arXiv subjects

Hadi Susanto

Publications and source records attributed to Hadi Susanto.

At least 19 recordsLinked to original sources

arXiv2026

Supratransmission in Lattices with Purely Nonlinear Coupling

Supratransmission is examined in nonlinear lattices with purely nonlinear coupling, extending the phenomenon to systems that lack a linear pass band. In contrast to standard lattices with mixed linear-nonlinear interactions, the present model has no linear spectrum, so energy propagation arises entirely from nonlinear effects. Asymptotic analysis yields a discrete $p$-Schr\"odinger (DpS) equation that {provides an accurate description in the weak- and intermediate-coupling regimes and offers qualitative insight in the strong-coupling regime}. Perturbation provides analytical approximations for the critical driving amplitude, explicitly showing its dependence on the driving frequency, coupling strength, and the nonlinearity exponent $p$. The analysis identifies a non-trivial dependence of the critical amplitude on $p$, with distinct trends in different coupling regimes. Numerical continuation and direct simulations {validate the theory in regimes where the asymptotic reduction is applicable and show good agreement across a wide range of parameters}. The results establish supratransmission in fully nonlinear lattices and clarify the associated energy-transport mechanisms, with relevance to mechanical lattices, tunable metamaterials, and nonlinear optical arrays.

nlin.PS↗

arXiv2026

Dynamics and stabilization of topological edge solitons in driven-damped nonlinear SSH lattices

We study topological edge solitons in a nonlinear Su--Schrieffer--Heeger (SSH) lattice subject to parametric driving and linear damping. Starting from a vertically driven pendulum chain, we derive an effective driven--damped nonlinear SSH model and investigate its stationary edge-localized states. Analytical calculations reveal the existence of two phase-locked dissipative edge-soliton families that emerge from the nonlinear continuation of the topological edge mode. Using numerical continuation and spectral stability analysis, we construct the corresponding nonlinear branches and determine their stability properties. We show that parametric driving and damping fundamentally modify the conservative edge-state family by generating two dissipative branches with markedly different stability characteristics: one branch remains predominantly unstable, whereas the other develops substantially larger stability regions and significantly weaker instability growth rates. Direct numerical simulations further demonstrate that the robust branch can remain strongly localized over long time intervals even when weakly unstable. Simulations of the full driven--damped Klein--Gordon pendulum chain confirm the persistence of the edge-localized dynamics predicted by the reduced model. These results identify parametric driving and damping as an effective mechanism for enhancing the robustness and persistence of nonlinear topological localization in active lattice systems.

nlin.PS↗

arXiv2026

Pearl supratransmission in a boundary-driven two-dimensional nonlinear Schr\"odinger equation with a hole

We investigate energy supratransmission in a boundary-driven two-dimensional nonlinear Schr\"odinger equation with a central hole. Harmonic forcing with azimuthal modulation generates standing-wave states whose existence and stability depend on the driving amplitude, the inner radius, and the imposed azimuthal charge. Bifurcation analysis shows that small inner radii produce strongly confined states with higher destabilization thresholds, whereas larger radii yield broader profiles and smoother transitions between stable and unstable branches. The cubic--quintic and saturable models exhibit similar qualitative behaviour but differ quantitatively in their critical amplitudes and parameter dependence. A variational approximation captures the dependence of the critical drive on the azimuthal charge and nonlinear parameters, and clarifies how the nonlinear response shapes the stationary states near the turning point. Time-dependent simulations show that supratransmission occurs through the emission of localized pulses, with nonzero azimuthal charge triggering symmetry breaking and producing two-dimensional localized excitations (pearls). Isosurface plots provide a complementary view of the resulting radial and angular excursions. These results establish a quantitative framework for supratransmission in two-dimensional geometries and are relevant to driven nonlinear systems in optics, Bose--Einstein condensates, and structured media.

nlin.PS↗

arXiv2026

Phase space integrity in neural network models of Hamiltonian dynamics: A Lagrangian descriptor approach

