IA Scholar Query: The Solution Operator of the Korteweg-de Vries Equation is Computable.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 06 Dec 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Singular perturbation analysis for a coupled KdV-ODE system
https://scholar.archive.org/work/izal7hybdrb35iwdxvd67l6jtm
Asymptotic stability is with no doubts an essential property to be studied for any system. This analysis often becomes very difficult for coupled systems and even harder when different timescales appear. The singular perturbation method allows to decouple a full system into what are called the reduced order system and the boundary layer system, to get simpler stability conditions for the original system. In the infinite-dimensional setting, we do not have a general result making sure this strategy works. This papers is devoted to this analysis for some systems coupling the Korteweg-the Vries equation and an ordinary differential equation with different timescales. More precisely, We obtain stability results and Tikhonov-type theorems.Swann Marxwork_izal7hybdrb35iwdxvd67l6jtmTue, 06 Dec 2022 00:00:00 GMTTransonic limit of traveling waves of the Euler-Korteweg system
https://scholar.archive.org/work/4ius2inhnbe5xd65lzno3enfb4
We prove the convergence in the transonic limit of two-dimensional traveling waves of the E-K system, up to rescaling, toward a ground state of the Kadomtsev-Petviashvili Equation. Similarly, in dimension one we prove the convergence in the transonic limit of solitons toward the soliton of the Korteweg de Vries equation.Marc-Antoine Vassenetwork_4ius2inhnbe5xd65lzno3enfb4Tue, 06 Dec 2022 00:00:00 GMTOn the stability of the Kawahara equation with a distributed infinite memory
https://scholar.archive.org/work/fun7nthdxrbfxhnwmbyrzeu2iy
This article will deal with the stabilization problem for the higher-order dispersive system, commonly called the Kawahara equation. To do so, we introduce a damping mechanism via a distributed memory term in the equation to prove that the solutions of the Kawahara equation are exponentially stable, provided that specific assumptions on the memory kernel are fulfilled. This is possible thanks to the energy method that permits to provide a decay estimate of the system energy.Roberto de A. Capistrano Filho, Boumediène Chentouf, Isadora Maria de Jesuswork_fun7nthdxrbfxhnwmbyrzeu2iyMon, 05 Dec 2022 00:00:00 GMTTwistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory
https://scholar.archive.org/work/nd57s3bbpzevzdflc6pyyorfim
We show that the approaches to integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations are closely related, at least classically. Following a suggestion of Kevin Costello, we start from holomorphic Chern-Simons theory on twistor space, defined with the help of a meromorphic (3,0)-form Ω. If Ω is nowhere vanishing, it descends to a theory on 4d space-time with classical equations of motion equivalent to the anti-self-dual Yang-Mills equations. Examples include a 4d analogue of the Wess-Zumino-Witten model and a theory of a Lie algebra valued scalar with a cubic two derivative interaction. Under symmetry reduction, these yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces holomorphic Chern-Simons theory to the 4d Chern-Simons theory with disorder defects studied by Costello Yamazaki. Finally we show that a similar reduction by a single translation leads to a 5d partially holomorphic Chern-Simons theory describing the Bogomolny equations.Roland Bittleston, David Skinnerwork_nd57s3bbpzevzdflc6pyyorfimMon, 05 Dec 2022 00:00:00 GMTAdmissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
https://scholar.archive.org/work/h7oa23m4ezhdhn3yxw5gnhhpqi
