Self-dual Network Equation Research Articles

Pseudo grid-based physics-informed convolutional-recurrent network solving the integrable nonlinear lattice equations

Traditional discrete learning methods involve discretizing continuous equations using difference schemes, necessitating considerations of stability and convergence. Integrable nonlinear lattice equations possess a profound mathematical structure that enables them to revert to continuous integrable equations in the continuous limit, particularly retaining integrable properties such as conservation laws, Hamiltonian structure, and multiple soliton solutions. The pseudo grid-based physics-informed convolutional-recurrent network (PG-PhyCRNet) is proposed to investigate the localized wave solutions of integrable lattice equations, which significantly enhances the model’s extrapolation capability to lattice points beyond the temporal domain. We conduct a comparative analysis of PG-PhyCRNet with and without pseudo grid by investigating the multi-soliton solutions and rational solitons of the Toda lattice and self-dual network equation. The results indicate that the PG-PhyCRNet excels in capturing long-term evolution and enhances the model’s extrapolation capability for solitons, particularly those with steep waveforms and high wave speeds. Finally, the robustness of the PG-PhyCRNet method and its effect on the prediction of solutions in different scenarios are confirmed through repeated experiments involving pseudo grid partitioning.

Relative voltage and current for the self-dual network equation characterized by Dbar data

The self-dual network equation is an important model in the electrical signals and the nonlinear, lumped and self-dual ladder-type network. The local ∂¯(Dbar) approach with non-canonical normalization condition is extended to investigate the self-dual network equation. The associated relative voltage and current for higher-order solitons are characterized by the Dbar data, including the discrete spectrum and norming constants. The dynamic behaviors are also discussed.

Discrete nonlocal N-fold Darboux transformation and soliton solutions in a reverse space-time nonlocal nonlinear self-dual network equation

In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of [Formula: see text]-fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal [Formula: see text]-symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals.

Discrete multi-soliton solutions and dynamics for a reverse-time nonlocal nonlinear self-dual network equation

Inspired by the idea of Ablowitz and Musslimani, a reverse-time nonlocal nonlinear self-dual network equation is proposed under certain nonlocal symmetry reduction. N-fold Darboux transformation technique is used to construct discrete unstable and stable multi-soliton solutions in zero seed background for this system. Based on the asymptotic and graphic analysis, structures of one-, two- and three-soliton solutions are shown graphically. It is found that the individual components in this nonlocal equation show unstable temporal soliton structures whereas the combined potential terms exhibit stable soliton structures. Dynamical behaviors for one-soliton solutions are investigated via numerical simulations. It is worth noting that the solutions of this nonlocal equation are clearly different from those of its corresponding local part. Results obtained in this paper may be helpful for the explanation of the propagation of electrical signals.

Modulational instability and dynamics of discrete rational soliton and mixed interaction solutions for a higher-order nonlinear self-dual network equation

In this paper, the higher-order nonlinear self-dual network equation is investigated. Firstly, an integrable lattice hierarchy associated with this equation is constructed from a discrete matrix spectral problem without the denominator. Secondly, the condition of modulational instability for this equation is given. Thirdly, the infinitely many conservation laws are constructed on the basis of its new Lax representation. Finally, the discrete generalised $$(m, N-m)$$ -fold Darboux transformation (DT) with non-zero constant as seed solution is used to derive new rational soliton (RS) and mixed interaction solutions. As an application, RS solutions can be obtained by the discrete generalised $$(1, N-1)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=1$$ ), and mixed interaction solutions of the usual sech-type soliton (US) and RS can be derived by the discrete generalised $$(2, N-2)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=2$$ ). We also perform the numerical simulations for such soliton solutions to explore their dynamical behaviours. Results given in this paper may have some prospective applications for understanding the propagation of electrical signals.

Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type* *Project supported by the National Natural Science Foundation of China (Grant Nos. 12071042 and 61471406), the Beijing Natural Science Foundation, China (Grant No. 1202006), and Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCP-B201704).

