Non-travelling Wave Research Articles

An investigation on explicit exact non-traveling wave solutions of the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation

Abstract This paper is devoted to investigating new non-traveling wave solutions for the (3 + 1)-dimensional potential Yu–Toda–Sasa–Fukuyama (YTSF) equation. By using the generalized variable separation method and extended three-wave approach, the process of solving the (3 + 1)-dimensional potential-YTSF equation is simplified and the interactions of multiple waves are revealed. With the aid of Maple, we derive thirty-six types new exact explicit non-traveling wave solutions with a like-parabolic tail. The main characteristic of these solutions is that they contain three arbitrary functions, which greatly enrich the diversity of solutions. This characteristic shows the novelty of our work. In particular, selecting suitable arbitrary functions, we can obtain traveling solutions, such as kink-wave solutions, solitary-wave solutions, kinky breather-wave solutions, singular solutions and periodic solutions. Then, some dynamical phenomena are exhibited by 3D representation, providing the complicated structure of the non-traveling wave solutions for the (3 + 1) dimensional potential-YTSF equation and their physical interpretation. In addition, our findings improve and extend the existing literature on related topics.

The new kink type and non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

In this paper, the new solitary wave solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation are obtained by Lie group symmetry method and the extended homoclinic test approach. Firstly, the equation can be reduced to (1+1)-dimensional partial differential equation by Lie group symmetry, and corresponding bilinear forms of the equation are given by symmetry functions. Secondly, the extended homoclinic test approach is employed to obtain the new kink type and singular solitary wave solutions. In addition, some new traveling and non-traveling wave solutions with arbitrary functions and oscillating tail are investigated through the special transformations for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation.

New exact solutions of nontraveling wave and local excitation of dynamic behavior for GGKdV equation

For GGKdV equation, solutions of the compatible KdV equation are obtained by using CKdVE method, and Lie point symmetry group of the equation is also obtained. Further, some new exact non-traveling wave solutions are obtained by using the equivalent transformation method and elliptic function method on solving the corresponding symmetric reduction equation, and local excitation modes of three kinds of solutions under three different groups of parameters are presented. Finally, the integrability in the sense of the CkdVE and the Lie Symmetric are proved, which shows the effectiveness of the organic combination of various kinds of nonlinear analytical methods. This CkdVE method communicated the mathematical relations of different nonlinear models. It is a new bridge between the known and unknown solutions of the nonlinear partial differential equations, and it is a new way to explore complex nonlinear complex phenomena.

Use of optimal subalgebra for the analysis of Lie symmetry, symmetry reductions, invariant solutions and conservation laws of the (3 + 1)-dimensional extended Sakovich equation

This paper investigates the [Formula: see text]-dimensional extended Sakovich equation, which represents an essential nonlinear scientific model in the field of ocean physics. The Lie symmetry analysis has been utilized for extracting the non-traveling wave solutions of the [Formula: see text]-dimensional extended Sakovich equation. These solutions are investigated through infinitesimal generators, which are obtained from Lie’s continuous group of transformations. As there are infinite possibilities for the linear combination of infinitesimal generators, so a one-dimensional optimal system of subalgebra has been established using Olver’s standard approach. Moreover, by considering the optimal system of subalgebra, the extended Sakovich equation is converted into a solvable nonlinear PDE through symmetry reductions. Finally, the conservation laws for the governing equation have been derived using Ibragimov’s generalized theorem and quasi-self-adjointness condition.

A New Composite Technique to Obtain Non-traveling Wave Solutions of the (2+1)-dimensional Extended Variable Coefficients Bogoyavlenskii–Kadomtsev–Petviashvili Equation

A New Composite Technique to Obtain Non-traveling Wave Solutions of the (2+1)-dimensional Extended Variable Coefficients Bogoyavlenskii–Kadomtsev–Petviashvili Equation

Further results about the non-traveling wave exact solutions of nonlinear Burgers equation with variable coefficients

