dispersionless framework, in analogy with variable–coefficient NLS and KdV–type models11,12,14. To the best of the authors’ knowledge, system 3 with a time-dependent coupling coefficient 
, together with its Lax pair, Darboux transformation, and the associated soliton-management analysis, is introduced here for the first time.
The main objectives of the present work are:
to construct a Lax pair for the time–dependent CD system and clarify the role of
in preserving integrability;to develop a Darboux transformation compatible with the temporal modulation and to derive Wronskian–type formulas for multi–soliton solutions in the commutative setting;
to analyze representative families of bright and dark solitons for several choices of
and to illustrate how temporal profiles such as linear ramps, localized pulses, and periodic drives control curvature, acceleration, and oscillatory behavior of the resulting nonlinear waves.
The paper is organized as follows. In Section Model and Lax pair we present the model in detail and obtain its Lax pair formulation. Section Darboux transformation and Wronskian solutions is devoted to the construction of the Darboux transformation and to Wronskian–type multi–soliton expressions. In Section Explicit time–managed soliton solutions we provide explicit one– and two–soliton solutions for several typical choices of 
and discuss their qualitative dynamics. Finally, Section Conclusions summarizes the main results and outlines possible directions for further research, including higher–dimensional and multicomponent generalizations (Fig. 1).
Fig. 1.
Workflow of the proposed analytical framework and final conclusion.
Model and Lax pair
We consider the system 3 with
 |
For convenience we write 
. Then the first equation of 3 can be rewritten as
 |
4 |
The second equation takes the form
 |
5 |
We seek a Lax pair of the Zakharov–Shabat type
 |
6 |
where 
and 
is a spectral parameter. A convenient choice is
 |
7 |
 |
8 |
where
 |
9 |
and
 |
10 |
Here a(t) being auxiliary scalar function, to be chosen in such a way that the zero–curvature condition
 |
11 |
reproduces 4–5. The derivation is straightforward but involves lengthy matrix algebra, and we omit the intermediate steps here. We emphasize that the time dependence is confined to the scalar function 
, so the structure of the Lax pair retains the standard dispersionless form, with the temporal modulation entering only through diagonal terms1,3.
Darboux transformation and Wronskian solutions
In this section, we construct a Darboux transformation for the Lax pair 6 associated with the system 3. The construction closely follows the classical Zakharov–Shabat scheme, with the main difference that the temporal modulation 
appears in the diagonal part of the Lax matrices18,26. Let 
be a fundamental solution of 6 for a given potential 
. We fix a spectral parameter 
and a corresponding eigenfunction
 |
12 |
We introduce a rank–one projector
 |
13 |
where 
denotes Hermitian conjugation. The one–fold Darboux matrix is chosen as
 |
14 |
which is a standard choice in the focusing case and preserves the reduction 
, 
provided the eigenfunction is taken consistently7,19.
The transformed wave function is defined by
 |
15 |
We require that 
satisfies a Lax pair of the same form as 6 with new fields 
:
 |
16 |
This covariance leads to
 |
17 |
Since U and V have the structure described in 7–9, and 
depends rationally on 
, comparison of the coefficients in 17 yields explicit formulas for 
and 
.
Using the projector 
and Darboux matrix 14, a direct calculation of the off–diagonal entries of 
gives
 |
18 |
 |
19 |
The diagonal part of 
yields the transformed real field
 |
20 |
Formulas 18–20 constitute the one–fold Darboux transformation for the system 3. Note that the modulation function 
does not appear explicitly in the transformation rules; it enters only through the dependence of the eigenfunction 
on (x, t) via the Lax pair.
Iteration and Wronskian representation
The Darboux transformation can be applied repeatedly. Let 
be distinct spectral parameters and let 
be eigenfunctions of the original Lax pair at 
. Using these eigenfunctions as seeds and iterating the transformation 14, one obtains N–fold transformed fields 
. In the commutative setting considered here, the resulting expressions can be written in terms of ordinary Wronskians of the eigenfunctions.
To describe the result, we introduce 2N scalar functions
 |
and define the 
Wronskian matrices
 |
21 |
Then the N–soliton solution on a constant background can be written schematically as
 |
22 |
 |
23 |
where 
, 
, and 
are 
matrices obtained by augmenting 
and 
with suitable additional columns built from the eigenfunctions and spectral parameters. The precise structure of these determinants follows the usual pattern of Wronskian solutions for coupled dispersionless and short pulse–type systems5,7.