We comprehensively study admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements parameterizing these systems. For each class from the constructed chain of nested gauged classes of such systems, we single out its singular subclass, which appears to consist of systems being similar to the elementary (free particle) system whereas the regular subclass is the complement of the singular one. This allows us to exhaustively describe the equivalence groupoids of the above classes as well as of their singular and regular subclasses. Applying various algebraic techniques, we establish principal properties of Lie symmetries of the systems under consideration and outline ways for completely classifying these symmetries. In particular, we compute the sharp lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by systems from each of the above classes and subclasses. We also show how equivalence transformations and Lie symmetries can be used for reduction of order of such systems and their integration. As an illustrative example of using the theory developed, we solve the complete group classification problems for all these classes in the case of two dependent variables.Vyacheslav M. Boyko, Oleksandra V. Lokaziuk, Roman O. Popovychwork_h7oa23m4ezhdhn3yxw5gnhhpqiMon, 05 Dec 2022 00:00:00 GMTThe defocusing NLS equation with nonzero background: Painlevé asymptotics in two transition regions
https://scholar.archive.org/work/4hrvq32sfvayzkawaajchjhlgu
In this paper, we address the Painlevé aymptotics in the transition region |ξ|:=|x/2t| ≈ 1 to the Cauchy problem of the defocusing Schrödinger equation with a nonzero background.With the ∂̅-generation of the nonlinear steepest descent approach and double scaling limit to compute the long-time asymptotics of the solution in two transition regions defined as 𝒫_± 1(x,t):={ (x,t) ∈ℝ×ℝ^+, 0<|ξ-(± 1)|t^2/3≤ C}, we find that the long-time asymptotics in both transition regions 𝒫_± 1(x,t) can be expressed in terms of the Painlevé II equation. We are also able to express the leading term explicitly in terms of the Ariy function.Zhaoyu Wang, Engui Fanwork_4hrvq32sfvayzkawaajchjhlguSat, 03 Dec 2022 00:00:00 GMTCoupling the SBA Method and the Elzaki Transformation to Solve Nonlinear Fractional Differential Equations
https://scholar.archive.org/work/gcyoscispbduvketzerkhdjcge
In this paper, we propose a new technique well adapted to the solution of nonlinear fractional differential equations. This technique combines the Elzaki transorm and the Some Blaise-Abbo (SBA) method. It allows to find the exact solution or an acceptable approximate solution of the equation.Kamate Adama, Bakari Abbo, Djibet Mbaiguesse, Youssouf Parework_gcyoscispbduvketzerkhdjcgeSat, 03 Dec 2022 00:00:00 GMTModified homotopy perturbation method and its application to analytical solitons of fractional-order Korteweg–de Vries equation
https://scholar.archive.org/work/p3ipfg5gwjgxlpm4sod45rgjj4
Background Experimentally brought to light by Russell and hypothetically explained by Korteweg–de Vries, the KDV equation has drawn the attention of several mathematicians and physicists because of its extreme substantial structure in describing nonlinear evolution equations governing the propagation of weakly dispersive and nonlinear waves. Due to the prevalent nature and application of solitary waves in nonlinear dynamics, we discuss the soliton solution and application of the fractional-order Korteweg–de Vries (KDV) equation using a new analytical approach named the "Modified initial guess homotopy perturbation." Results We established the proposed technique by coupling a power series function of arbitrary order with the renown homotopy perturbation method. The convergence of the method is proved using the Banach fixed point theorem. The methodology was demonstrated with a generalized KDV equation, and we applied it to solve linear and nonlinear fractional-order Korteweg–de Vries equations, which are in Caputo sense. The method's applicability and effectiveness were established as a feasible series of arbitrary orders that accelerate quickly to the exact solution at an integer order and are obtained as solutions. Numerical simulations were conducted to investigate the effect of Caputo fractional-order derivatives in the dispersion and propagation of water waves by varying the order $$\alpha$$ α on the $$[0,1]$$ [ 0 , 1 ] interval. Comparative analysis of the simulation results, which were presented graphically and discussed, reveals that the degree of freedom of the Caputo fractional-order derivative is vital to controlling the magnitude of environmental hazards associated with water waves when adjusted. Conclusion The proposed method is recommended for obtaining convergent series solutions to fractional-order partial differential equations. We suggested that applied mathematicians and physicists investigate this work to better understand the impact of the degree of freedom posed by Caputo fractional-order derivatives in wave dispersion and propagation, as physical applications can help divert wave-related environmental hazards.Adedapo Ismaila Alaje, Morufu Oyedunsi Olayiwola, Kamilu Adewale Adedokun, Joseph Adeleke Adedeji, Asimiyu Olamilekan Oladapowork_p3ipfg5gwjgxlpm4sod45rgjj4Fri, 02 Dec 2022 00:00:00 GMTStability of peakons of the Camassa–Holm equation beyond wave breaking
