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In this work, numerous rogue wave solution strategy (MRWSS) agreeing to the Hirota bilinear hypothesis is utilized. The said strategy to build different soliton wave solutions for the generalized Hirota-Satsuma-Ito (GHSI) condition is considered. These numerous soliton wave solutions incorporate first-order, second-order, third-order, and fourth-order waves solutions and so on, which are all depending on the starting theory of chosen capacities. For the lump solution with one most extreme or least esteem, the properties of the Hessian lattice are examined. Besides, the arrangements are contained the lump-two kink arrangements, cross-kink wave arrangements, periodic wave arrangements, and periodic-kink wave arrangements. The various classes of arrangements are gotten. We make utilize of the direct superposition procedure to find out N-soliton wave arrangements to the said condition. The appropriateness and viability of the obtained arrangements is detailed through the reenactment comes about within the shape of 3-D, density, and 2-D charts. The gotten comes about in this work are anticipated to open modern viewpoints for the traveling wave theory. [ABSTRACT FROM AUTHOR]

Abstract In this work, numerous rogue wave solution strategy (MRWSS) agreeing to the Hirota bilinear hypothesis is utilized. The said strategy to build different soliton wave solutions for the generalized Hirota-Satsuma-Ito (GHSI) condition is considered. These numerous soliton wave solutions incorporate first-order, second-order, third-order, and fourth-order waves solutions and so on, which are all depending on the starting theory of chosen capacities. For the lump solution with one most extreme or least esteem, the properties of the Hessian lattice are examined. Besides, the arrangements are contained the lump-two kink arrangements, cross-kink wave arrangements, periodic wave arrangements, and periodic-kink wave arrangements. The various classes of arrangements are gotten. We make utilize of the direct superposition procedure to find out N-soliton wave arrangements to the said condition. The appropriateness and viability of the obtained arrangements is detailed through the reenactment comes about within the shape of 3-D, density, and 2-D charts. The gotten comes about in this work are anticipated to open modern viewpoints for the traveling wave theory.

Nonlinear science is a key area of scientific research dedicated to studying the fundamental properties and common features of nonlinear phenomena. This study specifically investigates the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation in nonlinear dispersive systems. By employing the Hirota bilinear method along with the linear superposition principle to the bilinear form of this equation, various new wave structures are effectively derived, such as N-solitons, resonance-type solitons, hybrid interaction solutions, and resonant multi-wave solutions. To ensure the physical compatibility of the results, we generated 3D and density plots using appropriate parametric values, which will provide effective insights into the obtained wave solutions. Moreover, we provide a modulation instability analysis of the KP-BBM equation with its stability region. The novelty of this work lies in the fact that the results discussed are new and have not been previously explored for the given equation. Additionally, the methods applied are efficient and reliable for constructing new soliton wave solutions in nonlinear physical phenomena. We predict that the findings of this study will have significant applications in optical fibers, plasma physics, engineering and fluid dynamics. [ABSTRACT FROM AUTHOR]

This research mainly concerned with some new solutions of the (3 + 1)‐dimensional nonlinear evolution equation (NEE). First, we extract the resonant multiple soliton solutions (RMSSs) by taking advantage of the linear superposition principle (LSP) and weight algorithm (WA). Then the non‐singular complexiton and singular complexiton solutions are developed by introducing pairs of the conjugate parameters. Besides, the complex multiple‐soliton solutions are also explored with the aid of the bilinear approach. The graph descriptions of the attained solutions are paraded to show the dynamical properties. The outcomes of this work are all new and are desirous to bring some new perspective to the investigation of the complexiton solutions to the other partial differential equations (PDEs). [ABSTRACT FROM AUTHOR]

