Abundant closed-form wave solutions and dynamical structures of soliton solutions to the (3+1)-dimensional BLMP equation in mathematical physics

Schin Kumr , , Amit Kumr

a Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110 0 07, India

b Department of Mathematics, Sri Vankateswara College, University of Delhi, Delhi 110021, India

ABSTRACT The physical principles of natural occurrences are frequently examined using nonlinear evolution equa- tions (NLEEs).Nonlinear equations are intensively investigated in mathematical physics, ocean physics, scientific applications, and marine engineering.This paper investigates the Boiti-Leon-Manna-Pempinelli (BLMP) equation in (3+1)-dimensions, which describes fluid propagation and can be considered as a non- linear complex physical model for incompressible fluids in plasma physics.This four-dimensional BLMP equation is certainly a dynamical nonlinear evolution equation in real-world applications.Here, we im- plement the generalized exponential rational function (GERF) method and the generalized Kudryashov method to obtain the exact closed-form solutions of the considered BLMP equation and construct novel solitary wave solutions, including hyperbolic and trigonometric functions, and exponential rational func- tions with arbitrary constant parameters.These two efficient methods are applied to extracting solitary wave solutions, dark-bright solitons, singular solitons, combo singular solitons, periodic wave solutions, singular bell-shaped solitons, kink-shaped solitons, and rational form solutions.Some three-dimensional graphics of obtained exact analytic solutions are presented by considering the suitable choice of involved free parameters.Eventually, the established results verify the capability, efficiency, and trustworthiness of the implemented methods.The techniques are effective, authentic, and straightforward mathematical tools for obtaining closed-form solutions to nonlinear partial differential equations (NLPDEs) arising in nonlinear sciences, plasma physics, and fluid dynamics.

Keywords: Closed-form solutions Generalized exponential rational function method Solitons Generalized Kudryashov method Solitary waves

1.Introduction

The study of nonlinear partial differential equations (NPDEs) is used to describe nonlinear complex phenomena in diverse fields of nonlinear sciences, such as plasma physics, ocean physics, fluid dynamics, hydrodynamics, marine engineering, and many more.Many researchers and mathematicians use numerical and analyti- cal methods to study nonlinear PDEs and construct closed-form so- lutions, travelling wave solutions, solitary wave solutions, and the dynamics of solitary wave solutions.Exact closed-form solutions allow us to understand the characteristics and features of solitary wave solutions of complex physical phenomena and the dynam- ical behavior of established results for higher-dimensional nonlin- ear evolution equations.Many powerful methods for examining ex- act closed-form solutions of NPDEs have been introduced in recent decades by utilizing symbolic computation, such as the Hirota bi- linear technique, the Darboux transformations technique, the im- proved F-expansion technique, the solitary wave ansatz technique, the simple equation technique, the Lie symmetry method, the extended Tanh method, the unified method, and the extended Kudryashov method, and so on [1–42] .

This work focuses on the study of the (3+1)-dimensional Boiti- Leon-Manna-Pempinelli (BLMP) equation, which was proposed by Darvishi et al.[43]

whereu=u(x,y,z,t).This equation is an expansion of the (2+1) dimensional equation that represents the (3+1)-dimensional inter- action of the Riemann wave propagating along two spatial dimen- sions,x-axis, andy-axis [49,50] .Darvishi et al.[43] , who used multiple exp-function methods to study the double-waves, single- waves, and multiple solution solutions for the (3+1)-dimensional BLMP equation.Using the Painleve-B äcklund transformation, Xu [44] obtained four types of lump-kink solutions, including two N exponential functions and quadratic functions.Tang and Zai [45] also studied the BLMP equation via the multiple exp- function method and the scale transformation method.Ma et al.[46] derived the exact three-wave solutions via the Hirota bilinear method.Furthermore, Ma and Bai [47] derived rational solutions and soliton solutions, which included positions and the Wronskian technique.Tang and Zai [48] , who used the extended homoclinic test approach to obtain exact solutions, including periodic soliton solutions, kinky periodic solitary-waves, and kink waves.Liu et al.[49] constructed three-waves soliton solutions in terms of kinky periodic wave solutions, kink solutions, and periodic form solu- tions using Hirotas bilinear method.Mabrouk et al.[50] obtained new exact analytical solutions employing the group transformation method.

