Bilinear Method Research Articles

Multiple and Singular Soliton Solutions for Space–Time Fractional Coupled Modified Korteweg–De Vries Equations

The focus of this paper is on the nonlinear coupled evolution equations, specifically within the context of the fractional coupled modified Korteweg–de Vries (mKdV) equation, employing the conformable fractional derivative (CFD) approach. The primary objective of this paper is to thoroughly investigate the applicability of the Hirota bilinear method for deriving analytical solutions to the fractional mKdV equations. A range of exact analytical solutions for the fractional coupled mKdV equations is obtained. The findings in general indicate that the Hirota bilinear method is an effective approach for resolving the complexities associated with the fractional coupled mKdV equations.

Periodic line wave, rogue waves and the interaction solutions of the (2+1)-dimensional integrable Kadomtsev–Petviashvili-based system

This paper mainly investigates multiple types of solutions for the (2+1)-dimensional integrable Kadomtsev–Petviashvili-based system, which can describe the complex nonlinear wave phenomena observed in many physical systems, such as fluid mechanics, plasma physics and ocean dynamics. The Hirota’s bilinear method and the perturbation expansion skill are used to derive the periodic line wave solution and the interaction solution composed of the breather and the periodic line wave. By choosing appropriate parameters and employing the long wave limit of the soliton solution, two kinds of elementary rogue waves (RWs) are generated, which are kink-shaped line RW and W-shaped line RW. The interaction solutions among a lump, two lumps and two kinds of line RWs are obtained. Furthermore, the semi-rational solutions of the KP-based system are yielded, which include six types, namely (1) a line RW on a line soliton background, (2) a line RW on two line solitons background, (3) the line RW on the breather background, (4) a lump on the background with the periodic line wave, (5) the line RW on the background of the lump and the breather and (6) two lumps on the background with the periodic line wave. An effective analytical method related to the characteristic lines is presented to analyze the dynamical behaviors of the rogue waves and interaction waves. The method can be further extended to investigate other complex wave structures for the high-dimensional nonlinear integrable equations.

The bilinear neural network method for solving Benney–Luke equation

Benney–Luke equation, the estimation of water wave propagation on the water’s surface, is significantly important in studying the tension of water waves in physics. This paper focuses on the Bilinear Neural Network method (BNNM) to find the solutions to the Benney–Luke equation. Using the Hirota bilinear operator, the Benney–Luke equation is transformed into a bilinear expression and establishes models of neurons. The method construction is straightforward, well-organized, and effective for finding various exact analytic solutions of the Benny–Luke equation consisting of one-soliton solutions, two-soliton solutions, one-soliton hyperbolic wave solutions, two-soliton hyperbolic wave solutions, mixed-soliton hyperbolic solutions, and mixed-multi hyperbolic solutions by choosing the appropriate test functions. The solutions are rigorously depicted in three dimensions to observe the mutable behavior.

Soliton molecules, breather molecules and stability analysis of the three-coupled nonlinear Schrödinger equation

We investigate the three-coupled nonlinear Schrödinger (NLS) equation, which holds a significant physical implications in optical communications and biophysics. The multi-soliton solutions are derived utilizing the Hirota bilinear method. Multi-breather solutions are presented by assuming certain relationship for the parameters of the multi-solitons. By introducing the velocity resonance mechanism to multiple soliton/breather solutions, it is found that the three-coupled NLS model possesses abundant soliton molecule, breather molecule and breather-soliton molecule structures. The interactions among different types of soliton molecules which can be observed in marine and oceanic waters are investigated through numerical simulation. Ultimately, baseband modulation instability of the three-coupled NLS equation is analyzed. It is highly significance for the study of soliton formation and rogue waves in the three-coupled NLS equation.

Wronskian solutions, bilinear Bäcklund transformation, quasi-periodic waves and asymptotic behaviors for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

In this paper, we investigate the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is widely used in fluid mechanics and theoretical physics. The N-soliton solution is obtained via the Hirota’s bilinear method. The Wronskian solution is derived by using the Wronskian technique for the bilinear form. Through the exchange formula, we deduce the bilinear Bäcklund transformation consisting of four equations and six parameters. In order to consider the quasi-periodic wave having complex structure, one-, two- and three-periodic waves are investigated systemically by combining the Hirota’s bilinear method with Riemann theta function. Furthermore, the corresponding graphs of periodic wave are presented by considering the geometric properties between the characteristic lines. The propagation characteristics of periodic waves are investigated by virtue of the characteristic lines. Finally, the asymptotic relationships between quasi-periodic wave solutions and soliton solutions are established theoretically under a condition of the small amplitude limit. The analytical method used in this paper can be applied in other integrable systems.