The main orientation of the current research is to look into the generalized (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation (BKPE) for the water waves. By exerting the Cole-Hopf transform, we extract its Hirota bilinear equation. First, the weight algorithm (WA) together with the linear superposition principle(LSP) is carried out to look for the resonant multiple soliton solutions (RMSSs). Two different types of the RMSSs are obtained by introducing the parameterization of the wave numbers and frequencies. Second, the N-soliton solutions (NSSs) are also explored by using Hirota bilinear equation. On this basis, the resonance conditions of the soliton molecules on the (x, y)-, (x, z)- and (y, z)-planes are extracted and the soliton molecules are found. Finally, the ansatz function scheme, together with the symbolic computation is manipulated to look into the interaction solutions (ISs). Two different interaction solutions of the sin-cosh type and cos-cosh type are developed. A comparison between the RMSSs and the N-soliton solutions are elaborated in detail. Additionally, the dynamics of the solutions are displayed graphically to expound the physical interpretation. The proposed methods in this work can be also employed to inquire into the similar exact solutions of the other PDEs. [ABSTRACT FROM AUTHOR]

Generally, the principle of linear superposition is not applicable in nonlinear systems. The results of our researches prove that for some certain types of decomposed solutions of (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada(CDGKS) equation, the linear superposition principle holds. First, beginning with the potential (2+1)-dimensional CDGKS equation, six sets of decomposition equations are formulated by the formally variable separation approach. On the basis of these decompositions, the linear superposition solutions are then discussed. More specifically, the combination of two periodic wave solutions, soliton and periodic wave interaction solution and the multi-soliton solution are obtained.

This work aims to construct some novel exact solutions of the (2 + 1)-dimensional Sawada-Kotera equation (SKE). First, the resonant multiple soliton solutions (RMSSs) are discussed by employing the weight algorithm (WA) and linear superposition principle (LSP). Then, based on the RMSSs, the non-singular complexiton and singular complexiton solutions are developed by taking pairs of the conjugate parameters. Finally, the breather wave solutions (BWSs) of the sin-cosh type are explored through the symbolic computation and test function method. The dynamics of the corresponding extracted solutions are displayed graphically for showing the physical properties. The achievements of this article can provide some new insights to the (2 + 1)-dimensional SKE.

This paper is concerned with the new extended (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation (BLMPE), which is regarded as an extension of the well-known (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation and acts a key role in the incompressible fluid. By means of the Cole-Hopf transform, the Hirota bilinear equation of the studied equation is developed. Then based on the Hirota bilinear equation, we give a detailed investigation into the new exact solutions of the BLMPE. Firstly, the linear superposition principle combined with the weight algorithm is utilized to construct the resonant multiple wave solutions by introducing the parameterization for the wave numbers and frequencies. Secondly, the new homoclinic approach is applied to seek for the periodic wave solutions. Finally, the symbolic computation with the ansatz function scheme is adopted to construct the interaction solutions. The 3-D plots and 2-D contours are figured out to reveal the nonlinear dynamic behaviors of the different solutions. To the best of our knowledge, the results obtained in this work are all new and have not been reported elsewhere yet. The attained solutions can help us understand the nonlinear dynamic behaviors of the new extended (3 + 1)-dimensional BLMPE better. What's more, the proposed methods in this work can be also adopted to study the resonant multiple wave, periodic wave and interaction solutions of the other PDEs arising in physics. [ABSTRACT FROM AUTHOR]

Based on the Hirota bilinear form of a (2+1)-dimensional equation, breathers and resonant multiple waves as well as complexiton solutions are considered in this paper. First, the breather waves are constructed via employing the extend homoclinic test method. By calculation, two kinds of solutions are obtained. Through analysis, three pairs of breathers consisting of hyperbolic functions and trigonometric functions are derived. Furthermore, a rouge wave solution is deduced by applying the Taylor expansion method to a obtained breather wave. In addition, related figures are plotted to illustrate the dynamical features of these obtained solutions. Then, two types of the resonant multi-soliton solutions are obtained by applying the linear superposition principle to the the Hirota bilinear form. At the same time, 3D profiles and 2D density plots are presented to depict the intersection progression of wave motion. Finally, the complexiton solutions are constructed according to the yielded resonant multi-soliton solutions by further utilizing the linear superposition principle. By considering different domain fields, several types of complexiton solutions including the positive ones are derived. Moreover, related 3D and 2D figures are plotted for the obtained results in order to vividly exhibit their dynamics properties.