The key observation is that the modulation function 
does not modify the algebraic structure of the Wronskian representation; it only appears through the prefactor 
in 23 and through the time dependence of the eigenfunctions themselves. Therefore, once the eigenfunctions for a given 
are known, multi–soliton configurations can be generated systematically.
Explicit time–managed soliton solutions
One–soliton solution on a constant background
We now apply the one–fold Darboux transformation to the constant seed
 |
24 |
The corresponding eigenfunctions can be chosen in the form
 |
25 |
where the phase function is
 |
26 |
The exact form of 
depends on the precise normalization of the Lax pair; here we keep it symbolic in terms of the integral of 
.
Substituting the seed eigenfunctions 25 into the Darboux formulas 18–20 with 
and 
yields standard bright–soliton profiles in the 
–component and corresponding localized modulations in u. The resulting expressions can be written in closed form in terms of the real and imaginary parts of the phase 
, with all dependence on the modulation function 
entering only through the integral in 26. Since these formulas have the usual hyperbolic structure, we omit the intermediate algebraic steps and focus instead on explicit examples for particular choices of 
in the following subsections.
Before presenting the examples, we extract the key soliton characteristics from the general one-soliton solution. Setting 
with 
and using the Darboux formulas 18–20 applied to the seed 
, the real field takes the general form
 |
27 |
where 
and the central position is
 |
28 |
From 27 we read off the four fundamental soliton characteristics:
Amplitude:
; independent of t and of 
.Width:
; also independent of 
.Central position: X(t) given by 28, which depends directly on
. Different choices of 
therefore produce different space-time trajectories whilst the pulse shape remains fixed.Instantaneous velocity:
, which is directly proportional to 
. Thus 
acts as a velocity controller: a linearly growing 
produces uniformly accelerated motion; a bounded 
produces bounded displacement; and a periodic 
produces oscillatory motion.
The same structural analysis applies to the complex field 
, whose modulus 
forms a bright soliton centered at X(t) with the same width and trajectory. These observations confirm that temporal modulation reshapes only the embedding of the soliton in (x, t)-space, not its intrinsic profile; a finding that is further justified by the Hamiltonian analysis of Section Traveling–wave reduction and planar dynamical system.
Example 1: Linear ramp with 
For general parameters 
with 
the integral has the form 
, the general one-soliton solution is
 |
29 |
 |
30 |
where 
denotes the imaginary part of 
, and
 |
31 |
With the specific values 
, 
, 
. The instantaneous velocity 
is linear in t, so the soliton undergoes uniform acceleration. Integrating the velocity gives the soliton center
 |
which is quadratic (parabolic) in t. The amplitude and width are independent of 
. By choosing parameters 
, 
, 
, for which 
, 
, 
, one obtains
 |
confirming the parabolic trajectory visible in Fig. 2.
Fig. 2.
One–soliton solution for the linearly ramped coefficient 
.
Figure 2adisplays the three–dimensional profile of 
for 
. The soliton peak clearly bends as t increases, confirming the presence of time–dependent acceleration induced by the ramped coefficient 
. Figure 2bshows the corresponding surface plot of u(x, t), where the localized depression propagates along the same curved trajectory on top of the uniform background.
Example 2: Localized modulation 
We next fix the modulation function as 
and keep the same background and spectral parameters as in Example 1, namely 
In this case
 |
so the effective phase remains bounded as 
. The one–soliton solution generated by the Darboux transformation can be written in the explicit form
 |
32 |
and the corresponding real component is
 |
33 |
where
 |
34 |
with specific values 
, 
.
Figure 3ashows the three–dimensional plot of 
, while Fig. 3bdepicts the surface plot of u(x, t). The bright soliton in 
is strongly localized both in space and in time: its amplitude becomes largest when t is close to zero, where 
attains its maximum, and then gradually decays as 
. The corresponding depression in u exhibits the same localized behavior, confirming that the choice 
acts as a temporal window that confines the nonlinear interaction to a finite time interval.
Fig. 3.
One–soliton solution for the localized modulation
.
Example 3: Oscillatory modulation 
Finally we consider the periodic modulation 
again with the same background and spectral parameters as in Examples 1–2: 
In this case
 |
so the phase of the eigenfunction oscillates rather than growing monotonically with t. The one–soliton solution obtained from the Darboux transformation takes the explicit form
 |
35 |
where
 |
36 |
and the corresponding real component is
 |
37 |
where
 |
38 |
With 
, one has 
, 
, 
, so that 
is real and the argument of 
and 
simplifies to
 |
39 |
Figure 4ashows the three–dimensional plot of 
; the soliton crest exhibits a clear “zigzag” pattern in the (x, t)–plane, characteristic of periodically driven systems. The surface plot of u(x, t) in Fig. 4breveals a sequence of recurring dark structures on the uniform background 
, synchronized with the oscillations of 
. Such behavior is typical for soliton management scenarios under periodic modulation of the nonlinear coefficient11–13.
