Bilinear Equations Research Articles

Newly formed center-controlled rouge wave and lump solutions of a generalized (3+1)-dimensional KdV-BBM equation via symbolic computation approach

In this article, we investigate the generalized (3+1)-dimensional KdV-Benjamin-Bona-Mahony equation governed with constant coefficients. It applies the Painlevé analysis to test the complete integrability of the concerned KdV-BBM equation. The symbolic computational approach provides first-order, second-order rogue wave and lump solutions with center-controlled parameters. The rogue waves localized in space and time have a significant amplitude, and lumps are of rational form solution, localized decaying solutions in all space directions rationally. Utilizing a symbolic computation approach, we get the bilinear equation of the KdV-Benjamin-Bona-Mahony equation and show the center-controlled rogue waves and lumps. We employ the symbolic system software Mathematica to do the symbolic computations, form the first and second-order rogue waves, and lump solutions with appropriate values of constant coefficients. The KdV-Benjamin-Bona-Mahony equation analyses the evolution of long waves with modest amplitudes propagating in plasma physics and the motion of waves in fluids and other weakly dispersive mediums. Moreover, rogue waves and lumps occur in several scientific areas, such as fluid dynamics, optical fibers, dusty plasma, oceanography, water engineering, and other nonlinear sciences.

Lump-Type Solutions, Lump Solutions, and Mixed Rogue Waves for Coupled Nonlinear Generalized Zakharov Equations

This article studies diverse forms of lump-type solutions for coupled nonlinear generalized Zakharov equations (CNL-GZEs) in plasma physics through an appropriate transformation approach and bilinear equations. By utilizing the positive quadratic assumption in the bilinear equation, the lump-type solutions are derived. Similarly, by employing a single exponential transformation in the bilinear equation, the lump one-soliton solutions are derived. Furthermore, by choosing the double exponential ansatz in the bilinear equation, the lump two-soliton solutions are found. Interaction behaviors are observed and we also establish a few new solutions in various dimensions (3D and contour). Furthermore, we compute rogue-wave solutions and lump periodic solutions by employing proper hyperbolic and trigonometric functions.

Study on the Nonlinear Dynamics of the (3+1)-Dimensional Jimbo-Miwa Equation in Plasma Physics

The Jimbo-Miwa equation (JME) that describes certain interesting (3+1)-dimensional waves in plasma physics is studied in this work. The Hirota bilinear equation is developed via the Cole-Hopf transform. Then, the symbolic computation, together with the ansatz function schemes, are utilized to seek exact solutions. Some new solutions, such as the multi-wave complexiton solution (MWCS), multi-wave solution (MWS) and periodic lump solution (PLS), are successfully constructed. Additionally, different types of travelling wave solutions (TWS), including the dark, bright-dark and singular periodic wave solutions, are disclosed by employing the sub-equation method. Finally, the physical characteristics and interaction behaviors of the extracted solutions are depicted graphically by assigning appropriate parameters. The obtained outcomes in this paper are more general and newer. Additionally, they reveal that the used methods are concise, direct, and can be employed to study other partial differential equations (PDEs) in physics.

On the regional control problem of the bilinear wave equation

AbstractThe present research deals with regional optimal control problem of the bilinear wave equation evolving on a spatial domain . Such an equation is excited by bounded controls that act on the velocity term. It addresses the tracking of a desired state all over the time interval only on a subregion of with minimum energy. Then, we prove that an optimal control exists and is characterized as a solution to an optimality system. Algorithm for the computation of such a control is given and successfully illustrated through simulations.

On the Bilinear Second Order Differential Realization of an Infinite-Dimensional Dynamical System: An Approach Based on Extensions to M2-Operators

Considering the case of a continual bundle of controlled dynamic processes, the authors have studied the functional-geometric conditions of existence of non-stationary coefficients-operators from the differential realization of this bundle in the class of non-autonomous bilinear second-order differential equations in the separable Hilbert space. The problem under scrutiny belongs to the type of non-stationary coefficient-operator inverse problems for the bilinear evolution equations in the Hilbert space. The solution is constructed on the basis of usage of the functional Relay-Ritz operator. Under this mathematical problem statement, the case has been studied in detail when the operators to be modeled are burdened with the condition, which provides for entire continuity of the integral representation equations of the model of realization. Proposed is the entropy interpretation of the given problem of mathematical modeling of continual bundle dynamic processes in the context of development of the qualitative theory of differential realization of nonlinear state equations of complex infinite-dimensional behavioristic dynamical system.