This paper focuses on some novel exact solutions of the (3 + 1)-dimensional Kudryashov-Sinelshchikov equation (KSE). Based on the linear superposition principle (LSP) and weight algorithm (WA), we construct the complexiton solutions by introducing the pairs of the conjugate parameters. Besides, the Y-type soliton solutions are extracted by imposing the resonant conditions on the N-soliton solutions. Finally, two different types of the interaction solutions (ISs) are also explored by exerting the test function method and symbolic computation. The graphical representations of the corresponding solutions are presented to reveal the dynamical characteristics. To the best of our knowledge, the attained outcomes of this article are brand new and have never been reported in other literature, and can enable us make sense of the physical behaviors of the (3 + 1)-dimensional KSE better.

Copyright of Engineering Mechanics / Gongcheng Lixue is the property of Engineering Mechanics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this article, various exact solutions of the fifth-order variable coefficient KdV equation with higher-order dispersion term are studied. Because of the complexity of the exact solution of the variable coefficient t, it has a certain influence on the tension waves at the fluid interface on the gravity surface. First, the bilinear KdV equation is derived by using the Hirota bilinear method, and four mixed solutions consisting of positive quartic function, quadratic function, exponential function, and hyperbolic function are constructed. Second, the linear superposition principle is used to obtain the resonance multisoliton solution, and two cases are taken as examples to illustrate the study of resonance multi soliton solution. In addition, 3D images and contour images are drawn by mathematical symbol calculation and appropriate parameters, and the process of tension fluctuation is vividly explained by physical phenomena. The results obtained greatly expand the exact solution of the KdV equation in the existing literature and enable us to understand nonlinear dynamical systems more deeply.

Let T be a linear tree, and let S (T) denote the set of real symmetric matrices whose graph is T. If U (T) is the minimum number of eigenvalues with multiplicity 1 among matrices in S (T) and T ′ is a linear tree resulting from the addition of one vertex to T , we show that | U (T ′) − U (T) | ⩽ 1. We also determine the exact set of possible values of U (T) − U (T ′) , depending upon the manner in which the vertex is added to T to get T ′. These results are then used to give a new bound for U (T) , the diameter bound, and to improve an existing bound, 2 + D 2 (T). [ABSTRACT FROM AUTHOR]

Based on the Hirota bilinear form of a (2+1)-dimensional equation, breathers and resonant multiple waves as well as complexiton solutions are considered in this paper. First, the breather waves are constructed via employing the extend homoclinic test method. By calculation, two kinds of solutions are obtained. Through analysis, three pairs of breathers consisting of hyperbolic functions and trigonometric functions are derived. Furthermore, a rouge wave solution is deduced by applying the Taylor expansion method to a obtained breather wave. In addition, related figures are plotted to illustrate the dynamical features of these obtained solutions. Then, two types of the resonant multi-soliton solutions are obtained by applying the linear superposition principle to the the Hirota bilinear form. At the same time, 3D profiles and 2D density plots are presented to depict the intersection progression of wave motion. Finally, the complexiton solutions are constructed according to the yielded resonant multi-soliton solutions by further utilizing the linear superposition principle. By considering different domain fields, several types of complexiton solutions including the positive ones are derived. Moreover, related 3D and 2D figures are plotted for the obtained results in order to vividly exhibit their dynamics properties. [ABSTRACT FROM AUTHOR]

For a tree T, U (T) denotes the minimum number of eigenvalues of multiplicity 1 among all real symmetric matrices, whose graph is T. It is known that U (T) ≥ 2. A tree is linear if all its vertices of degree at least 3 lie on a single induced path, and k-linear if there are k of these high degree vertices. U (T) is known for generalized stars (1-linear trees), and U (T) is determined here for 2-linear trees. Three (and higher) linear trees offer considerable additional complications. [ABSTRACT FROM AUTHOR]

Optimal containment preview analysis and distributed design for multi-agent systems with general continuous-time linear dynamics and directed acyclic communication topology are considered. Firstly, the state augmentation technique and topology reconstruction method are applied to formulate the original problem as a set of internal-model based local optimal output regulation problem. Secondly, the linear superposition principle is employed to obtain the preview feed-forward compensations corresponding to multiple leaders. Thirdly, by expressing each preview compensation term as a form about the current state of the leader, the solvability of the local optimal output regulation problem, as well as the optimal regulation problems, are proved. On this basis, the sufficient conditions and the distributed optimal control strategy are derived, which ensure the convergence of the containment preview problem. Finally, the effectiveness of the distributed design is illustrated by a numerical experiment. [ABSTRACT FROM AUTHOR]