https://scholar.archive.org/work/4vaf6svrerhd3iddqmtqvmmju4
Using a generalized framework that consists of evolution of the solution to the Camassa–Holm equation and its energy measure, we establish the global-in-time orbital stability of peakons with respect to the perturbed (energy) conservative solutions to the Camassa–Holm equation. In particular, we extend the H1-stability result obtained by Constantin and Strauss [Commun. Pure Appl. Math. 53(5), 603–610 (2000)] globally-in-time even after the perturbed solutions experience wave breaking. In addition, our result also shows that the singular part of the energy measure of the perturbed solutions will remain stable for all times.Yu Gao, Hao Liu, Tak Kwong Wongwork_4vaf6svrerhd3iddqmtqvmmju4Thu, 01 Dec 2022 00:00:00 GMTThe Herglotz variational principle for dissipative field theories
https://scholar.archive.org/work/fdpb67qahzhavnmttibgegg4gu
In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg-De Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.Jordi Gaset and Manuel Lainz and Arnau Mas and Xavier Rivaswork_fdpb67qahzhavnmttibgegg4guWed, 30 Nov 2022 00:00:00 GMTUnique continuation and time decay for a higher-order water wave model
https://scholar.archive.org/work/xvcv7olfyjbo5jrllvyrr6n4v4
This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg -de Vries (KdV)--Benjamin-Bona-Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in [11], which combines energy estimates, multipliers and compactness arguments, the problem is reduced to prove the Unique Continuation Property (UCP) for weak solutions of the model. Then, this is done by deriving Carleman estimates for a system of coupled elliptic-hyperbolic equations.Ademir F. Pazoto, Miguel Sotowork_xvcv7olfyjbo5jrllvyrr6n4v4Wed, 30 Nov 2022 00:00:00 GMTNon-flat conformal blow-up profiles for the 1D critical nonlinear Schrödinger equation
https://scholar.archive.org/work/hoqcon777zdhrkqr5jxaeroxlm
For the critical one-dimensional nonlinear Schrödinger equation, we construct blow-up solutions that concentrate a soliton at the origin at the conformal blow-up rate, with a non-flat blow-up profile. More precisely, we obtain a blow-up profile that equals |x|+iκ x^2 near the origin, where κ is a universal real constant. Such profile differs from the flat profiles obtained in the same context by Bourgain and Wang [Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Sc. Norm. Super. Pisa Cl. Sci. 25 (1997)].Yvan Martel, Ivan Naumkinwork_hoqcon777zdhrkqr5jxaeroxlmWed, 30 Nov 2022 00:00:00 GMTOpen r-spin theory II: The analogue of Witten's conjecture for r-spin disks
https://scholar.archive.org/work/auaxfqsmjjeblnehr7hxl65l24
We conclude the construction of r-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open r-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the rth Gelfand-Dickey integrable hierarchy. This provides an analogue of Witten's r-spin conjecture in the open setting and a first step toward the construction of an open version of Fan-Jarvis-Ruan-Witten theory. As an unexpected consequence, we establish a mysterious relationship between open r-spin theory and an extension of Witten's closed theory.Alexandr Buryak and Emily Clader and Ran J. Tesslerwork_auaxfqsmjjeblnehr7hxl65l24Wed, 30 Nov 2022 00:00:00 GMTSome Comments on Unitary Qubit Lattice Algorithms for Classical Problems
https://scholar.archive.org/work/vfwxpygkzfhejf35hynyzsxtp4
