Bilinear Method Research Articles

Interaction of ion-acoustic solitons for multi-dimensional Zakharov Kuznetsov equation in Van Allen radiation belts

The propagation of nonlinear ion-acoustic waves (IAWs) in a magnetoplasma having superthermal inertialess electrons and inertial warm adiabatic ions is examined. In this regard, Zakharov Kuznetsov (ZK) equation is derived in order to investigate the characteristics of the ion-acoustic solitons (IASs) in the small but finite amplitude limit. Using the Hirota bilinear method, both single and two-solitons solutions of the two and three-dimensional ZK equations are obtained. According to the parameters of Van Allen radiation belts, the effects of superthermality, magnetic field, ion number density, and the ion temperature on the profile of both single and two-solitons are explored. The estimates of the spatial extent of the interaction of solitons in the outer radiation belts are also given.

Singular solitons interaction of dust ion acoustic waves in the framework of Korteweg de Vries and Modified Korteweg‐de Vries equations with (r,q) distributed electrons

AbstractTwo nonlinear evolution equations (NLEEs), namely, Korteweg‐de Vries (KdV) and modified Korteweg‐de Vries (MKdV) are derived from the dust hydrodynamic model of collisionless, unmagnetized dusty plasma with electrons following double spectral velocity distribution, that is, (r,q) distribution. Regular as well as singular analytic solutions of derived KdV equations and MKdV equations are obtained separately with the aid of the Hirota Bilinear method. Two singular soliton solutions of both KdV and MKdV equations are obtained separately. The efficiency and interactive approach of the heuristic Hirota Bilinear method leads to multiple singular soliton solutions for its beautiful algebraic technique which generates solitons solutions as well as singular solitons explicitly. Then the significance of multi singular soliton interaction has been studied for KdV singular solitons and for MKdV singular solitons in the critical parameter set.

A new extended (2+1)-dimensional Kadomtsev–Petviashvili equation with N-solitons, periodic solutions, rogue waves, breathers and lump waves

In this work, a new extended integrable (2+1)-dimensional Kadomtsev–Petviashvili equation is proposed and investigated, which models slowly varying perturbation wave in dispersion fluids. First, the WTC-Kruskal algorithm is applied to exploring the corresponding compatibility condition for this equation in Painlevé sense. Then, N-soliton, periodic, breather solution as well as the mixed form composing of breather and soliton(s) have been derived via Hirota bilinear method and symbolic computation. Moreover, we have derived the rational and semi-rational solutions in terms of “long wave” limit. The rational solutions can be classified as first order line rogue waves and lumps, while the semi-rational solutions have the forms: a hybrid of first order line rogue wave and one soliton, a hybrid of second order line rogue wave and one soliton, a hybrid of lump and soliton(s), a hybrid of lump and breather, a hybrid of lump and periodic solution.

Shifted nonlocal Kundu type equations: Soliton solutions

We study the shifted nonlocal reductions of the integrable coupled Kundu type system. We then consider particular cases of this system; namely Chen–Lee–Liu, Gerdjikov–Ivanov, and Kaup–Newell systems. We obtain one- and two-soliton solutions of these systems and their shifted nonlocal reductions by the Hirota bilinear method. We present particular examples for one- and two-soliton solutions of the reduced shifted nonlocal Chen–Lee–Liu, Gerdjikov–Ivanov, and Kaup–Newell equations.

Rational and semi-rational solutions to the Mel’nikov equation

We investigate rational and semi-rational solutions to the ( 2 + 1 ) -dimension Mel’nikov equation based on the Hirota bilinear method and the long wave limit approach. It is shown that the derived rational solutions display lump solitons in ( x , y ) -plane and line rogue waves in ( x , t ) -plane. As to the semi-rational solutions consisting of the hybrid of the soliton and rational solution, they exhibit the interaction between solitons and lumps in ( x , y ) -plane. While in ( x , t ) -plane the hybrid solutions demonstrate the interaction between line rogue waves and solitons. Dynamics of solutions are analysed with plots.