The multiple exp-function technique, as an extension to Hirota's perturbation algorithm, is employed to find N-soliton solutions of the 3D Vakhnenko–Parkes (3D-VP) equation. This is a straightforward and effective alternative method. Consequently, one- two- and three-soliton solutions besides the front wave solutions and their wave numbers are obtained, which are completely distinct from the other published literature. 3D and density plots were investigated via the help of the Maple package program by suitably choosing the free parameters in the solutions. Further, the physical meaning for these solutions is clarified. We mention that the propagation of high-frequency waves in a relaxing medium can be investigated by discussing the dynamical behavior of the obtained solutions for the 3D-VP equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized (2 + 1)-dimensional Hietarinta equation which is investigated by utilizing Hirota's bilinear method. Also, the bilinear form is obtained, and the N-soliton solutions are constructed. In addition, multi-wave and breather wave solutions of the addressed equation with specific coefficients are presented. Finally, under certain conditions, the asymptotic behavior of solutions is analyzed in both methods. Moreover, we employ the linear superposition principle to determine N -soliton wave solutions for the generalized (2 + 1)-dimensional Hietarinta equation. [ABSTRACT FROM AUTHOR]

The multiple exp‐function method and multiple rogue wave solutions method are employed for searching the multiple soliton solutions to the (3 + 1)‐dimensional Kadomtsev–Petviashvili–Boussinesq‐like (KP‐Boussinesq‐like) equation. The obtained solutions contain the first‐order, second‐order, third‐order, and fourth‐order wave solutions. At the critical point, the second‐order derivative and Hessian matrix for only one point are investigated, and the lump solution has one maximum value. Moreover, we employ the linear superposition principle to determine N‐soliton wave solutions of the KP‐Boussinesq‐like equation. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values. [ABSTRACT FROM AUTHOR]

In this article, we study the generalized (3+1)-dimensional nonlinear wave equation where is investigated in soliton theory and by employing the Hirota’s bilinear method the bilinear form is obtained, and the N-soliton solutions are constructed. In addition, some Lump solutions, Lump-kink solutions, Lump-two kink solutions, Lump-periodic solutions, its Interaction solutions, Cross-kink wave, Breather-type, Multi wave, Periodic wave solutions, and Solitary wave solutions of the addressed equation with specific coefficients are shown. Finally, under certain conditions the asymptotic behavior of the two-soliton solution is analyzed to prove that the collision of the two-soliton is elastic. Moreover, we employ the linear superposition principle to determine N-soliton wave solutions for the generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles.

In this work, we established some exact solutions for the (3 + 1)-dimensional potential-Yu-Toda-Sasa-Fukuyama (YTSF)-like equation with p=3 and p=5 which are considered based on the generalized Hirota bilinear method. Depending on the analysis of Hirota operator, a generalized bilinear differential equation of the YTSF-like equation type is formulated. Furthermore, through a specific computations with Maple software, M-soliton solutions, lump solution and three classes of interaction solutions of the YTSF-like equation expressed explicitly. Moreover, we employ the linear superposition principle (LSP) to determine N-soliton wave solutions of the (3 + 1)-dimensional YTSF-like equation. The studied equation describes the physical characterization of model for investigating the dynamics of solitons and nonlinear waves in fluid dynamics, plasma physics and weakly dispersive media. Moreover, a few key differences are presented, which exits in the literature and the current offer. The evaluated solutions are explained through various sketches in two, three-dimensional, density, and curve plots of their real, imaginary, and absolute value. The computational applied schemes’ performance is tested to illustrate their powerful, and effectiveness for handling many nonlinear evolutions equations.

