arXiv subjects

Hadi Susanto

Publications and source records attributed to Hadi Susanto.

At least 37 records · Page 2Linked to original sources

arXiv2025

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↗

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

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 Schrödinger 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 Grünwald-Letnikov method, which is a finite difference approximation of the Riemann-Liouville fractional derivative, and a Grünwald-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 Grünwald-Letnikov fractional derivative manages to reveal the phase space structure of fractional-order dynamical systems correctly.

nlin.CD↗

arXiv2024

New Power Method for Solving Eigenvalue Problems

We present a new power method to obtain solutions of eigenvalue problems. The method can determine not only the dominant or lowest eigenvalues but also all eigenvalues without the need for a deflation procedure. The method uses a functional of an operator (or a matrix) to select or filter an eigenvalue. The method can freely select a solution by varying a parameter associated to an estimate of the eigenvalue. The convergence of the method is highly dependent on how closely the parameter to the eigenvalues. In this paper, numerical results of the method are shown to be in excellent agreement with the analytical ones.

math.NA↗

arXiv2024

Transparent PT-symmetric nonlinear networks

We consider reflectionless wave propagation in networks modeled in terms of the nonlocal nonlinear Schrödinger (NNLS) equation on metric graphs, for which transparent boundary conditions are imposed at the vertices. By employing the ``potential approach" previously used for the nonlinear Schrödinger equation, we derive transparent boundary conditions for the NNLS equation on metric graphs. These conditions eliminate backscattering at graph vertices, which is crucial for minimizing losses in signal, heat, and charge transfer in various applications such as optical fibers, optoelectronic networks, and low-dimensional materials.

math-ph↗

arXiv2023

A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equations

We propose a new method, called a deep-genetic algorithm (deep-GA), to accelerate the performance of the so-called deep-BSDE method, which is a deep learning algorithm to solve high dimensional partial differential equations through their corresponding backward stochastic differential equations (BSDEs). Recognizing the sensitivity of the solver to the initial guess selection, we embed a genetic algorithm (GA) into the solver to optimize the selection. We aim to achieve faster convergence for the nonlinear PDEs on a broader interval than deep-BSDE. Our proposed method is applied to two nonlinear parabolic PDEs, i.e., the Black-Scholes (BS) equation with default risk and the Hamilton-Jacobi-Bellman (HJB) equation. We compare the results of our method with those of the deep-BSDE and show that our method provides comparable accuracy with significantly improved computational efficiency.

math.AP↗

arXiv2023

Higher-order composition of short- and long-period effects for improving analytical ephemeris computation

The construction of an analytic orbit theory that takes into account the main effects of the Geopotential is notably simplified when splitting the removal of periodic effects in several stages. Conversely, this splitting of the analytical solution into several transformations reduces the evaluation efficiency for dense ephemeris output. However, the advantage is twofold when the different parts of the mean-to-osculating transformation are composed into a single transformation. To show that, Brouwer's solution is extended to the second order of the zonal harmonic of the second degree by the sequential elimination of short- and long-period terms. Then, the generating functions of the different transformations are composed into a single one, from which a single mean-to-osculating transformation is derived. The new, unique transformation notably speeds up the evaluation process, commonly improving evaluation efficiency by at least one third with respect to the customary decomposition of the analytical solution into three different parts.

physics.class-ph↗

arXiv2022

Static Spacetimes In Higher Dimensional Scalar-Torsion Theories With Non-Minimal Derivative Coupling

In this paper we consider a class of static spacetimes in higher dimensional ($D \ge 4$) scalar-torsion theories with non-minimal derivative coupling and the scalar potential turned on. The spacetime is conformal to a product space of a two-surface and a $(D-2)$-dimensional submanifold. Analyzing the equations of motion in the theory we find that the $(D-2)$-dimensional submanifold has to admit constant triplet structures in which the torsion scalar is one of them. This implies that these equations of motion can be simplified into a single highly non-linear ordinary differential equation called the master equation. Then, we show that in this case the solution admits at least a naked singularity at the origin which is not a black hole. In the asymptotic region, the spacetimes converge to spaces of constant scalar curvature which are generally not Einstein. We also use perturbative method to linearize the master equation and construct the first order solutions. At the end, we establish the analysis of local-global existences of the master equation and then, prove the non-existence of regular global solutions.

gr-qc↗

arXiv2022

Static Black Holes in Higher Dimensional Einstein-Skyrme Models

In this paper we construct a class of hairy static black holes of higher dimensional Einstein-Skyrme theories with the cosmological constant $Λ\le 0$ whose scalar is an $SU(2)$ valued field. The spacetime is set to be conformal to $ \mathcal{M}^4 \times \mathcal{N}^{N-4}$ where $\mathcal{M}^4$ and $\mathcal{N}^{N-4}$ are a four dimensional spacetime and a compact Einstein $(N-4)$-dimensional submanifold for $N \ge 5$, respectively, whereas $N=4$ is the trivial case. We discuss the behavior of solutions near the boundaries, namely, near the (event) horizon and in the asymptotic region. Then, we establish local-global existence of black hole solutions and show that black holes with finite energy exist if their geometries are asymptotically Ricci-flat. At the end, we perform a linear stability analysis using perturbative method and give a remark about their stability.

hep-th↗

arXiv2021

Removing non-smoothness in solving Black-Scholes equation using a perturbation method