Lump interaction phenomena to the nonlinear ill-posed Boussinesq dynamical wave equation

In this manuscript, we discuss the dynamical behavior of ill-posed Boussinesq (IPB) dynamical wave equation. This equation depicts how long wave made in shallow water propagates due to the influence of gravity. The different wave structures are extracted by the utilization of Hirota's bilinear method (HBM) and different test function approaches. The wave structures in the forms of novel breather waves, lump solutions, two-wave solutions, and rogue wave solutions are obtained. Furthermore, with assistance of appropriate parameters, the acquired solutions' physical behavior is drawn out in 3 dimensional, and contour profiles. As a consequence of the obtained results, we can claim that the used methodology is simple, dynamic, extremely effective, and will serve as a useful tool for solving more severely nonlinear problems in a variety of domains, most notably ocean and coastal engineering. Additionally, our findings provide an important first step in comprehending the structure and physical behavior of complex structures. We hope that our findings will contribute significantly to our understanding of ocean waves. This work, we believe, is timely and will be of interest to a broad range of professionals involved in ocean engineering models. The obtained results are useful in grasping the fundamental scenarios of nonlinear sciences. With the help of Mathematica, the obtained results are verified by inserting them into the governing equation.

Lumps with their some interactions and breathers to an integrable (2 + 1)-dimensional Boussinesq equation in shallow water

In this paper, the lumps with their interactions (lump-single and lump-double stripes), and breather wave solutions are constructed to the new integrable (2 + 1)-dimensional Boussinesq equation via the Hirota bilinear method. The propagations of the obtained lumps, lump-stripes, and breather wave solutions of the aforesaid equation are presented through some 3D and 2D graphs. All of the graphs are given to show their correct wave profiles. The graphical outputs demonstrate that lump waves propagate in all directions throughout space. However, the amplitudes and forms of lump waves remain constant during their propagation in any direction. On the other hand, the lump wave travels along with the stripes for the lump and stripes solutions. Furthermore, two homoclinic test approaches are used to produce the breather wave solutions to the aforesaid equation. It is found that breather waves propagate in a specific direction on a constant period. In addition, the effects of constant coefficients of the model are presented also graphically. It is found that the amplitudes of all obtained wave solutions are changed with the change of constant coefficients of the model..The outcomes of the study can be useful for a better understanding of wave propagation dynamics in shallow water under gravity.

On the lump solutions, breather waves, two-wave solutions of (2 + 1)-dimensional Pavlov equation and stability analysis

Hirota’s bilinear method (HBM) has been successfully applied to the [Formula: see text]-dimensional Pavlov equation to analyze the different wave structures in this paper. The [Formula: see text]-dimensional Pavlov equation is used for the study of integrated hydrodynamic chains and Einstein–Weyl manifolds. In our research, we find new solutions in the forms of lump solutions, breather waves, and two-wave solutions. The modulation instability (MI) of the governing model is also discussed. Moreover, a variety of 3D, 2D, and contour profiles are used to illustrate the physical behavior of the reported results. Acquired findings are useful in understanding nonlinear science and its related nonlinear higher-dimensional wave fields. Through the use of Mathematica, the obtained results are verified by inserting them into the governing equation. The strengthening of representative calculations we’ve made gives us a strong and effective mathematical framework for dealing with the most difficult nonlinear wave problems.

Higher-order rogue wave solutions of the Sasa–Satsuma equation

Up to the third-order rogue wave solutions of the Sasa–Satsuma (SS) equation are derived based on the Hirota’s bilinear method and Kadomtsev–Petviashvili hierarchy reduction method. They are expressed explicitly by rational functions with both the numerator and denominator being the determinants of even order. Four types of intrinsic structures are recognized according to the number of zero-amplitude points. The first- and second-order rogue wave solutions agree with the solutions obtained so far by the Darboux transformation. In spite of the very complicated solution form compared with the ones of many other integrable equations, the third-order rogue waves exhibit two configurations: either a triangle or a distorted pentagon. Both the types and configurations of the third-order rogue waves are determined by different choices of free parameters. As the nonlinear Schrödinger equation is a limiting case of the SS equation, it is shown that the degeneration of the first-order rogue wave of the SS equation converges to the Peregrine soliton.

Lump solutions to a generalized nonlinear PDE with four fourth-order terms

Abstract A combined fourth-order (2 + 1)-dimensional nonlinear partial differential equation which contains four fourth-order nonlinear terms and all second-order linear terms is formulated. This equation covers three generalized KP, Hirota–Satsuma–Ito, and Calogero–Bogoyavlenskii–Schiff equations as examples, which have physical applications in the study of various nonlinear phenomena in nature. In terms of some settings of the coefficients, a class of lump solutions is constructed by the Hirota bilinear method and the solutions are calculated through the symbolic computation system of Maple. Meanwhile, the relation between the coefficients and the solution is explored. Two special lump solutions are generated by taking proper values for the involved coefficients and parameters, and their dynamic behaviors are studied, as illustrative examples. The primary advantage of the Hirota bilinear method is to transform a nonlinear equation into a bilinear one so that the targeted equation can be easily studied.

