Abstract
The hyperbolic nonlinear Schrödinger equation in the (3 + 1)-dimension depicts the evolution of the elevation of the water wave surface for slowly modulated wave trains in deep water. Many researchers have studied the applicability and practicality of this model, but the analytical approach has been virtually absent from the literature. We adapted the lie symmetry analysis method to obtain a new complex solution in this work. The obtained complex solution contains bright and dark solitons. Furthermore, modulation instability is applied to this model to explain the interplay between nonlinear and dispersive effects. As a result, the modulation instability condition and the explosive rate are also discussed.
1. Introduction
Hyperbolic partial differential equations arise in many different branches of science and engineering. For instance, they appear in numerous different contexts, including porous flow, surface water flow, landslide, faulting, fluid dynamics, circled fuel reactor, high-temperature hydrodynamics, electrodynamics, and electrodynamics. They also appear in transmission lines [1] and optics [2, 3].
The nonlinear Schrödinger equation is a key mathematical model that drives a wide range of scientific fields. It is integrable in one spatial dimension and can be solved exactly using the inverse scattering transform [4]. However, regardless of the type of diffraction operator, no exact solutions have been identified in two or three dimensions.
The hyperbolic nonlinear Schrödinger equation in (3 + 1)-dimensionwill be explored in this study, where the complex field is described by ; the space variables are , and , while the temporal variable is . As many important equations, the present one is used to describe the dynamics of optical soliton promulgation in mono-mode optical fibers. Also, in hydrodynamics, the present equation is used to describe the evolution of the elevation of the water wave surface for slowly modulated wave trains in deep water.
There is a great deal of literature about analytical and numerical solution methods for solving hyperbolic Schrödinger equation. The Jacobi elliptic solution, exponential function solutions, abundant exact solutions, and optical soliton solutions for the (3 + 1)-dimensional and (2 + 1)-dimensional Schrödinger equations are investigated by the various authors in [5–8]. The numerical solutions of the hyperbolic equations with the help of variable space operator, finite difference method, and spline method are established in [9–11]. The exact X-wave solutions of the (3 + 1)-dimensional nonlinear hyperbolic Schrödinger equation for supporting potential are reported by the authors in [12]. A class of bright-dark soliton solutions of the Schrödinger equation is obtained in [13]. The integral method presented in [14] for solving nonlinear Schrödinger equation in higher dimension. With the rapid advancement in the field of higher-dimensional hyperbolic Schrödinger equation, different new types of solutions have been obtained theoretically. This includes, for example, optical soliton [15], exact solution [16], novel complex wave solution [17], and optical solitary waves [18]. The theory of bifurcations of dynamical systems to study the dynamical behavior of traveling wave solutions has been given in [19]. The -expansion function methods and the extended sinh-Gordon equation expansion are used to construct various optical soliton and other solutions [20] to equation (1).
Motivated by the reason, (3 + 1)-dimensional hyperbolic Schrödinger equation (1) modulates a complex physical phenomenon which is useful for a better understanding of the fields, optics, and hydrodynamics. In this work, we are going to investigate the solution of equation (1) with the help of the Lie symmetry method [21, 22].
The work in this paper is discussed with the help of the following sections. Section 2 is devoted to the Lie symmetry analysis of the hyperbolic Schrödinger equation with the help of prolonged infinitesimals. In this section, we obtained the Lie symmetries, which play a key role in the reduction of the hyperbolic Schrödinger equation into (2 + 1)-dimensional equations. Furthermore, Section 2 also established the symmetries of the (2 + 1)-dimensional equation. These symmetries are further utilized to obtain a reduced (1 + 1)-dimensional equation in this section. In Section 3, solutions for the hyperbolic Schrödinger equation are obtained with the help of a reduced ordinary differential equation. Section 4 discussed the condition of stability using modulation instability (MI) analysis. Section 5 is devoted to the concluding remarks.