We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction, which may have potential applications in electric circuits. Nonlocal infinitely many conservation laws are constructed based on its Lax pair. Nonlocal discrete generalized (m, N – m)-fold Darboux transformation is extended and applied to solve this system. As an application of the method, we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized (1, N – 1)-fold Darboux transformation, respectively. By using the asymptotic and graphic analysis, structures of one-, two-, three- and four-soliton solutions are shown and discussed graphically. We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures. It is shown that the soliton structures are quite different between discrete local and nonlocal systems. Results given in this paper may be helpful for understanding the electrical signals propagation.

Nonlinear self-dual network equations: modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviours.

The nonlinear self-dual network equations that describe the propagations of electrical signals in nonlinear LC self-dual circuits are explored. We firstly analyse the modulation instability of the constant amplitude waves. Secondly, a novel generalized perturbation (M, N - M)-fold Darboux transform (DT) is proposed for the lattice system by means of the Taylor expansion and a parameter limit procedure. Thirdly, the obtained perturbation (1, N - 1)-fold DT is used to find its new higher-order rational solitons (RSs) in terms of determinants. These higher-order RSs differ from those known results in terms of hyperbolic functions. The abundant wave structures of the first-, second-, third- and fourth-order RSs are exhibited in detail. Their dynamical behaviours and stabilities are numerically simulated. These results may be useful for understanding the wave propagations of electrical signals.

Soliton interactions and their dynamics in a higher-order nonlinear self-dual network equation

Under investigation in this paper is a higher-order nonlinear self-dual network equation, which may simulate the wave propagation in a ladder type electric circuit. By means of the N-fold Darboux transformation and symbolic computation, the N-soliton solutions in determinant form are obtained. Based on the asymptotic and graphic analysis, the elastic interaction phenomena between/among two-, three- and four-soliton solutions are discussed, and some important physical quantities are accurately analyzed. Numerical simulations are used to explore the dynamical stability of one- and two-soliton solutions. Results might be helpful for understanding the propagation and interaction properties of electrical signals in a ladder type nonlinear self-dual network.

Rational solutions for three semi-discrete modified Korteweg–de Vries-type equations

In this paper, we consider three semi-discrete modified Korteweg–de Vries type equations which are the nonlinear lumped self-dual network equation, the semi-discrete lattice potential modified Korteweg–de Vries equation and a semi-discrete modified Korteweg–de Vries equation. We derive several kinds of exact solutions, in particular rational solutions, in terms of the Casorati determinant for these three equations, respectively. For some rational solutions, we present the related asymptotic analysis to understand their dynamics better.

Rational solutions for the potential nonlinear lumped self-dual network equation

In this letter we investigate the potential nonlinear lumped self-dual network equation, which is always used to model one-dimensional anharmonic chain of atoms. By the bilinear method we construct the rational solutions for the aforesaid equation. These solutions are presented in terms of Casoratian. The dynamical behaviors for the first three non-trivial rational solutions are analyzed with graphical illustration.

Infinite number of conservation laws and Darboux transformations for a 6-field integrable lattice system

In this paper, we study a 6-field integrable lattice system, which, in some special cases, can be reduced to the self-dual network equation, the discrete second-order nonlinear Schrödinger equation and the relativistic Volterra lattice equation. With the help of the Lax pair, we construct infinitely many conservation laws and a new Darboux transformation for system. Exact solutions resulting from the obtained Darboux transformation are presented by using a given seed solution. Further, we generate the soliton solutions and plot the figures of one-soliton solutions with properly parameters.

Degenerate Solutions of the Nonlinear Self-Dual Network Equation**Supported by the Natural Science Foundation of Zhejiang Province under Grant No. LY15A010005, the Natural Science Foundation of Ningbo under Grant No. 2018A610197, the NSF of China under Grant No. 11671219, K.C. Wong Magna Fund in Ningbo University

The N-fold Darboux transformation (DT) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero “seed” solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically, which provide excellent fit of exact trajectories. These orbits have a time-dependent “phase shift”, namely .