Lump and interaction solutions for a (2 + 1)-dimensional combined pKP-BKP equation in fluids

In this paper, a number of exact solutions of (2 + 1)-dimensional combined pKP-BKP equation are given. By taking into account the Hirota bilinear method and making use of quadratic function, a number of lump solutions are obtained. Combining quadratic function with exponential function, the interaction phenomenon between lump and soliton, and the lump-soliton solutions are obtained. When the trigonometric function is combined with the exponential function, the breather wave solutions are obtained. Finally, by picking appropriate parameter values, the dynamic properties of these exact solutions are presented by three-dimensional, density and contour plots.

Hybrid solutions of a (3 + 1)-dimensional generalized B-type Kadomtsev–Petviashvili equation

The simplified form of the Hirota bilinear method is used to investigate a [Formula: see text]-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation, and the five-soliton is formally derived. Applying the long-wave limit to the five-soliton and establishing the complex conjugate relations in the parameters of the five-soliton, five types of hybrid solutions of the [Formula: see text]-dimensional gBKP equation are obtained. Furthermore, the dynamical behaviors of the hybrid solutions are systematically analyzed via numerical simulations.

Lump and travelling wave solutions of a (3 + 1)-dimensional nonlinear evolution equation

In this paper, the (3+1)-dimensional nonlinear evolution equation is studied analytically. The bilinear form of given model is achieved by using the Hirota bilinear method. As a result, the lump waves and collisions between lumps and periodic waves, the collision among lump wave and single, double-kink soliton solutions as well as the collision between lump, periodic, and single, double-kink soliton solutions for the given model are constructed. Furthermore, some new traveling wave solutions are developed by applying the exp(−ϕ(ξ)) expansion method. The 3D, 2D and contours plots are drawn to demonstrate the nature of the nonlinear model for setting appropriate set of parameters. As a result, a collection of bright, dark, periodic, rational function and elliptic function solutions are established. The applied strategies appear to be more powerful and efficient approaches to construct some new traveling wave structures for various contemporary models of recent era.

Ion-acoustic wave structures in the fluid ions modeled by higher dimensional generalized Korteweg-de Vries–Zakharov–Kuznetsov equation

In this paper, the higher dimensional generalized Korteweg-de-Varies–Zakharov–Kuznetsov (gKdV-ZK) equation is under investigation. This model is used in the field of plasma physics which describes the effects of magnetic field on the weak ion-acoustic wave. We have applied two techniques, called as ϕ6-model expansion method and the Hirota bilinear method (HBM) to explore the diversity of wave structures. The solutions are expressed in the form of hyperbolic, periodic and Jacobi elliptic function (JEF) solutions. Moreover, the solitary wave solutions are also extracted. A comparison of our results to well-known results is made, and the study concludes that the solutions achieved here are novel. Additionally, 3-dimensional and contour profiles of achieved outcomes are drawn in order to study their dynamics as a function of parameter selection. On the basis of the obtained results, we can assert that the proposed computational methods are straightforward, dynamic, and well-organized, and will be useful for solving more complicated nonlinear problems in a variety of fields, particularly in nonlinear sciences, in conjunction with symbolic computations. Additionally, our discoveries provide an important milestone in comprehending the structure and physical behavior of complex structures. We hope that our findings will contribute significantly to our understanding of ocean waves. This study, we hope, is appropriate and will be of significance to a broad range of experts involved in ocean engineering models.

Application of Hirota operators for controlling soliton interactions for Bose-Einstien condensate and quintic derivative nonlinear Schrödinger equation

With the help of the Hirota bilinear method (HBM), we study soliton interactions of quasi-1D Bose-Einstein Condensate system (BECs) with dipole-dipole attraction and repulsion. BEC is an extended form of nonlinear Schrödinger equation (NLSE) and it consists of quadratic-cubic nonlinearities, linear gain or loss and time modulated dispersion. Due to its spatially varying coefficients property it has significance in the field of fluid dynamics, classical and quantum field theories, nonlinear optics and physics etc. We will also discuss soliton interactions with graphically descriptions for QDNLSE. We obtain some parabolic, anti parabolic, M-shaped, W-shaped, butterflies, bright, anti dark, V-shaped, S-shaped and other solitons for our governing models.