In this study, the (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama equation arising from the (3 + 1)-dimensional Kadomtsev–Petviashvili equation is investigated in detail by using two powerful approaches. First, the generalized resonant multi-soliton solution is generated via the simplified linear superposition principle. Second, after applying the simplest equation method, the generalized single solitary solution is extracted. The results show that the obtained solutions are perfect. The physical explanation of the obtained solutions is depicted in various 3D and 2D figures, which are used to illustrate that the interactions of resonant multi-soliton waves are inelastic. Ultimately, the study reveals that the inelastic interactions can be determined by the sign of the wave related number k i . [ABSTRACT FROM AUTHOR]

In the present paper, taking advantage of the bilinear method, we attained N-soliton solutions of the (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (gCH-KP) equation. By combining the long wave limit method and the complex conjugate method to N-soliton solutions, M-lump wave solutions including one-lump wave, two-lump wave and three-lump wave and two types of hybrid solutions including hybrid solution between lump wave and solitons and between M-lump wave and soliton were obtained. In addition, several groups of images exhibited their dynamic structure and physical properties with physically interpreted via symbolic computation. Finally, resonant multi-soliton wave solutions were obtained by carrying out the linear superposition principle and the progresses of wave motions with physical interpretation represented graphically. Furthermore, the received results have immensely augmented the exact solutions of (3+1)-dimensional gCH-KP equation on the available literature and enabled us to understand the nonlinear dynamic system deeply. In addition, the proposed methods will be a strong boost to the calculation method of nonlinear equations. [ABSTRACT FROM AUTHOR]

Generally, the principle of linear superposition is not applicable in nonlinear systems. The results of our researches prove that for some certain types of decomposed solutions of (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada(CDGKS) equation, the linear superposition principle holds. First, beginning with the potential (2+1)-dimensional CDGKS equation, six sets of decomposition equations are formulated by the formally variable separation approach. On the basis of these decompositions, the linear superposition solutions are then discussed. More specifically, the combination of two periodic wave solutions, soliton and periodic wave interaction solution and the multi-soliton solution are obtained. • It is proven that the linear superposition principle is suitable for some certain types of decomposed solutions of (2+1)-dimensional CDGKS equation. • Six sets of decomposition equations of the potential (2+1)-dimensional CDGKS equation are formulated by the formally variable separation approach. • Some specific linear superposition solutions of the (2+1)-dimensional CDGKS equation are obtained. [ABSTRACT FROM AUTHOR]

Herein, an innovative approach for determining the grinding media system of ball mill based on grinding kinetics and linear superposition principle was proposed. The optimal media ball size of −2 + 0.45 mm, −0.45 + 0.15 mm, and − 0.15 mm feeding samples were 40 mm, 30 mm, and 30 mm, respectively. According to the linear superposition principle, the media size ratio is equal to the yield of ores. The optimal media proportion was φ 50 mm: φ 40 mm: φ 30 mm = 28%: 28%: 44% via wet grinding experiment, in which the yield of qualified granular material (−0.074 + 0.03 mm) suitable for beneficiation was the largest in the ground products. The optimized space-filling factor and material media ratio was 35% and 1.0 through the wet grinding experiment. According to the energy consumption method, the mathematical function models of installed power and mill productivity vs. space-filling factors were established to calculate the installed power and productivity of an industrial ball mill (Φ 5.03 m × 6.40 m). Unlabelled Image • An novel approach for determining grinding media system of ball mill was proposed. • The optimum media ball size of three size fraction samples were obtained. • The media size ratio was ascertained as φ 50 mm: φ 40 mm: φ 30 mm = 28%: 28%: 44%. • The installed power and productivity of ball mill (Φ 5.03 m × 6.40 m) were calculated. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Henan University of Science & Technology, Natural Science is the property of Editorial Office of Journal of Henan University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, the generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles is investigated. It describes the dynamics of nonlinear waves in hydrodynamics. On the one hand, we construct a class of special N-wave solutions by applying the linear superposition principle, resonant multiple wave solutions and complexiton solutions are derived. On the other hand, the rational breather wave and rogue wave solutions of the equation are also obtained by employing the extended homoclinic test method. Finally, all these solutions are presented via 3-dimensional plots and density plots with choices some special parameters to show the dynamic characteristics. [ABSTRACT FROM AUTHOR]