We propose Lagrangian Descriptors (LDs) as a diagnostic framework for evaluating neural network models of Hamiltonian systems beyond conventional trajectory-based metrics. Standard error measures quantify short-term predictive accuracy but provide little insight into global geometric structures such as orbits and separatrices. Existing evaluation tools in dissipative systems are inadequate for Hamiltonian dynamics due to fundamental differences in the systems. By constructing probability density functions weighted by LD values, we embed geometric information into a statistical framework suitable for information-theoretic comparison. We benchmark physically constrained architectures (SympNet, H\'enonNet, Generalized Hamiltonian Neural Networks) against data-driven Reservoir Computing across two canonical systems. For the Duffing oscillator, all models recover the homoclinic orbit geometry with modest data requirements, though their accuracy near critical structures varies. For the three-mode nonlinear Schr\"odinger equation, however, clear differences emerge: symplectic architectures preserve energy but distort phase-space topology, while Reservoir Computing, despite lacking explicit physical constraints, reproduces the homoclinic structure with high fidelity. These results demonstrate the value of LD-based diagnostics for assessing not only predictive performance but also the global dynamical integrity of learned Hamiltonian models.

cs.LG↗

arXiv2026

Low-Energy and Low-Thrust Exploration Tour of Saturnian Moons with Full Lunar Surface Coverage

This study presents the trajectory design for a mission touring Saturn's Inner Large Moons (Rhea, Dione, Tethys, Enceladus, and Mimas) engineered to meet observational requirements, including full surface coverage, while ensuring low fuel consumption and compatibility with current power and propulsion technologies (radioisotope thermoelectric generators and Hall effect thrusters). The tour begins at Rhea and ends at Mimas, using a trajectory concept that alternates between extended observation phases around each moon and Saturn centered low-thrust spiral arcs to transition efficiently to the next target. The J2-perturbed Circular Restricted Three-Body Problem is adopted to design exploration paths, with halo orbits serving as staging points for heteroclinic and homoclinic loops that enable prolonged, repeated, and comprehensive surface reconnaissance (including critical regions such as Enceladus poles, where geological activity produces intense plumes). Stable and unstable hyperbolic invariant manifolds of the halo orbits act as departure and arrival gateways for propelled inter-moon transfers, modeled in an ephemeris-based framework including gravitational perturbations from the moons, the Sun, and Saturn's oblateness. The dynamical model setup is guided by a rigorous perturbation analysis to maximize computational efficiency while maintaining high fidelity trajectory design. A locally optimal guidance law minimizes propellant consumption. The proposed tour offers an alternative to traditional flyby missions, providing comparable total duration but greater observing time and reduced fuel requirements, and advances previous work by achieving both complete lunar surface coverage and high-fidelity modeling.

math.DS↗

arXiv2026

Maxwell Fronts in the Discrete Nonlinear Schr\"odinger Equations with Competing Nonlinearities

In discrete nonlinear systems, the study of nonlinear waves has revealed intriguing phenomena in various fields such as nonlinear optics, biophysics, and condensed matter physics. Discrete nonlinear Schr\"odinger (DNLS) equations are often employed to model these dynamics, particularly in the context of Bose-Einstein condensates and optical waveguide arrays. While the classical DNLS with cubic nonlinearity admits well-known solitonic solutions, the introduction of competing nonlinearities, such as quadratic-cubic and cubic-quintic terms, gives rise to new behaviors, including multistability and front formation. One such emergent structure, the Maxwell front, is characterized by stationary interfaces between two energetically equivalent steady states, occurring at a critical parameter known as the Maxwell point. This paper investigates the existence and stability of Maxwell fronts in DNLS models with competing nonlinearities. Specifically, we examine the quadratic-cubic nonlinearity, as found in the discrete quantum droplets equation, and the cubic-quintic nonlinearity, both of which exhibit multistability. We explore the persistence of Maxwell fronts in both the anticontinuum limit (where the coupling between lattice sites is weak) and the continuum limit (where the coupling is strong). The stability of these fronts is analyzed through linear stability analysis, utilizing eigenvalue counting arguments and exponential asymptotic techniques. Our results provide new insights into multistability, front dynamics, and the role of competing nonlinearities in discrete wave systems. The main contributions of this work include the characterization of Maxwell fronts in DNLS equations with competing nonlinearities, the analysis of their stability across different coupling regimes, and the application of novel asymptotic methods to investigate their behavior in the continuum limit.