On the breather waves, lump solutions, two-wave solutions of ([formula omitted]) dimensional Martínez Alonso–Shabat equation

We effectively utilize Hirota’s bilinear method (HBM) in this article to scrutinize the diverse wave profiles to the (3+1) dimensional Martínez Alonso–Shabat equation when the test function approaches are taken into consideration. Several novel breather waves, lump solutions, and two-wave solutions are obtained. Furthermore, with the assistance of appropriate parameters, the reported results’ physical behavior is sketched out in several 3 dimensional, 2 dimensional, and contour profiles. The obtained results are useful in grasping the fundamental scenarios of nonlinear sciences as well as the related nonlinear higher dimensional wave fields, both of which are important. With the help of Mathematica, the obtained results are verified by inserting them into the governing equation. Thus, our programmes, which are based on the fortress of representative calculations, provide a robust mathematical framework for resolving difficult nonlinear wave problems.

Dynamics and bidirectional lumps of the generalized Boussinesq equation with time-space dispersion term: Application of surface gravity waves

In this paper, we revisit the generalized Boussinesq equation that include a new time-space dispersion term. The Kudryashov expansion method is used to obtain singular and nonsingular bidirectional bell-shaped solutions. Also, a modified version of the rational sine-cosine method is proposed to extract bidirectional concave-periodic solutions. Then, the Hirota bilinear method and the Cole-Hopf transformation are implemented to extract bidirectional lump-solitons. On the other hand, the dynamics of Boussinesq’s parameters, nonlinearity and time-space dispersion, are investigated by using graphical analysis. Finally, the obtained solutions to the proposed Boussinesq model are verified by direct substitution in the governing equation.

Complex Plant Process Trait Evaluation Through Decomposition of Higher-order Interaction: A Case Study in Acclimation Responses of Cold-climate Hybrid Grapevine Through Bilinear and Multiway Methods

Traditionally, the structure of higher-order data in genotype-by-environment interaction requires simplification to use bilinear reduction models. Flexible multiway reduction models have been claimed to be more informative, as they allow exploration of individual trends and account for the covariance among data modes. In complex latent traits, such as acclimation response of grapevine (Vitis sp.), these methods may offer increased insight into plant adaptive processes. In a growth chamber study, data from seven phenotypic traits at 11 photoperiodic times in the presence of two temperatures of 30 accessions were analyzed. The four-way interaction among these data modes was isolated and further examined through bilinear singular value decomposition (SVD) and multiway Tucker decomposition models. A similar set of three latent process traits were identified regardless of model used. The Tucker decomposition model led to more concise clustering of wild-type accessions, was more interpretable, as trends could be evaluated separately, and had less indication of overfitting; therefore, the multiway method was preferred over the standard SVD bilinear method in the investigation of high-order interaction in acclimation response. This methodology may offer insight into other complex traits, such as phenolic development, drought tolerance, and horizontal disease resistance to improve breeding efforts as other individual mechanisms used by the organism are separated, quantified, and compared rather than the culmination of events as an end-product.

Interaction between kink solitary wave and rogue wave, new periodic cross-kink wave solutions and other exact solutions to the (4 + 1)-dimensional BLMP model

In this study, some new solutions to the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) model representing the wave propagation through incompressible fluids. In shallow water, the linearization of the wave structure needs more critical wave capacity conditions than in deep water, and the strong nonlinear features are visible. The obtained solutions may be applied in the demonstration of this model in some better way. These solutions are in the form of interaction between kink solitary wave and rogue wave, new periodic cross-kink wave solutions and other exact solutions with certain conditions. The achieved solutions are also verified by using the latest computational software such as MATHEMATICA. The modified integration techniques, the Hirota bilinear method and the extended (G′/G)-expansion method are utilized to secure the solutions. The obtained solutions are also explained by 3D, 2D and contour graphs. These results may helpful for the further study of the model in future.

Novel interaction phenomena of the new (2 + 1)-dimensional extended shallow water wave equation

Based on the Hirota bilinear method, we study some kinds of novel interaction solutions of the new integrable [Formula: see text]-dimensional extended shallow water wave equation. Using the symbolic computations, we obtain a rich variety of exact solutions such as the lump solutions, lump-kink solutions, breather solutions and two-kink solutions and these exact solutions are shown analytically and graphically by selecting the proper parameters. These new exact solutions contribute us to our understanding of the physical phenomena of the integrable nonlinear shallow water wave model.