Exact Description of the Discrete Breathers and Solitons Interaction in the Nonlinear Transmission Lines

For the nonlinear self-dual network equations and the equivalent Hirota lattice equation the pair collision processes of two discrete breathers, breather and one-parametric soliton (kink, antikink), breather and linear wave, one-parametric soliton and linear wave are described. The explicit expressions of the “kink-breather” and “breather-breather” solutions are constructed. The shifts of the center-of-masses and phases of the breather oscillations have been expressed in terms of the dynamical characteristics of linear and nonlinear excitations of the system.

Nonlinear periodic waves solutions of the nonlinear self-dual network equations

The new classes of periodic solutions of nonlinear self-dual network equations describing the breather and soliton lattices, expressed in terms of the Jacobi elliptic functions have been obtained. The dependences of the frequencies on energy have been found. Numerical simulations of soliton lattice demonstrate their stability in the ideal lattice and the breather lattice instability in the dissipative lattice. However, the lifetime of such structures in the dissipative lattice can be extended through the application of ac driving terms.

Construction of exact solutions for fractional-type difference-differential equations via symbolic computation

This paper deals with fractional-type difference-differential equations by means of the extended simplest equation method. First, an equation related to the discrete KdV equation is considered. Second, a system related to the well-known self-dual network equations through a real discrete Miura transformation is analyzed. As a consequence, three types of exact solutions (with the aid of symbolic computation) emerged; hyperbolic, trigonometric and rational which have not been reported before. Our results could be used as a starting point for numerical procedures as well.

Nonlinear Superposition Formula for the Hirota Lattice Equation

The nonlinear superposition formula for the Hirota lattice equation, which is equivalent to the system of nonlinear lumped self-dual network equations, was obtained. It is a recurrence relation allowing us to generate more complex soliton solutions using more simple soliton solutions. It was shown that, using the nonlinear superposition formula, one can derive breather and wobbling kink solutions. The wobbling kink solution of the Hirota lattice equation was obtained for the first time in this work.

Elastic Interaction and Conservation Laws for the Nonlinear Self-Dual Network Equation in Electric Circuit

The nonlinear self-dual network equation can describe the propagation of electrical signals in electric circuit. In this paper, the nonlinear self-dual network equation is investigated via Darboux transformation (DT) technique. N -fold DT and conservation laws for the nonlinear self-dual network equation are constructed on the basis of its Lax representation. N -soliton solutions in terms of determinant are derived with the resulting DT. Structures of the one-, two-, and three-soliton solutions are shown graphically. Elastic interaction phenomena between/among the two and three solitons are discussed for the nonlinear self-dual network equation, which might be helpful to understanding the propagation of electrical signals.

Breathers and limit solutions of the nonlinear lumped self-dual network equation

More general Casoratian conditions for the nonlinear lumped self-dual network equation are proved via its bilinear Bäcklund transformation. Limit solutions and breathers are the special cases in the solution list related to the new Casoratian conditions. By asymptotic analysis we give a clear description on 2-soliton interactions and the relationship between 2-soliton solution and the simplest limit solution. Dynamics of breathers and limit breathers are also investigated.

ADM-Padé technique for the nonlinear lattice equations

ADM-Padé technique is a combination of Adomian decomposition method (ADM) and Padé approximants. We solve two nonlinear lattice equations using the technique which gives the approximate solution with higher accuracy and faster convergence rate than using ADM alone. Bell-shaped solitary solution of Belov–Chaltikian (BC) lattice and kink-shaped solitary solution of the nonlinear self-dual network equations (SDNEs) are presented. Comparisons are made between approximate solutions and exact solutions to illustrate the validity and the great potential of the technique.

On Study of Solutions of Kac-van Moerbeke Lattice and Self-dual NetworkEquations

The closed form of solutions of Kac-van Moerbeke lattice andself-dual network equations are considered by proposingtransformations based on Riccati equation, using symboliccomputation. In contrast to the numerical computation oftravelling wave solutions for differential differenceequations, our method obtains exact solutions which have physical relevance.