• Based on the resonant multiple soliton solutions, the non-singular complexiton and singular complexiton solutions are developed by taking pairs of the conjugate parameters. • The breather wave solutions are also explored through the symbolic computation and test function method. • The graphical representations of the corresponding solutions are presented to reveal the dynamical characteristics. This work aims to construct some novel exact solutions of the (2 + 1)-dimensional Sawada-Kotera equation (SKE). First, the resonant multiple soliton solutions (RMSSs) are discussed by employing the weight algorithm (WA) and linear superposition principle (LSP). Then, based on the RMSSs, the non-singular complexiton and singular complexiton solutions are developed by taking pairs of the conjugate parameters. Finally, the breather wave solutions (BWSs) of the sin-cosh type are explored through the symbolic computation and test function method. The dynamics of the corresponding extracted solutions are displayed graphically for showing the physical properties. The achievements of this article can provide some new insights to the (2 + 1)-dimensional SKE. [ABSTRACT FROM AUTHOR]

In this paper, we mainly analyze the nonlinear dynamics behavior of the (2 + 1)-dimensional Sawada–Kotera (S–K) equation, which can be usually used to describe shallow water phenomena from natural science. First, the multiple resonant wave and complexiton solutions are constructed with the help of the linear superposition principle, under different domain fields, such as real and complex domain fields, respectively. Next, we apply a new ansatz method to obtain a class of rogue wave solutions (one-rogue wave and two-rogue wave solutions). Finally, the 3-dimensional and 2-dimensional density graphs are plotted for the yielded results in the above texts to better illustrate the dynamics processes to them. [ABSTRACT FROM AUTHOR]

In this paper, the simplified linear superposition principle is presented and employed to handle two versions of the fifth-order KdV equations, called the (2 + 1)-dimensional Caudrey–Dodd–Gibbon (CDG) equation and the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (KP) equation, respectively. Two general forms of resonant multi-soliton solutions are formally obtained. The paper proceeds step-by-step with increasing detail about the derivation process. Firstly, illustrate the algorithms of the linear superposition principle which paves the way for solving the wave related numbers. Then, demonstrate its application that exposes the proposed approach provides enough freedom to construct resonant multi-soliton wave solutions. Finally, some graphical representations of obtained solutions are portrayed by taking some definite values to free parameters, which describe various versions of inelastic interactions of resonant multi-soliton waves. The associated propagations may be related to large variety of real physical phenomena. [ABSTRACT FROM AUTHOR]

The Hirota bilinear method is a powerful tool for solving nonlinear evolution equations. Together with the linear superposition principle, it can be used to find a special class of explicit solutions that correspond to complex eigenvalues of associated characteristic problems. These solutions are known as complexiton solutions or simply complexitons. In this article, we study complexiton solutions of the the Hirota‐Satsuma‐Ito equation which is a (2 + 1)‐dimensional extension of the Hirota‐Satsuma shallow water wave equation known to describe propagation of unidirectional shallow water waves. We first construct hyperbolic function solutions and consequently derive the so‐called complexitons via the Hirota bilinear method and the linear superposition principle. In particular, we find nonsingular complexiton solutions to the Hirota‐Satsuma‐Ito equation. Finally, we give some illustrative examples and a few concluding remarks. [ABSTRACT FROM AUTHOR]

We present new complexiton solutions to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov (aNNV) equation by application of the Hirota direct method and the linear superposition principle. We first find hyperbolic function solutions to the corresponding bilinear equation and consequently derive the so-called complexitons. In particular, we construct nonsingular complexiton solutions from positive complexiton solutions of the bilinear form of the nonlinear equation. Finally, we give some illustrative examples and a few concluding remarks. [ABSTRACT FROM AUTHOR]