2. Lie Symmetry Analysis for Hyperbolic Schrödinger Equation
The purpose of this Section 2 is to establish the invariance condition for hyperbolic Schrödinger equation in [23]. This invariant condition plays a crucial role in the Lie symmetry analysis method. The first step in this direction is to use a complex functionto separate the real and imaginary components of equation (1). This complex function (2) converted equation (1) into the system of equations as
The one-parameter Lie group of transformations [24] of equation (3) is defined as follows:where is the parameter of the group. Furthermore, the identity element of the one-parametric Lie group (4) is defined by . Now, expanding (4) by Taylor series expansion about the identity element up to the terms of first order only, with the help of the initial conditions
We have the approximations of (4) up to the terms of first order only, as follows:
The operator in equation (6) is a differential operator that is given by the following expression as
The operator defined with the help of equation (7) plays the role of a vector field for the one-parametric group (4). Consequently, with the assistance of this operator (infinitesimal generator, symmetry generator), one can physically understand the behavior of hyperbolic Schrödinger equation’s solutions in the new coordinate system. The second-order prolongation of this useful operator is as follows (for more information, see [24, 25]):where the coefficients and in second-order prolongation formula are called the prolonged infinitesimals. These prolonged infinitesimals contain all the facts about the dependent variables of governing equation (for more details, see [26]). Now, the invariance condition, which is derived for equation (3) with the help of Lie group of transformations (4), can be formulated as follows:wherewith
2.1. Lie Symmetries of the Hyperbolic (3 + 1)-Dimensional Schrödinger Equation
In this subsection, our primary goal is to generate the Lie symmetries of the hyperbolic Schrödinger equation. Now, achieving this goal, invariant condition plays a crucial role in the Lie symmetry analysis method. Furthermore, under the Lie group of transformations (4), the invariant condition of equation (3) takes the form as follows:
Now, to obtain the infinitesimals for the hyperbolic nonlinear Schrödinger equation (3) with the help of invariant condition (12), the following steps are required.(1)In this step, substitute the value of extended infinitesimals into the invariant condition (12).(2)After substitution, collect the coefficient of different differentials. Furthermore, in this step, equate these coefficients equal to zero.(3)Consequently, step 2 produces a large system of linear partial differential equations in the coordinates , , , , , and .(4)Furthermore, this linear system of partial differential equations produced the solution in form of the functions , , , , , and as follows:where are arbitrary constants.
Theorem 1. The five-dimensional Lie-algebra for hyperbolic nonlinear Schrödinger equation (3) is admitted by the following five generators:
2.2. Reduced (2 + 1)-Dimensional Equation under the Subalgebra
In the quest for reduced equation [], [27] under this subalgebra, we start with the characteristic equation
The solution of the characteristic equation (15) corresponding to this subalgebra gives the similarity solutions and similarity variables for (3 + 1)-dimensional hyperbolic Schrödinger equation (3) as follows:
Therefore, (3 + 1)-dimensional hyperbolic equation (3) can be reduced into the (2 + 1)-dimensional hyperbolic partial differential equation with suitable transformation (16) as follows:
2.3. Lie Symmetry Analysis of Reduced (2 + 1)-Dimensional equation (17)
Furthermore, to get the reduced (1 + 1)-dimensional hyperbolic equation from the (2 + 1)-dimensional hyperbolic equation (17), we apply the Lie symmetry method to equation (17). Therefore, the one-parameter Lie group of transformations of equation (17) can be defined as follows:where is the parameter of the group. Furthermore, the identity element of the one-parametric Lie group (18) is defined by . Now, expanding (18) by Taylor series expansion about the identity element up to the terms of first order only, with the help of the initial conditions,
Furthermore, the Lie symmetry analysis method’s procedure is adopted as mentioned in Section 2. The infinitesimals for equation (17) with the help of a one-parametric Lie group (18) can be written as follows:where are arbitrary constants.
Theorem 2. The four-dimensional Lie-algebra for hyperbolic nonlinear Schrödinger equation (17) is admitted by the following four generators:
2.4. Reduced (1 + 1)-Dimensional Equation for Equation (17) under the Subalgebra
In the quest for a reduced (1 + 1)-dimensional equation under this subalgebra, we start with the characteristic equation
The solution of the characteristic equation (22) corresponding to this subalgebra gives the similarity solutions and similarity variables for the hyperbolic Schrödinger equation as follows:
Therefore, (2 + 1)-dimensional hyperbolic equation (17) can be reduced into the (1 + 1)-dimensional hyperbolic partial differential equation with suitable transformation (23) as follows:
3. Reduced Ordinary Differential Equation and Complex Solution of Hyperbolic Schrödinger Equation
In this section, first of all, the similarity solution and similarity variable are established. Furthermore, with the help of these similarity solutions and similarity variables, equation (24) is reduced to the ordinary differential equations [28]. This reduced equation is further utilized to find out the solution to the main equation (1). Consequently, adopting the procedure of the Lie symmetry method mentioned in the above sections, the similarity variable and similarity solutions for equation (24) in the form of trigonometric function can be written as
Now, by substituting the above transformation into equation (24), we obtained the reduced ordinary differential equation for equation (3) as follows:
On solving equation (26), the solutions of the above equation (26), as a function of and , take the form as follows:at , where is an arbitrary constant.