Black-Scholes equation as one of the most celebrated mathematical models has an explicit analytical solution known as the Black-Scholes formula. Later variations of the equation, such as fractional or nonlinear Black-Scholes equations, do not have a closed form expression for the corresponding formula. In that case, one will need asymptotic expansions, including homotopy perturbation method, to give an approximate analytical solution. However, the solution is non-smooth at a special point. We modify the method by {first} performing variable transformations that push the point to infinity. As a test bed, we apply the method to the solvable Black-Scholes equation, where excellent agreement with the exact solution is obtained. We also extend our study to multi-asset basket and quanto options by reducing the cases to single-asset ones. Additionally we provide a novel analytical solution of the single-asset quanto option that is simple and different from the existing expression.

q-fin.MF↗

arXiv2021

Embedded solitons in second-harmonic-generating lattices

Embedded solitons are exceptional modes in nonlinear-wave systems with the propagation constant falling in the system's propagation band. An especially challenging topic is seeking for such modes in nonlinear dynamical lattices (discrete systems). We address this problem for a system of coupled discrete equations modeling the light propagation in an array of tunnel-coupled waveguides with a combination of intrinsic quadratic (second-harmonic-generating) and cubic nonlinearities. Solutions for discrete embedded solitons (DESs) are constructed by means of two analytical approximations, adjusted, severally, for broad and narrow DESs, and in a systematic form with the help of numerical calculations. DESs of several types, including ones with identical and opposite signs of their fundamental-frequency and second-harmonic components, are produced. In the most relevant case of narrow DESs, the analytical approximation produces very accurate results, in comparison with the numerical findings. In this case, the DES branch extends from the propagation band into a semi-infinite gap as a family of regular discrete solitons. The study of stability of DESs demonstrates that, in addition to ones featuring the well-known property of semi-stability, there are linearly stable DESs which are genuinely robust modes.

nlin.PS↗

arXiv2020

Dynamics of laterally-coupled pairs of spin-VCSELs

A newly-developed normal mode model of laser dynamics in a generalised array of waveguides is applied to extend the spin-flip model (SFM) to pairs of evanescently-coupled spin-VCSELS. The effect of high birefringence is explored, revealing new dynamics and regions of bistability. It is shown that optical switching of the polarisation states of the lasers may be controlled through the optical pump and that, under certain conditions, the polarisation of one laser may be switched by controlling the intensity and polarisation in the other.

physics.optics↗

arXiv2019

Reduction of damped, driven Klein-Gordon equations into a discrete nonlinear Schrödinger equation: justification and numerical comparisons

We consider a discrete nonlinear Klein-Gordon equations with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein-Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.

nlin.PS↗

arXiv2019

Symmetry breaking bifurcations in the NLS equation with an asymmetric delta potential

We consider the NLS equation with a linear double well potential. Symmetry breaking, i.e., the localisation of an order parameter in one of the potential wells that can occur when the system is symmetric, has been studied extensively. However, when the wells are asymmetric, only a few analytical works have been reported. Using double Dirac delta potentials, we study rigorously the effect of such asymmetry on the bifurcation type. We show that the standard pitchfork bifurcation becomes broken and instead a saddle-centre type is obtained. Using a geometrical approach, we also establish the instability of the corresponding solutions along each branch in the bifurcation diagram

math.DS↗

arXiv2019

Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation

We study the integrable nonlocal nonlinear Schrödinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time ($\mathcal{PT}$) symmetric self-induced potential. We consider dynamics (including collisions) of moving solitons. Analytically we develop a collective coordinate approach based on variational methods and examine its applicability in the system. We show numerically that a single moving soliton can pass the origin and decay or be trapped at the origin and blows up at a finite time. Using a standard soliton ansatz, the variational approximation can capture the dynamics well, including the finite-time blow up, even though the ansatz is relatively far from the actual blowing-up soliton solution. In the case of two solitons moving towards each other, we show that there can be a mass transfer between them, in addition to wave scattering. We also demonstrate that defocusing nonlinearity can support bright solitons.

nlin.PS↗

arXiv2019

Local and Global Existence of Solutions to Scalar Equations on Spatially Flat Universe as a Background with Non-minimal Coupling

We prove the wellposedness of scalar wave equations on spatially flat universe as a background with nonminimal coupling with the scalar potential turned on by introducing the $k$-order linear energy and the corresponding energy norm. In the local case, we show that both the $k$-order linear energy and the energy norm are bounded for finite time with initial data in $H^{k+1}\times H^{k}$. Whereas in the global case, we have to add three assumptions related to the nonminimal coupling constant, the scale factor of spacetimes, and the form of the scalar that has to be a polynomial with a small positive parameter. Then, we show that the solution does globally exist with a particular decay estimate that depends on the scale factor of the spacetimes. Finally, we provide some physical models that support our general setup.

math-ph↗

arXiv2018

Variational approximations using Gaussian ansatz, false instability, and its remedy in nonlinear Schrödinger lattices

We study the fundamental lattice solitons of the discrete nonlinear Schrödinger (DNLS) equation and their stability via a variational method. Using a Gaussian ansatz and comparing the results with numerical computations, we report a novel observation of false instabilities. Comparing with established results and using Vakhitov-Kolokolov criterion, we deduce that the instabilities are due to the ansatz. In the context of using the same type of ansatzs, we provide a remedy by employing multiple Gaussian functions. The results show that the higher the number of Gaussian function used, the better the solution approximation.

nlin.PS↗