Highlights • The axisymmetric compatibility equation is introduced to establish characteristics theory replacing hypothesis. • The axisymmetric active earth pressure is determined by both boundary load and boundary velocity. • The dilatation angle can be considered in this method for axisymmetric active earth pressure. • The practical formula for axisymmetric active earth pressure and relevant parameter tables are provided. • The critical depth and the critical cohesion requiring no support are presented in circular excavation. Abstract The current theories and calculation method for axisymmetric active pressure have been established according to certain hypotheses. This paper presents a strict theory for this problem with the axisymmetric compatibility condition replacing the current hypotheses, and further develops a practical calculation method for engineering applications. The new governing equations are rigorously derived by substituting the limit stress components under the Drucker–Prager criterion into the equilibrium equations and substituting the strain rate obtained by the flow theory into the axisymmetric compatibility equation. The characteristic lines and characteristic relations are got and the numerical integration method for axisymmetric active pressure is established by introducing boundary conditions. The calculation and derivation indicate that the axisymmetric earth pressure not only depends on boundary load but is also related to boundary velocity. Based on the numerical integration method, the pressure coefficients are investigated in detail, and many new conclusions are found. Finally, a practical calculation formula for axisymmetric active earth pressure is presented based on the linear superposition principle and the function fitting of pressure coefficients. Six sets of experimental data are cited to verify and support the theory and practical method. [ABSTRACT FROM AUTHOR]

In this paper, the linear superposition principle and Hirota bilinear equations are simultaneously employed to handle two new (3+1)-dimensional Jimbo–Miwa equations. The corresponding resonant multi-soliton solutions and the related wave numbers are formally established, which are totally different from the previously reported ones. Moreover, the extracted N-soliton waves and dispersion relations have distinct physical structures compared to solutions obtained by Wazwaz. Finally, five graphical representations are portrayed by taking definite values to free parameters which demonstrates various versions of traveling solitary waves. The results show the proposed approach provides enough freedom to construct multi-soliton waves that may be related to a large variety of real physical phenomena and, moreover, enriches the solution structure. [ABSTRACT FROM AUTHOR]

• The complexiton solutions are obtained by the linear superposition principle and weight algorithm. • The Y-type soliton solutions are extracted by imposing the resonance conditions on the N-soliton solutions. • Two different types of the interaction solutions are also explored by exerting the test function method and symbolic computation. • The graphical representations of the corresponding solutions are presented to reveal the dynamical characteristics. • The attained outcomes of this article are all new and can enable us make sense of the physical behaviors of the (3 + 1)-dimensional KSE better. This paper focuses on some novel exact solutions of the (3 + 1)-dimensional Kudryashov-Sinelshchikov equation (KSE). Based on the linear superposition principle (LSP) and weight algorithm (WA), we construct the complexiton solutions by introducing the pairs of the conjugate parameters. Besides, the Y -type soliton solutions are extracted by imposing the resonant conditions on the N -soliton solutions. Finally, two different types of the interaction solutions (ISs) are also explored by exerting the test function method and symbolic computation. The graphical representations of the corresponding solutions are presented to reveal the dynamical characteristics. To the best of our knowledge, the attained outcomes of this article are brand new and have never been reported in other literature, and can enable us make sense of the physical behaviors of the (3 + 1)-dimensional KSE better. [ABSTRACT FROM AUTHOR]

In this letter, the linear superposition principle is used to discuss the ( 3 + 1 ) -dimensional Boiti–Leon–Manna–Pempinelli equation with bilinear derivatives. As a result, we obtain new resonant soliton and complexiton solutions by discussing two different cases involved the parameters. These solutions are a class of N -wave solutions of linear combinations of exponential traveling waves. [ABSTRACT FROM AUTHOR]

Linear superposition principle is applied to the Hirota bilinear form of a new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which leads to a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions. As a result, resonant multiple wave solutions are derived and plotted for the new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. [ABSTRACT FROM AUTHOR]

We aim to find exact solutions to some generalized KP- and BKP-type equations by the Wronskian technique and the linear superposition principle. By applying Wronskian identities of the bilinear KP hierarchy, three Wronskian formulations are first furnished, with all generating functions for matrix entries satisfying a system of combined linear partial differential equations. Second, we apply the linear superposition principle to exponential traveling wave solutions of the introduced generalized KP and BKP equations. Each exponential wave in the Wronskian and N-wave solutions satisfies the corresponding nonlinear dispersion relation. [ABSTRACT FROM AUTHOR]