Solution of the Continuous Time Bilinear Quadratic Regulator Problem by Krotov’s Method

This work contributes to the field of optimal control of bilinear systems. It concerns a continuous time, finite dimensional, bilinear state equation with a quadratic performance index to be minimized. The state equation is non-autonomous and comprises a deterministic, a-priori known excitation. The control trajectory is constrained to an admissible set without a specific structure. The performance index is a functional, quadratic in the state variables and control signals. Krotov's method is used for solving this problem by means of an improving sequence. To this end, the required sequence of an improving functions is formulated. Finally, the solution is encapsulated in an algorithm form and a numerical example of structural control problem is provided.

M-lump, rogue waves, breather waves, and interaction solutions among four nonlinear waves to new (3+1)-dimensional Hirota bilinear equation

M-lump, rogue waves, breather waves, and interaction solutions among four nonlinear waves to new (3+1)-dimensional Hirota bilinear equation

Dynamic analysis of lump solutions based on the dimensionally reduced generalized Hirota bilinear KP-Boussinesq equation

In this paper, a [Formula: see text]-dimensional generalized KP-Boussinesq equation is introduced and its associate Hirota bilinear form is also given. Based on finding the positive quadratic function solutions of the associate Hirota bilinear equation, the lump solutions of the proposed [Formula: see text]-dimensional generalized KP-Boussinesq equation and its corresponding reduced equations in [Formula: see text] dimensions are obtained. Furthermore, the sufficient and necessary conditions for guaranteeing the analyticity and rational localization of lump solutions are derived and expressed in the form of free parameters, which are involved in lump solutions and play a key role in controlling the dynamic properties of lump solutions. The localized properties are also analyzed and shown graphically.

New lump solutions to a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

The aim of this paper is to show the existence of three-wave lump solutions to a (3+1)-dimensional generalized CBS (gCBS) equation. Based on the Hirota method, the quadratic functions of the form f=f12+f22+f32+d with nondegenerate condition are applied to solve the corresponding bilinear equation and to generate lump solutions to the gCBS equation. We present two examples of such nonlinear equations and their lump solutions. Moreover, 3-dimensional plots and contour plots are exhibited for three reduced lump solutions.

Nonlinear Dynamic Behaviors of the (3+1)-Dimensional B-Type Kadomtsev—Petviashvili Equation in Fluid Mechanics

This paper provides an investigation on nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation, which is used to model the propagation of weakly dispersive waves in a fluid. With the help of the Cole—Hopf transform, the Hirota bilinear equation is established, then the symbolic computation with the ansatz function schemes is employed to search for the diverse exact solutions. Some new results such as the multi-wave complexiton, multi-wave, and periodic lump solutions are found. Furthermore, the abundant traveling wave solutions such as the dark wave, bright-dark wave, and singular periodic wave solutions are also constructed by applying the sub-equation method. Finally, the nonlinear dynamic behaviors of the solutions are presented through the 3-D plots, 2-D contours, and 2-D curves and their corresponding physical characteristics are also elaborated. To our knowledge, the obtained solutions in this work are all new, which are not reported elsewhere. The methods applied in this study can be used to investigate the other PDEs arising in physics.

Finite‐frequency output feedback MPC for Markov jump systems

AbstractThis paper investigates a general design framework of dynamic output feedback model predictive control (DOFMPC) for Markov jump systems within both time and frequency domain. Such a design with guaranteed H∞ and quadratic performance is formulated by a standard semi‐definite programming (SDP), and it is achieved by employing a special congruence transformation. The SDP condition greatly reduces the computational effort by eliminating bilinear matrix inequalities or equation constraints reported in existing references. Specifically, the H∞ norm of the transfer function is optimized within three types of frequency ranges on account of generalized Kalman–Yakubovic–Popov (GKYP) lemma. The quadratic index is optimized online via SDP. Finally, we verify the feasibility and effectiveness of the proposed method from both theoretical and practical point of view.