Fig. 4.
One–soliton solution for the oscillatory modulation 
.
Two-soliton solution for 
We again use the background 
, 
and the same reduction 
, 
. Let
 |
be two distinct spectral parameters and denote
 |
40 |
The integration in the eigenfunction phase gives precisely H(t).
For each 
we introduce the real and complex phases
 |
41 |
and the corresponding exponentials
 |
42 |
The two–fold Darboux transformation produces a solution which can be written in the rational form
 |
43 |
where the common denominator is expressed as a finite sum of products of 
and 
:
 |
44 |
The coefficients 
, 
, depend only on the spectral parameters and can be extracted directly from the Maple output.
The numerator of the complex field has the generic form
 |
45 |
while the numerator for the real field reads
 |
46 |
Here 
and 
are again explicit algebraic functions of 
and 
. Their full expressions are lengthy and are therefore omitted; they can be obtained symbolically with the help of Maple and used for numerical evaluation.
In the numerical example shown in Figures 5a
Fig. 5.
Two–soliton solution for the saturating modulation 
.
and 5bwe choose
 |
47 |
so that 
and the solitons are of bright type in the 
–component and dark type in the u–component. Substituting these values into 44–46 yields the explicit two–soliton profiles used for plotting.
Figure 5adisplays the modulus 
of the two–soliton solution for 
and the above choice of parameters. The denominator 44 contains the mixed exponential terms 
and 
, which encode the nonlinear interaction between the two elementary waves associated with 
and 
. As a consequence, the surface plot reveals two bright humps whose trajectories approach each other, interact in a finite time interval around 
, and then separate again. Far away from the interaction region, where 
and hence 
, each soliton travels almost along a straight line, indicating that the effective velocities tend to constant limits.
The corresponding real field 
is shown in Fig. 5b. According to 43 and 46, the quantity 
contains the same exponential combinations as 
, which leads to two dark–type structures riding on the constant background 
. During the interaction the depressions in u merge and distort the local profile, but for large positive and negative times they recover their original depths and widths. This behavior is characteristic of an elastic two–soliton collision.
Asymptotic behavior of the two–soliton solution
The real parts of the phases 
and 
control the localization of the individual solitons. Writing 
with 
, 
in our example, we have
 |
since H(t) is real and the prefactor of H(t) is purely imaginary. Thus the exponential factors 
, 
are localized along curves 
determined mainly by the imaginary parts of the phases, while the spatial decay away from these curves is governed by 
and 
.
In the limits 
one has
 |
so the time–dependent contribution to the phases saturates to a finite constant. The dominant t–dependence of the solution therefore comes from the linear term in t contained in the dispersionless part of the Lax pair. Consequently, in each of the four asymptotic regimes
 |
the mixed exponential terms in the numerator and denominator, containing products such as 
or 
, are exponentially small compared to the self terms 
26.[web:33] Retaining only the dominant contributions in 44–46 yields
 |
48 |
 |
49 |
which are precisely one–soliton profiles of the type obtained in Section "One–soliton solution on a constant background".
Let 
and 
denote the positions of the maximum of 
for the jth soliton in the limits 
and 
, respectively. Equating the real part of the phase to zero, 
, and differentiating with respect to t shows that the asymptotic velocities satisfy
 |
so that both solitons preserve their speeds after the interaction. The effect of the collision is encoded in the finite spatial shifts
 |
for any fixed time 
in the far field, and in additional phase shifts of the complex amplitude, which arise from the imaginary parts of the exponentials 
and 
. The invariance of the amplitudes and velocities, together with the finite shifts, confirms that the two–soliton interaction is elastic in the presence of the time–dependent coefficient 
.
To make the collision analysis more explicit, we now compute the position and phase shifts for the general two-soliton solution with spectral parameters 
(
, 
, 
) and background 
. Each soliton has a constant asymptotic velocity
 |
50 |
since for 
the phase integral 
saturates as 
. The solitons are therefore stationary in the far field and the “collision” corresponds to a finite-time interaction centered near 
where 
is largest.
The asymptotic position shift of soliton j due to the interaction is
 |
51 |
and the corresponding phase shift of the complex field 
is
 |
52 |
For the numerical values 
, 
, 
used in Fig. 5, one obtains 
, 
, confirming the small but nonzero position shifts visible in Figures 5a–5b. The invariance of amplitudes (
) and widths (
) before and after the interaction establishes that the collision is elastic. No radiation is generated and the localized structures persist for all times.