The linear superposition principle is applied to hyperbolic and trigonometric function solutions to generalized bilinear equations. We determine sufficient and necessary conditions for the existence of linear subspaces of hyperbolic and trigonometric function solutions to generalized bilinear equations. By using weights, three examples are given to show applicability of our theory. [ABSTRACT FROM AUTHOR]

The linear superposition principle can apply to the construction of resonant multiple wave solutions for a ( 3 + 1 )-dimensional nonlinear evolution equation. Two types of resonant solutions are obtained by the parameterization for wave numbers and frequencies for linear combinations of exponential traveling waves. The resonance phenomena of multiple waves are discussed through the figures for several sample solutions. [ABSTRACT FROM AUTHOR]

We apply the linear superposition principle to Hirota bilinear equations and generalized bilinear equations. By extending the linear superposition principle to complex field, we construct complex exponential wave function solutions first and then get complexions by taking pairs of conjugate parameters. A few examples of mixed resonant solitons and complexitons to Hirota and generalized bilinear differential equations are presented. [ABSTRACT FROM AUTHOR]

For the exponential traveling wave solutions to the Hirota bilinear equations, a sufficient and necessary criterion for the existence of linear superposition principle has been given. Motivated by this criterion, we propose a new Hirota bilinear equation via using a multivariate polynomial. Applying the linear superposition principle to this new Hirota bilinear equation, we finally find two types of resonant multiple wave solutions, among which, the resonant three-wave solutions are illustrated with three-dimensional plots. [ABSTRACT FROM AUTHOR]

Linear subspaces of hyperbolic and trigonometric function solutions to generalized bilinear equations are analyzed. Necessary and sufficient conditions are presented to apply the linear superposition principles. Applications of an algorithm using weights are made together with a few concrete examples. [ABSTRACT FROM AUTHOR]

Multi-wave resonant solutions of a class of nonlinear heat-type equations are investigated by their corresponding generalized bilinear equations. We develop a new technique for searching for resonant solutions to those generalized bilinear equations, using the idea of weights of dependent variables. The results show that generalized bilinear operators and generalized bilinear equations are powerful and irreplaceable tools for dealing with nonlinear differential equations. [ABSTRACT FROM AUTHOR]

In this article, the multiple exp-function method and the linear superposition principle are employed for constructing the exact solutions and the solitary wave solutions for the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. With help of Maple and by using the multiple exp-method, we can get exact explicit one-wave, two-wave, and three-wave solutions, which include one-soliton-, two-soliton-, and three-soliton-type solutions. Furthermore, we apply the linear superposition principle to find n-wave solutions of the CBS equation. Two cases with specific values of the involved parameters are plotted for each two-wave and three-wave solutions. [ABSTRACT FROM AUTHOR]

Evidently, the linear superposition principle can not be exactly established as a general principle in the presence of nonlinearity, and, at the first glance, there is no expectation for it to hold even approximately. In this letter, it is shown that the balance of different nonlinear effects describes what linear superpositions may occur in nonlinear systems. The heavenly equations are of significance in several scientific fields, especially in relativity, gravity, field theory, and fluid dynamics. A special type of implicit shock wave solution with three two-dimensional arbitrary functions of the general heavenly equation is revealed. Restrict the two-dimensional arbitrary functions to some types of one-dimensional arbitrary functions, it is found that the nonlinear effects can be balanced such that the "impossible" linear superposition solutions can be non-trivially constituted to new solutions of the general heavenly equation. • The linear superposition solution for the nonlinear general heavenly equation is studied. • The balance of nonlinear effects describes what linear superpositions may occur in nonlinear systems. • Decomposition may be a valuable tool for constructing linear superposition solutions of nonlinear systems. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents single and multi-potassium ion channel optical models based on optical point spread function and linear superposition principle. Its novelty lies in the dynamic description method of potential diffusion process which is generated by the whole membrane and nearby ion channel while potassium ion channels open. The main parameters and the corresponding equations of proposed model are studied. The dynamic spatiotemporal spread processes of membrane potential repolarization evoked by opened potassium ion channels are investigated as well. Furthermore, performance evaluation by computer simulation and experimental results proves the correctness and effectiveness of the idea above. [Copyright &y& Elsevier]