The primary motivation for this research is to obtain closed- form solutions to the Boiti-Leon-Manna-Pempinelli (BLMP) equa- tion using two efficient methods: the generalized exponential rational function (GERF) technique [51–55] and the generalized Kudryashov method [59–61] .As explained by wave transforma- tions via symbolic computations, the aforementioned methods are very efficient, robust, and straightforward.Some families of closed- form solutions/exact solutions and rational form function solutions of the BLMP equation are investigated physically and analytically with the help of soft symbolic computation via Wolfram Mathe- matica.The newly closed-form solutions are demonstrated in the different shapes of dark-bright solitons, oscillating periodic soli- tons, singular solitons, combo solitons, and solitonic interactions between singular soliton forms with kink waves, periodic solitons, and kink singular waves.These newly established results are com- pletely novel and have never been discussed in literature.More- over, the diverse range of characteristics, dynamics, and appropri- ate free known constants of these new solutions are displayed us- ing three-dimensional illustrations, with the suitable preference of the involved rational functions, trigonometric functions, hyperbolic functions, and other constant parameters.Investigation of such types of closed-form solutions is extremely valuable in the field of advanced research and development.

The generated closed-form solitary wave solutions in this article are expressed in terms of hyperbolic and trigonometric functions, as well as exponential rational functions with arbitrary constant parameters.The dynamics of various types of closed-form solu- tions of the Boiti-Leon-Manna-Pempinelli (BLMP) equation are im- portant in nonlinear sciences, plasma physics, fluid dynamics, and marine engineering.Furthermore, these new solutions may ensure the physical and dynamical behavior of the equation, which aids us in understanding the mechanism of the higher-dimensional nonlin- ear equation.The exact solitary wave solutions in the Soliton the- ory interact with each other without losing their amplitude and velocity.For instance, their identities and shapes do not vary af- ter their mutual interactions.Furthermore, the newly formed so- lutions and their graphical representations show different dynam- ical structures of solitary waves, which is extremely important for learning more about nonlinear partial differential equations (NPDEs) [62–64] .

The article is organized as: In the second section of this ar- ticle, the main important steps of applying the GERF technique and the generalized Kudryashov method, the description, and the basic methodology are discussed.The several closed-form solu- tions of the BLMP equation are obtained in sections 3 and 4 us- ing the proposed methods.In Section 5 , we have given a physical and graphical representation of some obtained solutions via three- dimensional graphics.These graphics can assist us with a better understanding of their dynamical structures.Finally, the conclud- ing remarks are given in the last section.

2.Methodology

2.1. The GERF method

Ghanbari and Inc [51] introduced the generalized exponential rational function (GERF) method, also known as the integration method, to obtain the exact closed-form solutions to the nonlinear Schrödinger equation.The GERF method has recently been used to solve a variety of nonlinear wave equations [52–57] .A brief methodology of the GERF method for obtaining the exact wave so- lutions of nonlinear PDEs is stated in this section.

Let us consider nonlinear PDEs in the following form

whereFis the function ofu(x1,x2,x3,t)and its partial derivatives.The following main steps of the GERF method are framed as

Step 1:

We use the wave transformations,

whereX=a1x1+a2x2+a3x3+btanda1,a2,a3andbare ar- bitrary constants.Substituting (3) into (2) , we obtain an ordinary differential equation as

whereΨ′denotes the derivative ofΨwith respect to X.

Step 2:

The solitary wave solutions of the Eq.(4) is given by

where the positive integerNcould be determined by applying the homogeneous balancing principle andΠ(X)is given as

whereδ1,δ2,δ3,δ4andξ1,ξ2,ξ3,ξ4are real (or complex) numbers.