A fourth-order nonlinear equation studied by using a multivariate bilinear neural network method

A fourth-order nonlinear equation studied by using a multivariate bilinear neural network method

Instability analysis and face‐to‐face collision of dust acoustic waves in dusty plasmas: Perspective Jupiter environment

AbstractThe collisionless unmagnetized five components dusty plasmas have been considered to study the significance of the face‐to‐face collision of single‐ and multi‐solitons, phase shifts due to face‐to‐face collision, and instability of the dust acoustic waves in Jupiter's environment. To study the consequence of face‐to‐face collision extended Poincaré–Lighthill–Kuo method is used to derive the two‐sided nonlinear Korteweg–De Vries (KdV) equation. Besides, to investigate the dust acoustic wave's instability, the nonlinear Schrödinger equation has been derived following the procedure of Slathia et al. The multi‐soliton solutions are determined by employing the Hirota bilinear method. The concerned parameters in the issue play a crucial role in the phase shift (generated by the face‐to‐face collision); the collisional process of single‐, double‐, triple‐, and quadruple‐soliton; and the generation of dust acoustic solitons. Furthermore, the plasma parameters are crucial in the formation of dark solitons and envelop shocks.

Geometric realization of the Sasa-Satsuma equation on the symmetric space SU(3)/U(2)

The analytic property of the Sasa-Satsuma equation has been well-explored via using an array of mathematical tools (such as the inverse scattering transformation, the Hirota bilinear method and the Darboux transformation). This paper devotes to exploring geometric properties of this equation via the zero curvature representation in terms of the language in Yang-Mills theory. The generalized Landau-Lifshitz type model of Sym-Pohlmeyer moving curves evolving in the symmetric Lie algebra g=k⊕m with initial data being suitably restricted is gauge equivalent to the Sasa-Satsuma equation. This gives a geometric realization of the Sasa-Satsuma equation.

Superposition Formula of arbitrary functions to a (3+1)-demensional Boiti–Leon–Manna–Pempinelli equation

The (3+1)-dimensional Boiti–Leonon–Manna–Pempinelli equation was studied by using Hirota bilinear method. In this paper, we successfully derived the multi-solutions of this equation consisting of the superpositions of M arbitrary functions. A variety of new solutions can be obtained by choosing different values of the parameter M and test functions f in this paper, i.e., the localized solutions generated by M-positive quadratic functions, the mixed solutions consisting of the localized wave solutions and the kink wave solutions, and the mixed solutions consisting of a trigonometric function and exponential functions. Meanwhile, their corresponding physical plots were displayed to analyze their physical significance and dynamic properties. Some solutions from previous studies were found when M took a specific value and the test functions took particular functions. Thus, the received solutions are more general compared to the previous studies.

Research on vibration fatigue behavior of blade structures based on infrared thermography

In recent years, infrared thermography has been widely used in mechanical testing because of its non-contact, rapidity and high accuracy. In order to explore the advantages of infrared thermography in vibration fatigue monitoring of blade structures, the characterization of the vibration fatigue behavior of blade structures is carried out based on infrared thermography with the validation and assistance of laser vibrometry. Firstly, the numerical simulation and experimental analysis were used to determine the size and location of the maximum stress in blade structures. Afterwards, the linear relationship between the stress and amplitude(σν-A curve) of blade structures was calibrated by laser vibrometry, and thus the S-N curve was fitted. Eventually, the heat dissipation law and crack extension process for vibration fatigue of blade structures was analyzed by infrared thermography, and the fatigue limit of the blade structure was predicted by combining infrared thermography and the Luong bilinear method. It is shown that the heat dissipation law and crack extension process for vibration fatigue of blade structures can be effectively analyzed by infrared thermography. Infrared thermography saves some economic and time costs for fatigue limit prediction, and the method is within 2.6% of the results predicted by the Basquin model. This paper provides a certain reference basis for the research methodology for vibration fatigue behavior of blade structures.