Considering now two solitons with distinct velocities, we choose 
and 
with 
. For 
(as in Fig. 6), the instantaneous velocity of soliton j is
 |
53 |
so the two solitons move with different mean velocities 
superimposed by the same oscillatory component. The collision is still elastic (no radiation is emitted) but the oscillatory drive produces a periodic sequence of near-collisions rather than a single interaction event.
Fig. 6.
Two–soliton solution for the oscillatory modulation
.
For the oscillatory modulation 
with 
and the same spectral parameters 
, 
, the two–soliton solution exhibits a strikingly different behavior from the saturating case 
. The three–dimensional surface of 
in Fig. 6ashows a sequence of bright peaks arranged along an oscillatory path in the (x, t)–plane. The time–dependent coefficient periodically enhances and weakens the effective nonlinearity, which leads to a repeated focusing and defocusing of the two interacting solitons. As t increases, the crests of 
drift back and forth in space, producing a multi–humped “zigzag” pattern rather than a single smooth bend.
The corresponding plot of 
in Fig. 6bdisplays two dark–type structures on the background 
that undergo periodic modulation in both depth and position. Whenever 
attains a local maximum, the depressions in u sharpen and move closer together, indicating stronger nonlinear coupling between the components. When 
is near its minima, the dips broaden and move apart, and the background dominates. Overall, the interaction remains elastic in the sense that no radiation is observed and the localized structures persist for all times, but the periodic drive imprints a pronounced breathing–like behavior on the two–soliton configuration. This example illustrates how an oscillatory time–dependent coefficient can be used to induce repeated collisions and controlled reshaping of coupled dispersionless solitons.
Traveling–wave reduction and planar dynamical system
Hamiltonian phase-plane structure
In order to analyze the effect of the time–dependent coefficient 
on traveling–wave solutions of the variable–coefficient CD system 3, it is natural to study the corresponding reduced ordinary differential equations. Introducing the new time variable
 |
one readily checks that 
and hence 
. Substituting this into 3 shows that, in the variables 
, the system assumes the autonomous form
 |
54 |
Thus the explicit time dependence generated by 
can be removed by a reparametrization of time; the price to pay is that the relation between the new time 
and the physical time t becomes nonlinear and depends on the chosen modulation.
We now look for traveling waves in the autonomous system 54 of the form
 |
55 |
with phase
 |
56 |
where 
are real constants. Substituting 55–56 into 54 and separating real and imaginary parts leads to the relations
 |
57 |
and the scalar second–order equation
 |
58 |
where a prime denotes differentiation with respect to 
and A is an integration constant arising from 57. The real field U is reconstructed from V via
 |
59 |
Introducing the variables
 |
60 |
the ODE 58 is equivalent to the planar dynamical system
 |
61 |
with coefficients
 |
62 |
System 61 is Hamiltonian:
 |
63 |
where
 |
64 |
The equilibrium points are given by
 |
65 |
and the Jacobian matrix at (Y, Z) has determinant
 |
66 |
Thus, 
is a saddle when 
and a center when 
, while 
are centers for 
and saddles for 
. In particular, for 
and 
the potential V(Y) has a double–well structure with two minima and one maximum at the origin, while for 
and 
it has a single well with a unique minimum at 
. These phase–plane configurations coincide with those obtained for standard coupled dispersionless equations with constant coefficients, as discussed in26.
The crucial observation is that the coefficients 
and 
in 62 depend only on the traveling-wave parameters A, c, 
and are completely independent of 
. In other words, after the time reparametrization 
, the traveling-wave dynamical system 61 has exactly the same Hamiltonian structure, equilibria, and bifurcation scenarios as in the constant-coefficient case.
Conditions for the reparametrization to be a homeomorphism. The validity of the equivalence between the variable-coefficient system 3 and the autonomous system 54 depends on whether the map 
is a homeomorphism (specifically, a diffeomorphism) on the relevant time domain. This holds if and only if 
is strictly of one sign:
uniformly (e.g., 
or 
): the map 
is a strictly increasing diffeomorphism on all of 
, and the two systems are globally topologically conjugate. The phase-portrait comparison is globally valid.Linear ramp
, 
: the map is a diffeomorphism on 
but fails at 
where 
. The equivalence of phase portraits and the conclusions of this section therefore hold on the restricted domain 
(i.e., 
for 
). The explicit soliton solutions of Section One–soliton solution on a constant background are obtained independently and are valid for all 
; only the phase-portrait comparison is restricted.Oscillatory modulation
(or 
with
): the map 
is periodic and non-injective, so it is not a global homeomorphism. In this case the reparametrization does not establish global topological conjugacy between the variable-coefficient and autonomous systems. The phase portraits of the autonomous system provide only a local description, valid on each sub-interval where 
and locally monotone. All soliton solutions in Section Explicit time–managed soliton solutions are derived via the Darboux transformation and hold for all t; they are not affected by this limitation. For 
with 
(so that 
always), global conjugacy is restored.