Step 3:A collection of algebraic equations can be determined by balancing the coefficient ofΨ(X)and its derivative and set- ting them to zero using (5) and (4) , as well as (6) .The parametersP0,Pn, andQncan be evaluated by nonlinear algebraic equations using the Mathematica software, and then the desired solutions to the NPDEs can be determined by substituting all the values in the main equation.

2.2. The generalized Kudryashov method

In this section, the brief methodology of the generalized Kudryashov method for extracting consistent solutions to NPDEs is discussed.The generalized Kudryashov method has been used to obtain consistent exact analytic solutions to complex nonlinear evolution equations [58–62] .We start with an NPDE with the in- dependent variables(x1,x2,x3,t)as shown below.

The given nonlinear PDEs (7) can be transformed into the ordi- nary differential equation with respect to the new variablesξby using the following wave transformationu(x,y,z,t)=G(ξ)whichξ=κx+μy-νt+λz:

for which, we observe exact analytic solution of ordinary Eq.(8) in the following form

wherepi,qjare constant-parameters to be obtained (pi0,qj0 ) andΦ(ξ)is the solution of the Bernoulli’s differential equa- tion

which is written as

Based on the homogeneous balance principle, the positive numbersLandMin (9) can be computed by balancing the degrees among the nonlinear terms and most leading order derivatives occurring in Eq.(8) .We obtain a polynomial involving the powers ofΦ(ξ)by applying (9) to (8) and (10) .By balancing the coefficients ofΦ(ξ)and setting them to zero, an algebraic system can be determined.The unknown parameterspi(i= 1,2,L),qj(j= 1,2,M)can be eval- uated using Mathematica, and the desired solutions to the NPDE (7) are constructed.

3.Application of the GERF method to BLMP equation

The prime objective of this section is to acquire the exact ana- lytic solutions of BLMP (1) .Let us assume wave transformation

whereκ,μ,νandλare arbitrary constants.Putting the value ofuwithXfrom (12) to (1) , we get the reduced equation

Furthermore, we assume that above reduced Eq.(13) has the fol- lowing solution:

The closed-form wave solutions of the equation can be obtained in the following cases.We continue with the previous section’s method, employing the GERF method to obtain closed-form wave solutions.

Family 1:

Taking [δ1;δ2;δ3;δ4] = [1 ;-1 ;1 ;1] and [ξ1;ξ2;ξ3;ξ4] = [1 ;-1 ;1 ;-1] , Eq.(6) transform into

Π(X)= tanh(X)

We solve algebraic equations with the mathematical software Mathematica Wolfram, and then we get the constant parameter values as

Case 1.1

Substituting all the value from (15) in (14) , one receives

Consequently, the achieved exact analytic solutions of the gov- erning BLMP

Case 1.2

Substituting all the values from (17) in (14) , one receives

Ψ(X)=P0-2κtanh(X).

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 1.3

Substituting all the value from (19) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Family 2:

Taking [δ1;δ2;δ3;δ4] = [i;-i;1 ;1] and [ξ1;ξ2;ξ3;ξ4] = [i;-i;i;-i] , Eq.(6) transform into the following form

Π(X)= -tan(X)

Case 2.1

Substituting all the value from (21) in (14) , one receives

Ψ(X)=P0+ 2κtan(X).

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 2.2

Substituting all the value from (23) in (14) , one receives

Consequently, the established exact analytic solutions of the gov- erning BLMP

Case 2.3

Substituting all the value from (25) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Family 3:

Taking [δ1;δ2;δ3;δ4] = [1 +i; 1 -i; 1 ; 1] and [ξ1;ξ2;ξ3;ξ4] = [i;-i;i;-i] , Eq.(6) transform into

Π(X)= 1 -tan(X)

Case 3.1

Substituting all the values from (27) in (14) , one receives

Ψ(X)=P0-2κ(1 -tan(X)).