Abstract: A qubit lattice algorithm (QLA) for normal incidence of an rectangular electromagnetic pulse onto a dielectric slab is examined and shows that the transmission coefficient is indeed augmented over the Fresnel boundary value infinite plane wave result by the square root of the ratio of the refractive indices of the two media. For an oscillatory wave packet, this transmission coefficient is further increased. As the QLA is not fully unitary, due to one evolution operator being Hermitian, first steps are taken in correcting a similar problem of determining a fully unitary QLA for the Korteweg-de Vries equation. This is achieved by appropriate perturbation of the unitary collision angle.Paul Anderson, Lillian Finegold-Sachs, George Vahala, Linda Vahala, Abhay K. Ram, Min Soe, Efstratios Koukoutsis, Kyriakos Hizandiswork_vfwxpygkzfhejf35hynyzsxtp4Wed, 30 Nov 2022 00:00:00 GMTLong-time asymptotics of a complex cubic Camassa-Holm equation
https://scholar.archive.org/work/3fzvw3rcgbejrnfjkdncz5qs7u
In this paper, we study the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation m_t=b u_x+1/2[m(|u|^2-|u_x|^2)]_x-1/2 m(u u̅_x-u_xu̅), m=u-u_x x, where b is an arbitrary real constant. l Long-time asymptotics of the equation is obtained through the ∂̅-steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed corresponding Riemann-Hilbert problem (RHP). Then, we present long time asymptotic expansions of the solution u(y,t) in different space-time solitonic regions of ξ=y/t. The half-plane (y,t):-∞ 0 is divided into four asymptotic regions: ξ∈(-∞,-1), ξ∈ (-1,0), ξ∈ (0,1/8) and ξ∈ (1/8,+∞). When ξ falls in (-∞,-1)∪ (1/8,+∞), no stationary phase point of the phase function θ(z) exists on the jump profile in the space-time region. In this case, corresponding asymptotic approximations can be characterized with an N(Λ)-solitons with diverse residual error order O(t^-1+2ε). eight stationary phase points on the jump curve as ξ∈ (-1,0) and ξ∈ (0,1/8), respectively. The corresponding asymptotic form is accompanied by a residual error order O(t^-3/4).Hongyi Zhang, Yufeng Zhang, Zhijun Qiaowork_3fzvw3rcgbejrnfjkdncz5qs7uWed, 30 Nov 2022 00:00:00 GMTKoopman analysis of the periodic Korteweg-de Vries equation
https://scholar.archive.org/work/hhnvk6kck5ggzmomytp5xbvtie
The eigenspectrum of the Koopman operator enables the decomposition of nonlinear dynamics into a sum of nonlinear functions of the state space with purely exponential and sinusoidal time dependence. For a limited number of dynamical systems, it is possible to find these Koopman eigenfunctions exactly and analytically. Here, this is done for the Korteweg-de Vries equation on a periodic interval, using the periodic inverse scattering transform and some concepts of algebraic geometry. To the authors' knowledge, this is the first complete Koopman analysis of a partial differential equation which does not have a trivial global attractor. The results are shown to match the frequencies computed by the data-driven method of dynamic mode decomposition (DMD). We demonstrate that in general DMD gives a large number of eigenvalues near the imaginary axis, and show how these should be interpretted in this setting.Jeremy P Parker, Claire Valvawork_hhnvk6kck5ggzmomytp5xbvtieWed, 30 Nov 2022 00:00:00 GMTA Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
https://scholar.archive.org/work/b4muktoz2rdfph7ypuot6bbqce
The increasing demand for bandwidth and long-haul transmission has led to new methods of signal processing and transmission in optical fiber communication systems. The nonlinear Fourier transform is one of the most recent methods proposed, and is able to represent an integrable nonlinear Schrödinger equation (NLSE) channel in terms of its continuous and discrete spectrum, to overcome the limitation of the bandwidth imposed by the Kerr effect on silica fibers. In this paper, we will propose and investigate the Boffetta-Osburne method for the direct nonlinear Fourier implementation, and the Gel'fand-Levitan-Marchenko equation for the inverse nonlinear Fourier, as only the continuous part of the nonlinear spectrum will be used to encode information. A novel methodology is proposed to improve their numerical implementation with respect to the NLSE, and we analyze in detail how the improved algorithm can be used in a real optical system, by investigating three different modulation schemes. We report increased performance transmission and consistency in the numerical results when the proposed methodology is applied. Our results show that b-modulation will increase the Q-factor by 2 dB with respect to the other two modulations. The improvement results with our proposed methodology suggest that b-modulation applied only to a continuous part of the nonlinear spectrum is a very effective method for maximizing both the transmission bandwidth and distance in optical fiber communication systems.Julian Hoxha, Wael Hosny Fouad Aly, Erdjana Dida, Iva Kertusha, Mouhammad AlAkkoumiwork_b4muktoz2rdfph7ypuot6bbqceTue, 29 Nov 2022 00:00:00 GMTTautological relations and integrable systems