Nonlinear Burgers equation with variable coefficients (BEVC) which involves mathematical physics in plasma and fluid dynamics and so on. Investigating the exact solutions of BEVC to show the different dynamics of wave phenomena becomes a hot topic both on many mathematicians and physicists. In this paper, by (G′G2)-expansion and the Jacobian elliptic functions (JEFs) two different methods, we found that various forms for exact wave solutions of BEVC. To our best knowledge, we found a variety of new solutions that have not been studied in previous articles such as u13,u14,u511,u512,u513,u514. The most important thing is that there are double Jacobian elliptic functions ideas in finding solution process, which has not been seen before in seeking for nonlinear BEVC. These new exact soliton solutions contain variable coefficients derived in the form of trigonometric function, rational function, and Jacobian elliptic function, hyperbolic function. The obtained results showed many different types such as annihilation, parabolic kink, curved shaped kink, tine shaped, shock solitons and so on. Furthermore, the above obtained solutions for Burgers equation with variable coefficients are different from Mohanty et al. (2022), Zayed and Abdelaziz (2010).

Bright and dark solitons for the fifth-order nonlinear Schrödinger equation with variable coefficients

In this work, a fifth-order nonlinear Schrödinger equation with variable coefficients is investigated, which represents the diffusion of ultrashort pulses in incompatible optical fibers. With the aid of two direct functions, some non-traveling wave bright and dark soliton solutions are obtained. Via the G′/G-expansion method, the non-traveling wave hyperbolic-type and trigonometric-type solutions are studied. Finally, we show the dynamic properties and characteristics of these derived results by using some three-dimensional figures and contour figures.

Exact Solutions and Non-Traveling Wave Solutions of the (2+1)-Dimensional Boussinesq Equation

By the extended (G′G) method and the improved tanh function method, the exact solutions of the (2+1) dimensional Boussinesq equation are studied. Firstly, with the help of the solutions of the nonlinear ordinary differential equation, we obtain the new traveling wave exact solutions of the equation by the homogeneous equilibrium principle and the extended (G′G) method. Secondly, by constructing the new ansatz solutions and applying the improved tanh function method, many non-traveling wave exact solutions of the equation are given. The solutions mainly include hyperbolic, trigonometric and rational functions, which reflect different types of solutions for nonlinear waves. Finally, we discuss the effects of these solutions on the formation of rogue waves according to the numerical simulation.

Abundant explicit non-traveling wave solutions for the (2+1)-dimensional breaking soliton equation

By combining the generalized variable separation method with the extended homoclinic test approach (EHTA) explicit exact non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are constructed. With the aid of symbolic computation, a series of new non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are expressed explicitly. These non-traveling wave solutions are new solutions with three arbitrary functions, which have a more general form than that in the previous literatures. The result obtained here reveals the complex structure of the solutions of the (2+1)-dimensional breaking soliton equation. The previous results obtained in literatures can be regarded as special cases here. When some arbitrary functions included in these solutions are taken as some special functions, exact periodic solitary wave, cross soliton-like wave, periodic cross-kink wave, periodic two-solitary wave are presented.

Study on double-periodic soliton and non-traveling wave solutions of integrable systems with variable coefficients

An improved test function method for finding double-periodic soliton and non-traveling wave solutions of integrable systems with variable coefficients is proposed and applied to solve the variable-coefficient Kadomtsev–Petviashvili equation, which describes waves in ferromagnetic media and matter-wave pulses in Bose–Einstein condensates.

New non-traveling wave solutions for the (2+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equation

In the paper, we mainly study the (2+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equation (VC-DJKM equation) by combining the extended homoclinic test approach and the generalized variable separation method. Applying this thought and with the aid of symbolic computation, we present thirty six kinds of new exact non-traveling wave solutions of the (2+1)-dimensional VC-DJKM equation including kink-like solution, singular solitary wave-like solution, periodic solitary wave-like solution, kinky breather wave-like solution, and so on. These results all have a like-parabolic tail which reveals the complex structure of solution and maybe give a prediction of physical phenomenon. The generalized variable separation method greatly enriches the types and structures of solutions. Moreover, when some functions in the generalized variable separation form take 0 or 1, it will degenerate into the variable separation form of multiplication or addition. As the special case of VC-DJKM equation, the corresponding results of the (2+1)-dimensional DJKM equation with constant coefficients are also given.