Consequently, the established exact solitary wave solutions of the governing BLMP

Case 3.2

Substituting all the values from (29) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 3.3

Substituting all the values from (31) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Family 4:

Taking [δ1;δ2;δ3;δ4] = [2 +i; 2 -i; 1 ; 1] and [ξ1;ξ2;ξ3;ξ4] = [ -i;i;-i;i] , Eq.(6) transform into the following form

Π(X)= 2 + tan(X)

.

Case 4.1

Substituting all the values from (33) in (14) , one receives

Ψ(X)=P0+ 2κ(2 + tan(X)).

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 4.2

Substituting all the values from (35) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Family 5:

Taking [δ1;δ2;δ3;δ4] = [1 -i; -1 -i; -1 ; 1] and [ξ1;ξ2;ξ3;ξ4] = [i; -i;i; -i] , Eq.(6) convert into the fol- lowing form

Π(X)= -1 + cot(X)

Case 5.1

Substituting all the values from (37) in (14) , one receives

Ψ(X)=P0-2κ(cot(X)-1).

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 5.2

Substituting all the values from (39) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Family 6:

Taking [δ1;δ2;δ3;δ4] = [ -1 -i; 1 -i; -1 ; 1] and [ξ1;ξ2;ξ3;ξ4] = [i; -i;i; -i] , Eq.(6) transform into the following form

Π(X)= 1 + cot(X)

Case 6.1

Substituting all the values from (41) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 6.2

Substituting all the values from (56) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Family 7:taking [δ1;δ2;δ3;δ4] = [1 ; 2 ; 1 ; 1] and [ξ1;ξ2;ξ3;ξ4] = [1 ;0 ;1 ;0] , Eq.(6) shows

Case 7.1

Substituting all the values from (46) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 7.2

Substituting all the values from (48) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Family 8:

Taking [δ1;δ2;δ3;δ4] = [2 ;3 ;1 ;1] and [ξ1;ξ2;ξ3;ξ4] = [1 ;0 ;1 ;0] , Eq.(6) transform into the following form

Case 8.1

Substituting all the values from (50) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 8.2

Substituting all the values from (52) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Family 9:

Taking [δ1;δ2;δ3;δ4] = [2 ;1 ;1 ;1] and [ξ1;ξ2;ξ3;ξ4] = [1 ;0 ;1 ;0] , Eq.(6) transform into the following form

Case 9.1

Substituting all the values from (54) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 9.2

Substituting all the values from (56) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Family 10:

Taking [δ1;δ2;δ3;δ4] = [ -2 -i; 2 -i; -1 ; 1] and [ξ1;ξ2;ξ3;ξ4] = [i; -i;i; -i] , Eq.(6) transform into the following form

Π(X)= 2 + cot(X)

Case 10.1

Substituting all the value from (58) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

Case 10.2

Substituting all the value from (61) in (14) , one receives

Consequently, the achieved exact analytic solutions of the govern- ing BLMP

4.Application of the generalized Kudryashov method to BLMP equation

In order to investigate some novel exact wave solutions, we choose new wave transformation for the nonlinear PDEs (1) is de- scribed as:

whereν1refer the velocity of the soliton which transform the NPDEs (1) into an ordinary differential equation given below:

we get the following relation using homogeneous balancing as:

Employing the balancing principal in Eq.(65) , we haveM= 1 and the relation (66) revealsL= 2 .Therefore, Eq.(9) transform in the following form as

whereinp0,p1,p2,q0, andq1are arbitrary parameters.By mak- ing use of the solution (67) in (65) along with Bernoulli Eq.(10) , yield a polynomial inΦ(ξ).Picking the coefficient of same power ofΦ(ξ)and fixing them to zero, furnish the consequential alge- braic system.

Set 1:

p0 =p2=and the remaining pa- rameters are free.