Lump and soliton on certain spatially-varying backgrounds for an integrable (3+1) dimensional fifth-order nonlinear oceanic wave model

The dynamics of nonlinear waves with controllable dispersion, nonlinearities, and background continues to be an exciting line of active research in recent years. In this work, we focus to investigate an integrable (3+1)-dimensional nonlinear model describing the evolution of water waves with higher-order temporal dispersion by characterizing the dynamics of lump and soliton waves on different spatially-varying backgrounds. Particularly, we construct some explicit lump wave and kink-soliton solutions for the considered model through its trilinear and bilinear equations with appropriate forms of the polynomial and exponential type initial seed solutions, respectively. Additionally, we obtain hyperbolic and periodic waves through bilinear Bäcklund transformation. We explore their propagation and transformation/modulation dynamics due to different spatial backgrounds through categorical analysis and clear graphical demonstrations for a complete understanding of the resulting solutions. Our analysis shows that the lump solution results into a simultaneous existence of well-localized spike and declining (coupled bright-dark wave) structures on a constant background, while the soliton solution exhibits kink and anti-kink wave patterns along different spatio-temporal domains with controllable properties. When the arbitrary spatial background is incorporated appropriately, we can witness the manifestation of soliton into manipulated periodic and interacting dynamical structures. On the other hand, the lump wave results into the coexistence, interacting wave and breather formation due to periodic and localized type arbitrary spatial backgrounds. The present results will be an important addition to the context of engineering nonlinear waves with non-vanishing controllable backgrounds in higher-dimensional models. Further, the present study can be extended to investigate several other nonlinear systems to understand the physical insights of the variable background in their dynamics.

Rational solutions of an extended (2+1)-dimensional Camassa-Holm- Kadomtsev-Petviashvili equation in liquid drop

<abstract><p>This paper investigates rational solutions of an extended Camassa-Holm-Kadomtsev-Petviashvili equation, which simulates dispersion's role in the development of patterns in a liquid drop, and describes left and right traveling waves like the Boussinesq equation. Through its bilinear form and symbolic computation, we derive some multiple order rational and generalized rational solutions and analyze their dynamic features, such as the connection between rational solution and bilinear equation, scatter behavior, moving path, and exact location of the soliton. The obtained solutions demonstrate two wave forms: multi-lump and multi-wave that consist of three, six and eight lump waves or two, three and four line waves. Moreover, different from the multi-wave solitons, stationary multiple dark waves are presented.</p></abstract>

Bilinearization-reduction approach to integrable systems

The paper is a review of the bilinearization-reduction method which provides an approach to obtain solutions to integrable systems. Many integrable coupled systems can be bilinearized and their solutions are presented in terms of double Wronskians (or double Casoratians in discrete case). The bilinearization-reduction method is based on bilinear equations and solutions in double Wronskian/Casoratian form. For those integrable equations that are reduced from coupled systems, one can first solve the unreduced coupled system, obtaining their solutions in double Wronskian/Casoratian form, then, implement suitable reduction techniques, so that solutions of the reduced equation can be obtained as reductions of those of the unreduced coupled system. The method proves effective in solving not only classical integrable equations but also the nonlocal ones. The so-called nonlocal integrable equations were introduced by Ablowitz and Musslimani via reductions with reverse-space (or reverse-time, or reverse-space-time). Note that this method particularly provides a convenient bilinear approach to solve nonlocal integrable systems. In this review, the nonlinear Schrödinger hierarchy and the differential-difference nonlinear Schrödinger equation are employed as demonstrative examples to elaborate this method. These two examples will be pedagogically helpful in understanding the reduction technique. The reduction is implemented by imposing suitable constraints on the basic column vectors of the double Wronskian/Casoratian. Realizations of the constraints are converted to solve a set of matrix equations which varies with the constraints. Special solutions of the matrix equations are provided, which are also helpful in understanding the eigenvalue structure of the involved spectral problems corresponding to the considered equations. Other examples include the Fokas-Lenells equation and the nonlinear Schrödinger equation with nontrivial background. Since many nonlinear equations with physical significance are integrable as reductions of integrable coupled systems, the paper provides a review as well as an introduction about the bilinearization-reduction method that can be used to solve these nonlinear integrable models.

High-order rational-type solutions of the analogous (3+1)-dimensional Hirota-bilinear-like equation.

In this article, a new dynamical system equation named the (3+1)-dimensional Hirota-bilinear-like equation (HBLE) was constructed. The generalized Hirota bilinear method was applied to obtain this new HBLE in (3+1) dimensions. This new HBLE possesses a similar bilinear form to the original (3+1)-dimensional Hirota bilinear equation, but with additional nonlinear terms. A set of high-order rational solutions is constructed for the given equation, generated from polynomial solutions to the associated generalized bilinear equation. The analyticity conditions of the resulting solutions were investigated and six groups of general solutions were derived. In addition, the shape and surface of the high-order rational function solutions and their dynamic behaviors were studied by utilizing Maple.