https://scholar.archive.org/work/szgkqx4vmffw7p7k7v4pczeeey
We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus g with n marked points. A large part of these relations has a surprisingly simple form: the tautological classes involved in the relations are given by stable graphs that are trees and that are decorated only by powers of the psi-classes at half-edges. We show that the proposed conjectural relations imply certain fundamental properties of the Dubrovin-Zhang (DZ) and the double ramification (DR) hierarchies associated to F-cohomological field theories. Our relations naturally extend a similar system of conjectural relations, which were proposed in an earlier work of the first author together with Guéré and Rossi and which are responsible for the normal Miura equivalence of the DZ and the DR hierarchy associated to an arbitrary cohomological field theory. Finally, we prove all the above mentioned relations in the case n=1 and arbitrary g using a variation of the method from a paper by Liu and Pandharipande, this can be of independent interest. In particular, this proves the main conjecture from our previous joined work together with Hernández Iglesias. We also prove all the above mentioned relations in the case g=0 and arbitrary n.Alexandr Buryak, Sergey Shadrinwork_szgkqx4vmffw7p7k7v4pczeeeyMon, 28 Nov 2022 00:00:00 GMTPhysics-Informed Neural Networks (PINNs)-Based Traffic State Estimation: An Application to Traffic Network
https://scholar.archive.org/work/jqcki6oys5bzzkpg5zg2fveukq
Traffic state estimation (TSE) is a critical component of the efficient intelligent transportation systems (ITS) operations. In the literature, TSE methods are divided into model-driven methods and data-driven methods. Each approach has its limitations. The physics information-based neural network (PINN) framework emerges to mitigate the limitations of the traditional TSE methods, while the state-of-art of such a framework has focused on single road segments but can hardly deal with traffic networks. This paper introduces a PINN framework that can effectively make use of a small amount of observational speed data to obtain high-quality TSEs for a traffic network. Both model-driven and data-driven components are incorporated into PINNs to combine the advantages of both approaches and to overcome their disadvantages. Simulation data of simple traffic networks are used for studying the highway network TSE. This paper demonstrates how to solve the popular LWR physical traffic flow model with a PINN for a traffic network. Experimental results confirm that the proposed approach is promising for estimating network traffic accurately.Muhammad Usama, Rui Ma, Jason Hart, Mikaela Wojcikwork_jqcki6oys5bzzkpg5zg2fveukqSun, 27 Nov 2022 00:00:00 GMTSoliton-mean field interaction in Korteweg-de Vries dispersive hydrodynamics
https://scholar.archive.org/work/cpimpvy54vebbigihnia4zwgra
The propagation of localized solitons in the presence of large-scale waves is a fundamental problem, both physically and mathematically, with applications in fluid dynamics, nonlinear optics and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg-de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A comprehensive review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton-rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite-gap description are used to describe soliton-rarefaction wave and soliton-dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the Inverse Scattering Transform. For transmitted solitons, far-field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave-mean field interactionMark J. Ablowitz, Justin T. Cole, Gennady A. El, Mark A. Hoefer, Xu-dan Luowork_cpimpvy54vebbigihnia4zwgraSun, 27 Nov 2022 00:00:00 GMT