Consequently, substituting equation (27) into equations (24), the solution of the hyperbolic Schrödinger equation (24) is given as follows:at , where is an arbitrary constant.
Further, substituting equation (28) into the similarity solutions (23) and using the form of similarity variables and , the solution of the hyperbolic Schrödinger equation (17) is given as follows:where and is an arbitrary constant.
Furthermore, substituting equations (29) into the similarity solutions (16) and using the form of similarity variables , and , solutions of the hyperbolic Schrödinger equation (3) are given as follows:where and is an arbitrary constant.
Finally, by substituting equation (30) into equation (2), the solution of the hyperbolic (3 + 1)-dimensional Schrödinger equation (1) can be written as
4. Modulation Instability of Hyperbolic (3 + 1)-Dimensional Schrödinger Equation
As a result of the interplay between the nonlinear and dispersive effects, many systems of nonlinear partial differential equations exhibit an instability that leads to modulation of the steady state [29]. Modulation instability is a well-known term for this phenomenon [30, 31]. The modulation instability of the hyperbolic (3 + 1)-dimensional Schrödinger equation is discussed in this section. The state solution of the hyperbolic Schrödinger equation (1) is of the typewhere is the incident power, and is the nonlinear phase shift induced by self-phase modulation. The perturbation is examined by utilizing linear stability analysis.
Substituting equation (32) in equation (1) and linearizing in , we obtainwhere indicates the complex conjugate. However, because of the complex conjugate term , the Fourier components at frequencies and are coupled. Therefore,where denotes the wave number and represents the frequency of the perturbation. Furthermore, substituting equation (34) into (33), we obtained the following dispersion relation
The steady-state stability of the dispersion relation (35) depends on the stimulated Raman scattering, group velocity dispersion, and self-phase modulation when
The steady state is stable against the small perturbations.
Furthermore, if
MI occurs. Indeed, the steady-state solution becomes unstable. Indicating that the perturbation at that frequency would grow exponentially along with the fiber with the power gain .
It is easy to see that is the modulation instability explosive rate. We derive that MI exits in the range satisfyingwith .
5. Discussion and Concluding Remarks
In this article, the authors studied the invariant and Lie symmetry analysis of the (3 + 1)-dimensional hyperbolic Schrödinger equation. In the evolution of the research, the first step is to obtain the infinitesimal with the application of the Lie symmetry method to drive the similarity variable. After that, similarity solutions are obtained in a particular form. Finally, the authors made the following concluding remarks about the work:(1)To the best knowledge of the authors, 2D and 3D equations are not solved exactly yet. In this work, we are able to find a particular type of solution of reduced equation (26) for the particular value of ; that is,(2)The reduced (1 + 1)-dimensional equation (24) generates a similarity variable in the form of trigonometric functions. This indicates that the applied method preserves the phenomenon in the correct form generated by the hyperbolic Schrödinger equation.(3)Although the obtained solution in equation (31) is particular one, this solution can play an essential role in choosing the initial data for a numerical scheme to investigate the problem numerically.(4)The depicted solution in Figure 1 clearly demonstrates that the solutions are in good agreement with the phenomenon depicted by the (3 + 1)-dimensional hyperbolic Schrödinger equation.(5)With the help of modulation instability, we successfully derive the range of the instability and stability of the (3 + 1)-dimensional hyperbolic Schrödinger equation.
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Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Vikas Kumar was responsible for original draft preparation. Ram Jiwari was responsible for methodology. Aloev Rakhmatullo Djurayevich was responsible for investigation. Mirzoali Urazaliyevich Khudoyberganov was responsible for review and editing.
Acknowledgments
The work was supported by the Government of India, Department of Science Technology (DST), New Delhi, India, with grant No. INT/UZBEK/P-05.