After inserting all the above parameters into (67) provides

Hence, the leading exact analytic solutions of the BLMP (1)

After inserting all the above parameters into (67) provides

Hence, the leading exact analytic solutions of the BLMP (1)

Set 3:

After inserting all the above parameters into (67) provides

Hence, the leading exact analytic solutions of the BLMP (1)

After inserting all the above parameters into (67) provides

Hence, the leading exact analytic solutions of the BLMP (1)

After inserting all the above parameters into (67) provides

Hence, the leading exact analytic solutions of the BLMP (1)

After inserting all the above parameters into (67) provides

Hence, the leading exact analytic solutions of the BLMP (1)

5.Results and discussion

The following section discusses the graphical representation of newly formed results as well as the effect of free known constant parameters on them.To extract exact wave solutions from the gov- erning higher-dimensional model (1) , two powerful integration ap- proaches are used, one of which is the GERF method, and the other is the generalized Kudryashov method.These closed-form solutions provide different solitonic wave structures, like periodic solitons, multi-wave solitons, interaction between multiple solitons with a kink wave profile, periodic wave solitons and combo solitons so- lutions.To explain the physical interpretation of the established closed-form solutions through 3D postures by choosing the differ- ent values of free known-constant parameters, the observations are given in the following manner:

Fig.1 : The figure depicts two distinct localized excitation wave structures for the solution (20) for the appropriate values of the parametersP0= 23,κ= 1.3,μ= 1.1,λ= 1.3,andz= 21.Fig.1 (a) shows the interaction between multisoliton and kink wave profile att= 0.21 ; aftert= 0.38 , the dynamical structure of kink wave into lump-ptype wave profile is shown in 1 (b).We ob- serve that the multi-wave solitons with kink waves smoothly prop- agate in the positive direction of thexandyaxes as timetgoes on.

Fig.1.The closed-form solution (20) shows new solitary waves in three dimensions.

Fig.2 : A periodic multisoliton wave profiles have been investi- gated for the solution (26) via two different perspective views with time variation att= 1 andt= 3 .The periodic solitons have a par- ticularly localized formation with a temporally periodic structure, which is important when one reflects on the localization-type ap- pearances in the theory of solitons.The Fig.2 is sketched by con- sidering the appropriate values of parametersP0= 10,κ= 7,μ= 0.3,λ= 1.2,andz= 17.

Fig.2.The closed-form solution (26) shows new solitary waves in three dimensions.

Fig.3 : The graphical representation of this figure in two dif- ferent 3D plots reflects the multisoliton wave profiles in thex-yplanes for the solution (30) .The dynamical structures show that velocities, shapes, and the amplitudes of the multiple soliton wave profiles are kept invariant during the appropriate values of the pa- rametersP0= 2,κ= 1,μ= 0.3,λ= 0.2,andz= 11.

Fig.3.The closed-form solution (30) shows new solitary waves in three dimensions.

Fig.4 : This graph depicts the graphical behavior of the ob- tained solution (34) with respect to timet.Att= 1 andt= 2.1 , the multisoliton wave profiles were observed.The amplitude, shapes and velocities do not change in the multisoliton wave profile.The graphs are plotted for the appropriate parameter valuesP0= 2,κ= 1.2,μ= 9.63,λ= 2.5,andz= 36.

Fig.4.The closed-form solution (34) shows new solitary waves in three dimensions.

Fig.5 : The graphical structures in this figure depict the solved (57) with the distinct timetvalues.The periodic multisoliton Lump-type wave profiles has been pointed att= 0.001 andt= 8 .To find the best dynamical views of the graph, we use the appro- priate values of the parametersP0= 13,κ= 1,μ= 0.003,λ= 6,andz= 3.

Fig.5.The closed-form solution (57) shows new solitary waves in three dimensions.

Fig.6 : The graphics of the analytic solution (68) show multi- soliton wave profiles.The periodic oscillating multi-soliton profile has been observed att= 1 and it annihilated aftert= 98 .We dis- covered that periodic solitons travel in the positive direction of the x and y axes and maintain their shape while moving.The appropri- ate parameter values are taken asp1= 5,η= -ν1= 1,μ1= 0.2,λ1= 1.022,q0= 12,q1= 3,z= 11 for numerical simulation.