Least-square Fitting on the Circle using Optimal Control

Filtering of circular data from noisy measurements is well known to be a hard problem because of the ambiguity of the wrapped phase and further complicated by the constraint that filtered states need to be on the circle. Probabilistic models focus on specifying the dynamics of the random phase and then estimate these states recursively using Bayes’ formula, while deterministic approaches normally define a cost function containing the error between the states and measurements and then minimize it over all allowed state paths. In this paper, we construct a deterministic filter on the circle by minimizing the least square error based on Pontryagin's minimum principle, where the optimal state trajectory is described by a bilinear differential equation with deterministic optimal control input. The effectiveness of the proposed filter is shown through numerical experiments.

International Stability under Iterated Sanctions and Counter-sanctions

In this paper, some discrete mathematical models for dynamics of two antagonistic parties (opponents) under iterated sanctions and counter-sanctions are proposed. The modeling approach is inspired by the Osipov-Lanchester bilinear model for warfare. The criteria for model stability are derived both for the full information case and for the stochastic and nonlinear uncertainty case. The controlled version of the model described by bilinear difference equations is proposed, and a statement concerning controllability of the model by small controlling actions is formulated. The results are interpreted in terms of international stability preservation. The risks of global instability caused by further increase of sanctions intensity are formulated. A possible way of international stabilization control by small actions based on mutual trust of parties is proposed and discussed.

A new (3+1) dimensional Hirota bilinear equation: Painlavé integrability, Lie symmetry analysis, and conservation laws

This study's subject is a (3 + 1) dimensional new Hirota bilinear (NHB) equation that appears in the theory of shallow water waves. We investigate how particular dispersive waves behave in an NHB equation. In this regard, we first test the Painlavé integrability using the WTC-Kruskal method and second perform Lie point symmetry analysis on NHB. The algorithm outputs the Lie algebra, the symmetry reductions, and group invariant solutions. Using Lie point symmetries, NHB transforms into an ordinary differential equation. The integration architectures to solve this equation are the Bernoulli sub-ODE, 1/ G, and modified Kudryashov methods. We plot their graphs and observe how the solutions behaved in understanding the physical phenomenon. Additionally, we discuss the model utilizing nonlinear self-adjointness and Ibragimov's approach to generate conservation laws for each Lie symmetry generator.

BDF2 schemes for optimal parameter control problems governed by bilinear parabolic equations

SummaryIn this paper, BDF2 schemes combined with the finite element method are proposed to solve control constrained optimal parameter control problems governed by bilinear parabolic equations, where the control variable acts as a reaction coefficient. The discretize‐then‐optimize (DO) approach is adopted. First, the state equation is discretized and two different discrete formulas are utilized for the objective functional. Then, by using the Lagrange multiplier method, two fully discrete first‐order necessary optimality systems are deduced and named as DO1 and DO2 systems, respectively. It can be found that the optimize‐then‐discretize (OD) and discretize‐then‐optimize (DO) approaches coincide in the case of the DO2 system. Furthermore, a priori error estimates are derived for the state, co‐state and control variables of the DO2 system, which show that the DO2 system has second‐order accuracy in time. Finally, three numerical examples are presented to illustrate the theoretical findings.

Integrability features of a new (3+1)-dimensional nonlinear Hirota bilinear model: multiple soliton solutions and a class of lump solutions

PurposeThis paper aims to propose a new (3+1)-dimensional integrable Hirota bilinear equation characterized by five linear partial derivatives and three nonlinear partial derivatives.Design/methodology/approachThe authors formally use the simplified Hirota's method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space.FindingsThe Painlevé analysis shows that the compatibility condition for integrability does not die away at the highest resonance level, but integrability characteristics is justified through the Lax sense.Research limitations/implicationsMultiple-soliton solutions are explored using the Hirota's bilinear method. The authors also furnish a class of lump solutions using distinct values of the parameters via the positive quadratic function method.Practical implicationsThe authors also retrieve a bunch of other solutions of distinct structures such as solitonic, periodic solutions and ratio of trigonometric functions solutions.Social implicationsThis work formally furnishes algorithms for extending integrable equations and for the determination of lump solutions.Originality/valueTo the best of the authors’ knowledge, this paper introduces an original work with newly developed Lax-integrable equation and shows new useful findings.