Propagation and interaction between special fractional soliton and soliton molecules in the inhomogeneous fiber

IntroductionFractional nonlinear models have been widely used in the research of nonlinear science. A fractional nonlinear Schrödinger equation with distributed coefficients is considered to describe the propagation of pi-second pulses in inhomogeneous fiber systems. However, soliton molecules based on the fractional nonlinear Schrödinger equation are hardly reported although many fractional soliton structures have been studied. ObjectivesThis paper discusses the propagation and interaction between special fractional soliton and soliton molecules based on analytical solutions of a fractional nonlinear Schrödinger equation. MethodsTwo analytical methods, including the variable-coefficient fractional mapping method and Hirota method with the modified Riemann–Liouville fractional derivative rule, are used to obtain analytical non-travelling wave solutions and multi-soliton approximate solutions. ResultsAnalytical non-travelling wave solutions and multi-soliton approximate solutions are derived. The form conditions of soliton molecules are given, and the dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed in the periodic inhomogeneous fiber and the exponential dispersion decreasing fiber. ConclusionAnalytical chirp-free and chirped non-traveling wave solutions and multi-soliton approximate solutions including soliton molecules are obtained. Based on these solutions, dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed. These theoretical studies are of great help to understand the propagation of optical pulses in fibers.

New non-traveling wave solutions for (3+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equation

<abstract><p>In this paper, we investigate non-traveling wave solutions of the (3+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa (VC-DJKM) equation, which describes the real physical phenomena owing to the inhomogeneities of media. By combining the extended homoclinic test approach with variable separation method, we obtain abundant new exact non-traveling wave solutions of the (3+1)-dimensional VC-DJKM equation. These results with a parabolic tail or linear tail reveal the complex structure of the solutions for (3+1)-dimensional VC-DJKM equation. Moreover, the tail in these solutions maybe give a prediction of physical phenomenon. When arbitrary functions contained in these non-traveling wave solutions are taken as some special functions, we can get the kink-type solitons, singular solitary wave solutions, and periodic solitary wave solutions, and so on. As the special cases of our work, the corresponding results of (3+1)-dimensional DJKM equation, (2+1)-dimensional DJKM equation, (2+1)-dimensional VC-DJKM equation are also given.</p></abstract>

ABUNDANT NEW NON-TRAVELING WAVE SOLUTIONS FOR THE (3+1)-DIMENSIONAL BOITI-LEON-MANNA-PEMPINELLI EQUATION

<abstract> Seeking exact solutions of higher-dimensional nonlinear partial differential equations has recently received tremendous attention in mathematics and physics. In this paper, we investigate exact solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation which describes nonlinear wave propagation in incompressible fluid. Firstly, by means of extended homoclinic test approach, we get eight kinds of non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Then, combining the improved tanh function method and new ansatz solutions, we obtain abundant new exact non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. These results include not only many results obtained in other literatures, but also some new exact non-traveling wave solutions. Moreover, the exact kink wave solutions, periodic solitary wave solutions and singular solitary wave solutions are given when arbitrary functions contained in these solutions are taken as some special functions. </abstract>

Dynamics of localized wave solutions for the coupled Higgs field equation

By virtue of Hirota bilinear form of the coupled Higgs field equation, some higher-order rogue wave, the Ma breather, the Akhmediev breather and the general breather solutions are constructed through symbolic computation. With the help of the contour line method, we investigate the localization characters of the first-order rogue wave, which is different from the case of its in nonlinear Schrodinger equation. The Ma breather and the Akhmediev breather solutions of the coupled Higgs field equation possess time-periodic and space-periodic, respectively. The results of this paper show us the generation and evolutions of novel nontraveling wave with several interesting structures.

Abundant new non-travelling wave solutions for the (3+1)-dimensional potential-YTSF equation

By combining the extended homoclinic test approach (EHTA) with the method of separation of variables explicit exact non-traveling wave solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation is constructed. With the aid of symbolic computation, abundant new explicit exact solutions of the (3+1)-dimensional potential-YTSF equation are obtained. The results obtained her e reveal the complex structure of the solutions of the (3+1) dimensional potential-YTSF equation. Many of the previous results obtained in other literature can be seen as special cases here. When some arbitrary functions included in these solutions are taken as some special functions, exact periodic solitary wave, periodic soliton, periodic cross-kink wave, periodic two-solitary wave are presented.

Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients

The Lie point symmetries are reported by performing the Lie symmetry analysis to the Ablowitz-Kaup-Newell-Suger (AKNS) equation with time-dependent coefficients. In addition, the optimal system of one-dimensional subalgebras is constructed. Based on this optimal system, several categories of similarity reduction and some new invariant solutions for the equation are obtained, which include power series solutions and travelling and non-traveling wave solutions.

Double-wave solutions and Lie symmetry analysis to the (2 + 1)-dimensional coupled Burgers equations

This paper investigates the (2 + 1)-dimensional coupled Burgers equations (CBEs) which is an important nonlinear physical model. In this respect, by making use of the generalized unified method (GUM), a series of double-wave solutions of the (2 + 1)-dimensional coupled Burgers equations are derived. The Lie symmetry technique (LST) is also utilized for the symmetry reductions of the (2 + 1)-dimensional coupled Burgers equations and extracting a non-traveling wave solution. Through some figures, we discussed the wave structures of the double-wave solutions of the CBEs for different values of parameters in these solutions.

MT–HVdc Systems Fault Classification and Location Methods Based on Traveling and Non-Traveling Waves—A Comprehensive Review

Estimation of fault classification and location in a multi-terminal high voltage direct current (MT–HVdc) transmission system is a challenging problem and is considered to be a fundamental maneuver of dc grid protection. This research paper critically reviews traveling and non-travelling wave methods of classification and location of dc faults in multi-terminal HVdc transmission systems. Detailed mathematical analysis of MT–HVdc systems composed of high grounding resistance, cable and overhead line segments, and bipolar coupled transmission network under healthy and faulty conditions, are evaluated. The gravity of this research paper addresses benefits and shortcomings of traveling and non-traveling wave methods and futuristic techniques of fault classification and location.

Low-Power (1.5 pJ/b) Silicon Integrated 106 Gb/s PAM-4 Optical Transmitter

As next-generation data center optical interconnects aim for 0.8 Tb/s or 1.6 Tb/s, serial rates up to 106 GBd are expected. However doubling the bandwidth of current 53 GBd 4-level pulse-amplitude modulation (PAM-4) transmitters is very challenging, and perhaps unfeasible with compact, non-travelling wave, lumped modulators or lasers. In our previous work, we presented an integrated 4:1 optical serializer with electro-absorption modulators (EAMs) in each path. Transmitter (TX) functionality was shown up to 104 GBd non-return-to-zero (NRZ) On-Off Keying (OOK) or PAM-4. However the performance for PAM-4 was limited by the distortion introduced by the EAM non-linearity. We also presented a real-time, DSP-free 128 Gb/s PAM-4 link with a silicon photonic transmitter using binary driven EAMs in a Mach-Zehnder interferometer (MZI) configuration. By combining two of such half-rate (53 GBd) transmitters in an integrated 2:1-serializer, improved 106 GBd PAM-4 performance is expected without needing to compensate the inherent modulator non-linearity and without requiring faster modulators or drivers. In this paper, we present a Silicon integrated 53 GBd PAM-4 TX as a candidate for integration into 106 GBd PAM-4 2:1 serialized TX. The presented TX consists of two EAMs in an MZI configuration, wirebonded to a low-power 55 nm 4-channel SiGe BiCMOS driver, operating at 1.5 pJ/b (excluding laser). With a reference receiver (RX), transmission at or below the KP4-FEC threshold is shown beyond 1km standard single-mode fiber (SSMF) and up to 2 km non-zero dispersion-shifted fiber (NZ-DSF) at 1565 nm. Furthermore, the integrated TX was combined with an Si-integrated RX consisting of the same EAM component, wirebonded to a 55 nm SiGe BiCMOS transimpedance amplifier (TIA). Both TX and RX were wirebonded on an RF-PCB, with electrical connectivity through transmission lines and 6-inch 50 GHz multi-coax cable connectors. With this electrically connectorized all-EAM TX and RX, PAM-4 link operation is shown up to 40 GBd at 3.9 pJ/b (excluding laser), without using DSP or equalization.