nlin.PS↗

arXiv2026

New logarithmic power nonlinear Schr\"odinger equations with super-Gaussons

We introduce a new class of nonlinear Schr\"odinger equations with a logarithmic-power nonlinearity that admits exact localized solutions of super-Gaussian form. The resulting stationary states possess flat-top profiles with sharp edges and are referred to as super-Gaussons, in analogy with the Gaussian Gaussons of the classical logarithmic NLS (log-NLS). The model, which we call the logarithmic-power NLS (logp-NLS), is parameterized by an exponent $p\geq1$ that controls the degree of flatness of the soliton core and the sharpness of its decay. Mathematically, $p$ interpolates between the standard log-NLS ($p=1$) and increasingly flat-top profiles as $p$ increases, while physically it governs the stiffness of an underlying logarithmic-power compressibility law. The proposed equation is constructed so as to admit super-Gaussian stationary states and can be interpreted a posteriori within a generalized pressure-law framework, thereby extending the log-NLS. We investigate the dynamics of super-Gaussons in one spatial dimension through numerical simulations for various values of $p$, demonstrating how this parameter regulates both the internal structure of the soliton and its collision dynamics. The logp-NLS thus generalizes the standard log-NLS by admitting a broader family of localized states with distinctive structural and dynamical properties, suggesting its relevance for flat-top solitons in nonlinear optics, Bose-Einstein condensates, and related nonlinear media.

nlin.PS↗

arXiv2026

Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm

This work investigates shallow physics-informed neural networks (PINNs) for solving forward and inverse problems governed by nonlinear partial differential equations (PDEs). By formulating PINN training as a nonlinear least-squares problem, the Levenberg-Marquardt (LM) algorithm is used to efficiently optimize the network parameters. Exact analytical expressions for neural-network derivatives with respect to the input variables are derived, revealing the relationships between the network output and its spatial and temporal derivatives and providing a clearer interpretation of the PINN architecture. These expressions are then used to derive explicit formulas for the Jacobian matrix required by LM. The proposed approach is evaluated on the Burgers, Schr\"odinger, Allen-Cahn, and three-dimensional Bratu equations. Numerical results show that LM substantially outperforms BFGS, L-BFGS, and Adam in convergence speed, accuracy, and final loss values. Comparisons with deeper networks further demonstrate that shallow LM-PINNs can achieve higher accuracy with substantially fewer parameters, emphasizing the importance of considering network architecture and optimization strategy jointly. The explicit analytical Jacobian also provides computational and memory advantages that are particularly relevant to large-scale PINNs. Overall, these results suggest that, for a broad class of PDEs, shallow PINNs combined with effective second-order optimization can provide accurate and computationally efficient solutions to both forward and inverse problems.

math.NA↗

arXiv2025

Uncontrolled geostationary satellites: mapping periodic transitions to chaos with Lagrangian Descriptors

Uncontrolled geostationary satellites abandoned near an unstable equilibrium point of the equator experience irregular transitions between dynamical states (continuous circulation, long and short libration). They are caused by the interaction between the longitudinal dynamics, governed by the tesseral harmonics of the geopotential, and the orbital precession forced by Earth's oblateness and lunisolar perturbations. The transitions are extremely sensitive to small perturbations, making the long-term evolution unpredictable. Recently, a Monte Carlo analysis of trajectories starting in the immediate vicinity of the 165 degrees E unstable equilibrium point, revealed that the evolution to chaos is not gradual. It occurs via sudden episodes of disorder at specific points of the precession cycle, when the orbital inclination is minimal. Due to the high cost of the statistical analysis, the results were limited to a single initial longitude. This paper applies modified versions of the diameter Lagrangian descriptor to reduce the computational burden. This enables mapping the dynamical behavior over the complete range of longitudes where transitions between modes of motion are possible, considering both unstable equilibrium points (165 degrees E and 15 degrees W). It is found that the episodes of chaos remain linked to the orbital inclination cycle, but their timing depends on the initial spacecraft longitude. As the initial position moves farther away from the unstable point, the transitions take place at higher values of the orbital inclination. The longitudes where the transitions occur at maximum inclination correspond to the boundaries of the chaotic region.

nlin.CD↗

arXiv2025

Spontaneous symmetry-breaking in the nonlinear Schr\"odinger equation on star graphs with inhomogeneities

We investigate the nonlinear Schr\"odinger equation on a three-edge star graph, where each edge contains a linear localized inhomogeneity in the form of a Dirac delta linear potential. Such systems are of significant interest in studying wave propagation in networked structures, with applications in, e.g., Josephson junctions. By reducing the system to a set of finite-dimensional coupled ordinary differential equations, we derive explicit conditions for the occurrence of a symmetry-breaking bifurcation in a symmetric family of solutions. This bifurcation is shown to be of the transcritical type, and we provide a precise estimate of the bifurcation point as the propagation constant, which is directly related to the solution norm, is varied. In addition to the symmetric states, we explore non-positive definite states that bifurcate from the linear solutions of the system. These states exhibit distinct characteristics and are crucial in understanding solutions of the nonlinear system. Furthermore, we analyze the typical dynamics of unstable solutions, showing their behavior and evolution over time. Our results contribute to a deeper understanding of symmetry-breaking phenomena in nonlinear systems on metric graphs and provide insights into the stability and dynamics of such solutions.