The modulation 
reappears in the mapping between the traveling-wave coordinate 
and the physical variables (x, t). Under the conditions where the reparametrization is a diffeomorphism, a given phase-plane orbit generates different space-time trajectories for different choices of 
. Thus, 
does not alter the intrinsic traveling-wave dynamics, but it reshapes how these waves are embedded in the (x, t)-plane; in full consistency with the soliton-trajectory analysis of Section Explicit time–managed soliton solutions.
Elliptic-function periodic waves and limiting solitons
The planar system 61 possesses a conserved Hamiltonian 
with potential
 |
67 |
For the double-well case (
, 
), the potential has two minima at 
and one local maximum at 
. The bounded periodic orbits encircling a single minimum yield cnoidal-wave solutions
 |
68 |
where the amplitude 
, wavenumber 
, and elliptic modulus 
are determined by the energy level 
via
 |
69 |
The spatial period is 
, where K(k) is the complete elliptic integral of the first kind.
As 
, 
and the cnoidal wave reduces to the bright soliton profile 
, recovering the one-soliton solutions of Section Explicit time–managed soliton solutions.
As 
, 
and the solution approaches a small-amplitude harmonic wave. For the single-well case (
, 
), the general bounded solutions involve the 
-function:
 |
70 |
which is always positive and has period 
. In the limit 
this reduces to the dark-soliton profile 
with a non-zero background.
Via the time reparametrization, the cnoidal-wave solution in the 
-frame corresponds in the physical (x, t)-frame to a periodic wave whose spatial profile is fixed by 68 but whose spatial period, measured in real time, becomes time-dependent. A linearly growing 
therefore compresses the period monotonically, whilst 
causes periodic alternation between compression and dilation of the wave train. This provides a dispersionless analogue of period-doubling phenomena observed in periodically driven nonlinear oscillators.
Conclusions
We have introduced and analyzed a coupled dispersionless system in 
dimensions with a time-dependent coefficient in the nonlinear coupling. A Zakharov–Shabat Lax pair was constructed in which the modulation 
appears only in scalar diagonal terms, confirming that the model remains integrable in the presence of temporal inhomogeneity. Building on this structure, we developed a Darboux transformation compatible with the time dependence and derived Wronskian-type formulas for multi-soliton solutions in the commutative setting.
Using these expressions, we obtained representative families of bright and dark traveling waves for several choices of 
, including profiles that mimic linear ramps, localized pulses and periodic drives. The resulting solutions demonstrate how temporal modulation can be used to manage the amplitudes and velocities of the coupled fields and to generate curved, accelerating and oscillatory soliton trajectories in the physical 
-plane. A complementary analysis based on a nonlinear reparametrization of time showed that the traveling-wave reduction of the modulated system leads to a planar Hamiltonian dynamical system whose equilibria, potential wells and phase-plane structure coincide with those of the constant-coefficient coupled dispersionless equations. Thus the modulation does not introduce new qualitative phase-plane phenomena; instead, it reshapes the embedding of the same traveling-wave orbits in space–time.
The framework developed here can be extended in several directions. One possibility is to consider multi-component or higher-dimensional coupled dispersionless systems with time-dependent coefficients, where richer interaction patterns and management scenarios are expected. Another is to incorporate additional spatially varying coefficients alongside 
, allowing simultaneous temporal and spatial control of dispersionless waves. It would also be natural to combine the present Darboux and Wronskian techniques with data-driven approaches, such as Lax-pair-informed neural networks, in order to address parameter identification, robustness and stability questions in inhomogeneous media. These directions will be pursued in future work.
Author contributions
H.W.A.R. conceived the study and developed the methodology, performed the formal analysis and wrote the original draft of the manuscript. Y.Y. contributed to validation and mathematical analysis and participated in reviewing and editing the manuscript. A.A. contributed to the physical modeling, visualization, reviewing and editing the manuscript. All authors reviewed and approved the final manuscript.
Data availability
All data generated or analysed during this study are included in this published article.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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All data generated or analysed during this study are included in this published article.