Solitons, nonlinear wave transitions and characteristics of quasi-periodic waves for a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation in fluid mechanics and plasma physics

A (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation describing many nonlinear phenomena in fluid dynamics and plasma physics is considered. The N-solitons and breathers are obtained by basing on its Hirota’s bilinear form and taking the complex conjugate condition on parameters of N-solitons. What is more, breathers can be transformed into a series of nonlinear localized waves by the mechanism of breather transformation. Then through the multi-dimensional Riemann-theta function and the bilinear method, the high-dimensional complex three-periodic wave solutions are constructed systematically, which are the generalization of one-periodic wave and two-periodic wave solutions. By a limiting procedure, the asymptotic relations between the quasi-periodic waves and solitons are strictly established. Additionally, a novel analytical method of characteristic line is introduced to analyze statistically the dynamical characteristics of the quasi-periodic waves. The analytical method employed in this paper can be further extended to investigate the other complex high-dimensional nonlinear integrable equations.

Solving the variable coefficient nonlinear partial differential equations based on the bilinear residual network method

Solving the variable coefficient nonlinear partial differential equations based on the bilinear residual network method

Multimodal Features Alignment for Vision–Language Object Tracking

Vision–language tracking presents a crucial challenge in multimodal object tracking. Integrating language features and visual features can enhance target localization and improve the stability and accuracy of the tracking process. However, most existing fusion models in vision–language trackers simply concatenate visual and linguistic features without considering their semantic relationships. Such methods fail to distinguish the target’s appearance features from the background, particularly when the target changes dramatically. To address these limitations, we introduce an innovative technique known as multimodal features alignment (MFA) for vision–language tracking. In contrast to basic concatenation methods, our approach employs a factorized bilinear pooling method that conducts squeezing and expanding operations to create a unified feature representation from visual and linguistic features. Moreover, we integrate the co-attention mechanism twice to derive varied weights for the search region, ensuring that higher weights are placed on the aligned visual and linguistic features. Subsequently, the fused feature map with diverse distributed weights serves as the search region during the tracking phase, facilitating anchor-free grounding to predict the target’s location. Extensive experiments are conducted on multiple public datasets, and our proposed tracker obtains a success score of 0.654/0.553/0.447 and a precision score of 0.872/0.556/0.513 on OTB-LANG/LaSOT/TNL2K. These results are satisfying compared with those of recent state-of-the-art vision–language trackers.

Lump–kink and hybrid solutions of the extended (3+1)-dimensional potential KP equation in fluid mechanics

In this paper, the extended (3+1)-dimensional potential KP equation in fluid mechanics is studied through Hirota bilinear method. Many types of hybrid solutions, such as the lump–kink solution, lump-two kink solution and periodic lump solution are obtained by assuming different functions in the bilinear equation. The interaction solution between lump and triangular periodic wave is also derived by combining sine and cosine functions with quadratic functions. Dynamical structures of these exact solutions are depicted by presenting the corresponding three-dimensional, two-dimensional structures and density graphs. These diverse interaction solutions could be helpful for understanding physical phenomena in fluid mechanics.

Interaction solutions of (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation via bilinear method

Using the bilinear neural network method (BNNM) and the symbolic computation system Mathematica, this paper explains how to find an exact solution for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation. In terms of activation function and weight coefficient, BNNM is a more appealing option for users than traditional symbolic computation methods. It is possible to develop a wide range of solutions and expand the classes of exact solutions by modifying the activation function. The activation function’s versatility allows it to generate a wide range of solutions with several theoretical and practical uses. The analytical solution is obtained by using a double layer type, while the rogue wave solution and mixed solutions are obtained by using a single layer type. The evolution of these waves is then illustrated using various 3D graphs, 2D graphs, and density plots.