Fig.6.The closed-form solution (68) shows new solitary waves in three dimensions..

Fig.7 : The graphical representation of the obtained solution (69) withrespecttotimetisshowninthisfigure.TheLump-form wave profiles have been observed att= 1 andt= 1.6 .We also no- ticed that the lump-type solitary wave travels along the positive direction of the x and y-axes, and maintains its shape unchanged for the moment.The 3D-plots are exhibited for the appropriate pa- rameter valuesp1= 13,η= 5,ν1= 11,μ1= 0.1,λ1= 6,q0= 1,q1= -4,z= 1.6 .

Fig.7.The closed-form solution (69) shows new solitary waves in three dimensions.

Fig.8 : The annihilation wave structure has been recorded for the solution (70) in this graph.Aftert= 10 , the oscillating peri- odic wave profile was converted into a stationary wave profile by adjusting the parametersp1= 3,p2= 2,η= -2,κ1= 1.1,μ1= 1.2,λ1= 1.033,q0= 1,z= 2.6 .

Fig.8.The closed-form solution (70) shows new solitary waves in three dimensions..

6.Conclusion

In summary, we have obtained abundant closed-form solutions to the higher-dimensional BLMP equation, which is one of the most important physical models in plasma physics, ocean physics, opti- cal physics, fluid dynamics, marine engineering, nonlinear sciences, and engineering physics.Exact closed-form solutions always give a detailed explanation of the dynamical behaviors of the most under- studied complex physical phenomena.On the other hand, for some more complicated equations, seeking such types of solutions is ex- tremely difficult and even not possible in several situations.In this work, we have derived closed-form solitary wave solutions for a (3+1)-dimensional nonlinear BLMP equation by utilizing two ef- ficient mathematical techniques, namely the GERF technique and the generalized Kudryashov method.In the dynamics of closed- form solutions, the acquired exponential rational-form solutions are also very supportive and encouraging.Consequently, soliton so- lutions, bright-dark solitons, singular solitons, combo singular soli- tons, breather-form solitons, kink singular solitons, and computa- tional soliton solutions and other soliton solutions of the BLMP equation are obtained in various forms.Several of the solutions obtained are expressed in terms of hyperbolic, trigonometric, and exponential rational functions with arbitrary constant parameters, which differ significantly from the author’s work [64] .The effec- tiveness, dependability, and simplicity of current techniques are supported by computerized symbolic computational work.These techniques can be constructively used in many nonlinear evolution equations that arise in nonlinear sciences, mathematical physics, and marine engineering.

6.1. Future scope

For the higher-dimensional nonlinear BLMP Eq.(1) , we stud- ied the newly obtained solutions using the GERF method and the generalized Kudryashov method.The solutions that have been re- ported by the GERF process are available in various ways, such as trigonometric functions, trigonometric and hyperbolic functions and rational solutions that employ the best value of constant ar- bitrary parameters.The generalized Kudryashov method, on the other hand, yields hyperbolic form and trigonometric form solu- tions.All of the different types of general solutions and closed- wave solutions were obtained for the first time by successfully ap- plying the GERF method and the generalized Kudryashov method to the higher-dimensional BLMP equation, which has been very popular in plasma physics and fluid dynamics in recent years.One of the most compelling reasons for choosing applied methods is that they may provide a rich variety of analytical solutions for certain coefficient and power values in the solution function.The most intriguing applications of the BLMP equation will be bene- ficial to young researchers who want to investigate our analytical results in applied mathematics, plasma physics, nonlinear sciences, fluid dynamics, and other disciplines.In future investigations, the generalized Kudryashov method and the GERF technique can be applied to nonlinear systems and nonlinear evolution equations that mathematically model many phenomena in ocean physics and marine engineering and contribute to the physical interpretation of the obtained solutions.

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

Acknowledgments

Under the project scheme MATRICS (MTR/2020/0 0 0531), the Science and Engineering Research Board, SERB-DST, India is fund- ing this research.Sachin Kumar, the author, has received this re- search grant.