nlin.PS↗

arXiv2025

Exponential asymptotics of quantum droplets and bubbles

This research investigates the formation and stability of localized states, known as quantum droplets and bubbles, in the quadratic-cubic discrete nonlinear Schr\"odinger equation. Near a Maxwell point, these states emerge from two fronts connecting the bistable equilibria. By adjusting a control parameter, we identify a "pinning region" where multiple stable states coexist and are interconnected through homoclinic snaking. We analyze the system's behavior to uncover the underlying mechanisms under strong coupling conditions. Using exponential asymptotics, we determine the pinning region's width and its dependence on coupling strength, revealing an exponentially small relationship between them. Additionally, we employ eigenvalue counting to establish the stability of these states by computing the critical eigenvalue of their corresponding linearization operator, proving onsite fronts unstable and intersite fronts stable. These theoretical results are validated through numerical simulations, which show excellent agreement with our analytical predictions.

nlin.PS↗

arXiv2025

Exponential asymptotics of dark and bright solitons in the discrete nonlinear Schr\"odinger equation

We investigate the existence and linear stability of solitons in the nonlinear Schr\"odinger lattices in the strong coupling regime. Focusing and defocusing nonlinearities are considered, giving rise to bright and dark solitons. In this regime, the effects of lattice discreteness become exponentially small, requiring a beyond-all-orders analysis. To this end, we employ exponential asymptotics to derive soliton solutions and examine their stability systematically. We show that only two symmetry-related soliton configurations are permissible: onsite solitons centered at lattice sites and intersite solitons positioned between adjacent sites. Although the instability of intersite solitons due to real eigenvalue pairs is known numerically, a rigorous analytical account, particularly for dark solitons, has been lacking. Our work fills this gap, yielding analytical predictions that match numerical computations with high accuracy. We also establish the linear stability of onsite bright solitons. While the method cannot directly resolve the quartet eigenvalue-induced instability of onsite dark solitons due to the continuous spectrum covering the entire imaginary axis, we conjecture an eigenvalue-counting argument that supports their instability. Overall, our application of the exponential asymptotics method shows the versatility of this approach for addressing multiscale problems in discrete nonlinear systems.

nlin.PS↗

arXiv2025

Physics-informed neural networks for high-dimensional solutions and snaking bifurcations in nonlinear lattices

This paper introduces a framework based on physics-informed neural networks (PINNs) for addressing key challenges in nonlinear lattices, including solution approximation, bifurcation diagram construction, and linear stability analysis. We first employ PINNs to approximate solutions of nonlinear systems arising from lattice models, using the Levenberg-Marquardt algorithm to optimize network weights for greater accuracy. To enhance computational efficiency in high-dimensional settings, we integrate a stochastic sampling strategy. We then extend the method by coupling PINNs with a continuation approach to compute snaking bifurcation diagrams, incorporating an auxiliary equation to effectively track successive solution branches. For linear stability analysis, we adapt PINNs to compute eigenvectors, introducing output constraints to enforce positivity, in line with Sturm-Liouville theory. Numerical experiments are conducted on the discrete Allen-Cahn equation with cubic and quintic nonlinearities in one to five spatial dimensions. The results demonstrate that the proposed approach achieves accuracy comparable to, or better than, traditional numerical methods, especially in high-dimensional regimes where computational resources are a limiting factor. These findings highlight the potential of neural networks as scalable and efficient tools for the study of complex nonlinear lattice systems.

math.NA↗

arXiv2025

Model-free Forecasting of Rogue Waves using Reservoir Computing

Recent research has demonstrated Reservoir Computing's capability to model various chaotic dynamical systems, yet its application to Hamiltonian systems remains relatively unexplored. This paper investigates the effectiveness of Reservoir Computing in capturing rogue wave dynamics from the nonlinear Schr\"{o}dinger equation, a challenging Hamiltonian system with modulation instability. The model-free approach learns from breather simulations with five unstable modes. A properly tuned parallel Echo State Network can predict dynamics from two distinct testing datasets. The first set is a continuation of the training data, whereas the second set involves a higher-order breather. An investigation of the one-step prediction capability shows remarkable agreement between the testing data and the models. Furthermore, we show that the trained reservoir can predict the propagation of rogue waves over a relatively long prediction horizon, despite facing unseen dynamics. Finally, we introduce a method to significantly improve the Reservoir Computing prediction in autonomous mode, enhancing its long-term forecasting ability. These results advance the application of Reservoir Computing to spatio-temporal Hamiltonian systems and highlight the critical importance of phase space coverage in the design of training data.