Interaction of ion acoustic multi-solitons in a magnetised superthermal plasma in the presence of dust

In this investigation, a magnetised plasma consisting of inertial ions fluid, charged immobile dust grains in the background and superthermally distributed electrons is considered to study head-on collision between dust ion acoustic multi-solitons. The extended Poincaré–Lighthill–Kuo (PLK) method is employed for the derivation of two KdV equations. Further, in order to determine the multi-solitons solutions of the two KdV equations, the Hirota bilinear method is used. The phase shifts due to head-on collision of multi-solitons have also been determined. The combined influence of superthermality of electrons, density ratio, magnetic field strength etc. on the characteristics of dust ion acoustic solitons, time evolution and the corresponding phase shifts of multi-solitons has been studied numerically. The findings of this investigation might be of great importance to elucidate and provide a comprehensive framework for the understanding of various nonlinear structures in space and astrophysical environments especially in Saturn’s magnetosphere.

FVCNet: Detection obstacle method based on feature visual clustering network in power line inspection

AbstractPower line inspection is an important means to eliminate hidden dangers of power lines. It is a difficult research problem how to solve the low accuracy of power line inspection based on deep neural network (DNN) due to the problems of multi‐view‐shape, small‐size object. In this paper, an automatic detection model based on Feature visual clustering network (FVCNet) for power line inspection is established. First, an unsupervised clustering method for power line inspection is proposed, and applied to construct a detection model which can recognize multi‐view‐shape objects and enhanced object features. Then, the bilinear interpolation method is used to Feature enhancement method, and the enhanced high‐level semantics and low‐level semantics are fused to solve the problems of small object size and single sample. In this paper, FVCNet is applied to the MS‐COCO 2017 data set and self‐made power line inspection data set, and the test accuracy is increased to 61.2% and 82.0%, respectively. Compared with other models, especially for the categories that are greatly affected by multi‐view‐shape, the test accuracy has been improved significantly.

The periodic soliton solutions for a nonlocal nonlinear Schrödinger equation with higher-order dispersion

A nonlocal nonlinear Schr¨odinger (NNLS) equation with fourth-order dispersion and cubic-quintic nonlinearities has been studied analytically and numeri- cally. Under the constraint conditions, auxiliary functions are introduced, and explicit one- and two-soliton solutions are obtained by the Hirota bilinear method. Accord- ing to the solutions, the propagation dynamics of soliton pulses are investigated. The influences of different parameters on the dynamics of one- and two-soliton solutions have been analyzed. The results show that the two-soliton solution exhibits diverse dy- namic characteristics under the suitable parameter selections. In addition, the stability of one- and two-soliton solutions against the constraint conditions deviations and under the initial perturbations are also studied numerically.

Various nonlinear characteristics of breather/rogue waves and controllable interaction phenomena for a new KdV equation with variable coeffcients

In this paper, we investigate and analyze various nonlinear phenomena of a new (2+1)-dimensional KdV equation with variable coefficients, and successfully obtain breather/rogue wave solutions and interaction solutions of the KdV equation by using the bilinear neural network method and symmetry transformation. Subsequently, we analyze the dynamical characteristics and evolution process of these obtained solutions through the 3-D animations, and find a series of interesting nonlinear phenomena concerning breather/rogue waves, such as fission, regeneration, annihilation, collision, and controllable interaction phenomena on nonzero backgrounds. This paper provides a more intuitive understanding for the nonlinear phenomena of these obtained solutions, and these nonlinear phenomena have potential application value in fluid dynamics, elastic mechanics and other fields of nonlinear science.

Novel interaction solutions to the (3+1)-dimensional Hirota bilinear equation by bilinear neural network method

Solving differential equations is an ancient and very important research topic in theory and practice. The exact analytical solution to differential equations can describe various physical phenomena such as vibration and propagation wave. In this paper, the bilinear neural network method (BNNM), which uses neural network to unify all kinds of classical test function methods, is employed to obtain some new exact analytical solutions of the ([Formula: see text])-dimensional Hirota bilinear (HB) equation. Based on the Hirota form of the HB equation, we constructed four kinds of new solutions which contain the breather solution, rogue wave solution, breather lump-type soliton solution and the interaction solution between the periodic waves and two-kink wave by introducing a series of test functions in both single-layer and multi-layer neurons such as [4–2–2] and [4–2–3] neural network models. In addition, we compared them to those that had already been published. It is clear that our results are not consistent with those found in these publications. Their corresponding dynamic features are vividly demonstrated in some 3D, contour, x-curves and y-curves plots. The results obtained demonstrate the potential of the proposed methods to solve other nonlinear partial differential equations in fields.