cs.CE↗

arXiv2025

Nonlinear states of the conservative complex Swift-Hohenberg equation

We consider the conservative complex Swift-Hohenberg equation, which belongs to the family of nonlinear fourth-order dispersive Schr\"odinger equations. In contrast to the well-studied one-dimensional dissipative Swift-Hohenberg equation, the complex variant introduces a wide array of largely unexplored solutions. Our study provides a fundamental step in understanding the complex characteristics of this equation, particularly for typical classes of solutions-uniform, periodic, and localized states-and their relationship with the original dissipative model. Our findings reveal significant differences between the two models. For instance, uniform solutions in the conservative model are inherently unstable, and periodic solutions are generally unstable except within a narrow parameter interval that supports multiple localized states. Furthermore, we establish a generalized Vakhitov-Kolokolov criterion to determine the stability of localized states in the conservative equation and relate it to the stability properties of the dissipative counterpart.

nlin.PS↗

arXiv2025

Supratransmission of rogue wave-like breathing solitons

Supratransmission is a fascinating and counterintuitive nonlinear wave phenomenon that enables energy transmission through frequency band gaps. Recent studies have suggested that supratransmission in a damped-driven Klein-Gordon equation can lead to the formation of rogue waves. In this work, we reduce the same model to a damped and driven discrete nonlinear Schr\"odinger equation and demonstrate that the claimed rogue waves are unstable solitons exhibiting breathing dynamics. Through extensive numerical simulations, we thoroughly investigate and analyze the behavior of solitons and supratransmission in these lattices. Additionally, we perform asymptotic analysis to explain the underlying mechanism responsible for supratransmission. Our findings provide new insights into the dynamics of nonlinear waves and their stability in such systems.

nlin.PS↗

arXiv2025

Using Lagrangian descriptors to reveal the phase space structure of dynamical systems described by fractional differential equations: Application to the Duffing oscillator

We showcase the utility of the Lagrangian descriptors method in qualitatively understanding the underlying dynamical behavior of dynamical systems governed by fractional-order differential equations. In particular, we use the Lagrangian descriptors method to study the phase space structure of the unforced and undamped Duffing oscillator when its time evolution is governed by fractional-order differential equations. In our study, we implement two types of fractional derivatives, namely the standard Gr\"unwald-Letnikov method, which is a finite difference approximation of the Riemann-Liouville fractional derivative, and a Gr\"unwald-Letnikov method with a correction term that approximates the Caputo fractional derivative. While there is no issue with forward-time integrations needed for the evaluation of Lagrangian descriptors, we discuss in detail ways to perform the non-trivial task of backward-time integrations and implement two methods for this purpose: a `nonlocal implicit inverse' technique and a `time-reverse inverse' approach. We analyze the differences in the Lagrangian descriptors results due to the two backward-time integration approaches, discuss the physical significance of these differences, and eventually argue that the nonlocal implicit inverse implementation of the Gr\"unwald-Letnikov fractional derivative manages to reveal the phase space structure of fractional-order dynamical systems correctly.

nlin.CD↗

arXiv2024

A finite difference method with symmetry properties for the high-dimensional Bratu equation

Solving the three-dimensional (3D) Bratu equation is highly challenging due to the presence of multiple and sharp solutions. Research on this equation began in the late 1990s, but there are no satisfactory results to date. To address this issue, we introduce a symmetric finite difference method (SFDM) which embeds the symmetry properties of the solutions into a finite difference method (FDM). This SFDM is primarily used to obtain more accurate solutions and bifurcation diagrams for the 3D Bratu equation. Additionally, we propose modifying the Bratu equation by incorporating a new constraint that facilitates the construction of bifurcation diagrams and simplifies handling the turning points. The proposed method, combined with the use of sparse matrix representation, successfully solves the 3D Bratu equation on grids of up to $301^3$ points. The results demonstrate that SFDM outperforms all previously employed methods for the 3D Bratu equation. Furthermore, we provide bifurcation diagrams for the 1D, 2D, 4D, and 5D cases, and accurately identify the first turning points in all dimensions. All simulations indicate that the bifurcation diagrams of the Bratu equation on the cube domains closely resemble the well-established behavior on the ball domains described by Joseph and Lundgren [1]. Furthermore, when SFDM is applied to linear stability analysis, it yields the same largest real eigenvalue as the standard FDM despite having fewer equations and variables in the nonlinear system.

math.NA↗