Lie symmetries and reductions via invariant solutions of general short pulse equation

Abstract

Around 1880, Lie introduced an idea of invariance of the partial differential equation (PDE) under one-parameter Lie group of transformation to find the invariant, similarity, or auto-model solutions. Lie symmetry analysis (LSA) provides us an algorithm to search for point symmetries for solving related linear systems for infinitesimal generators. Actually, point symmetries lead us to one-parameter family of solutions from a known solution. LSA is a program that provides us the exact solutions for the non-linear differential equations (DEs) in analogy of the program designed by Galois for algebraic polynomial equations. In this paper, we have carried out the LSA for computing the similarity solutions (symmetries) of the non-linear short pulse equation (SPE) for the cases when h(u) = e^u, k(u) = u_xx, , and k(u) = u_xx. In addition, an optimal system of one-dimensional sub-algebra has been set up. The reductions and invariant solutions for the generalized SPE are calculated corresponding to this optimal system as well. Reductions reduce the non-linear PDE or system of PDEs into non-linear reduced ordered ODE or system of PDEs. This helps to solve these systems of PDEs to reduced form. Graphical behavior of the transformed points of the 1-parameter solution functions have drawn.

1 Introduction

Galois used the group theory to discuss the solvability of algebraic polynomial equations. Sophus Lie used the same idea foe differential equations and devised a comprehensive program now known as Lie symmetry analysis (LSA). In his attempt, he also developed the theory of Lie groups with broad applications in many areas of mathematics, physics, and in applied sciences [1, 2]. [3] have explained the procedure of symmetry reductions and exact solutions for the non-linear PDEs using three different methods that are direct, classical, and non-classical. [4] used LSA for systems of non-linear PDEs including the solutions, for system of non-linear coupled PDEs in real physical application, for the unsteady liquid and gas flow in long pipelines, for approximated long wave equations in shallow water and for the general dispersive long-wave equation.

Non-linear short pulse equation (SPE) represents the propagation of ultra-short pulses (light pulses) in optical fibers of silica [5]. Propagation of pulses in optical fibers was first depicted by the cubic non-linear Schrodinger equation (NLSE) which are used to provide the actual fiber connections and refer new systems of fiber communication to attain very high data transmission [6, 7]. Research studies on a large scale have been performed for the propagation of ultra-short pulses (very narrow pulses) that permit high quality fast data transmission [6, 8]. In case of short pulses (or ultra-short pulses), the rationality of NLSE lacks due to the breakdown, [9]. Also, the higher order terms that are involved in cubic NLSE cause difficulties, see Figure 1, for the behavior of NLSE as an output [10]. Therefore, determined the SPE which provides more accurate approximation of the solution of Maxwell’s equation in non-linear media rather than the NLSE [6]. The SPE has vast applications in many applied fields such as systems of impulse, mechanics, neural networks, and in many other fields of sciences. Determined the symmetries of SPE and travelling wave solution by parametric representation and power series process, respectively, [11]. Evaluated the symmetry reductions and conservation laws by using the direct method for SPE, [12]. Authors also determined the Lie symmetries for SPE through the non-local system. Established the results for well-posedness of solutions which are bounded for homogenous IBVP and Cauchy problem connected with SPE, [5]. Matsuno constructed multiple exact solutions by using the direct method for three novel PDEs related with generalizations of SPE that are integrable, [13]. He gave the parametric representation of multi-soliton solutions of generalized SPE. LSA has been used by many mathematicians to explore the results related to the exact solutions of non-linear PDEs which depict physical phenomena [14]. Discussed the class of non-linear PDEs having an arbitrary order [15]. Authors estimated the determining equations for non-classical symmetries by using compatibility of original equations with invariant surface conditions.

FIGURE 1

2 Results

In the present section, we give our main results with computations.

2.1 Lie symmetries of SPE for the case of h(u) = e^u and k(u) = u_xx

Eq. 2 becomesConsider the one parameter Lie group of point transformations for Eq. 3.where ϵ ∈ R is the group parameter.The group generator of (4) is defined in the following vector form as:where λ, μ and ν are infinitesimal functions of group variables. The second prolongation of the infinitesimal generator along with coefficients has the following form:where D_x and D_t are the total derivatives.

Apply the second prolongation of the infinitesimal generator Eq. 5 onto Eq. 3. Then, in order to calculate symmetry of Eq. 3, we have the equation of the following form:

Solving this equation

Substitute the values of ν^xx, ν^xt and Eq. 3 which leads to an under-determined system of equations given as:

The solution of the aforementioned determining equations gives the coefficient functions in the form of and are arbitrary constants. Thus, the Lie algebra of the infinitesimal symmetries for the case n = 1 isTheorem 3.1 The set of these generators is closed under the one parameter Lie groups which are generated by infinitesimal generators W_i for i = 1, 2, and 3 are given in the following table. The entries give the transformed points exp(ϵW_i)(x, t, u) = (x*, t*, u*).where ϵ ∈ R is the group parameter.Theorem 3.2 If satisfies Eq. 3, then, u⁽ⁱ⁾(i = 1, 2, and 3) are solutions of Eq. 3:where , (i = 1,2, and 3), ϵ ≪ 1 is any positive number.

The Eq. 13 provides a class of solutions for Eq. 3 depending upon the parameter ϵ and general function where (i = 1, 2, 3). The Figures 2–5 show the graphical view of the functions uⁱ, (i = 1, 2, 3) where uⁱ attained from Lie symmetry groups W_i. These graphs are formed by letting different general functions in place of in Eq. 13. The graphs are constructed from the maple.

FIGURE 2

FIGURE 3

FIGURE 4

FIGURE 5

For first equation in Eq. 13, letting the general trigonometric function in place of along-with ϵ = 0.000005 and abscissa x = −5 to 5, ordinate t = −5 to 5.

Figure 2 shows the graphical behavior of Eq. 15.letting another general value of function . The function becomes

Figure 3 shows the graphical view of Eq. 16.

For second equation of Eq. 13, considering a general function for the same values of ϵ = 0.000005 and aforementioned coordinates for Eq. 14.

Figure 4 shows its graphical view.

For last equation of Eq. 13, we let a general logarithmic function and for similar values of ϵ, x, and t coordinates.

Its graph is in Figure 5.

2.2 Optimal system of subalgebras

In this part, we will find the optimal system of one dimensional Lie subalgebras for Eq. 3 by using the adjoint representation. The corresponding commutator table and the adjoint table are as follows:Commutator Table: Adjoint Table:

Let us take a generatorCase No.1 For β₁ ≠ 0, the generator turns to

Applying on Y givesfurthermore, proceeding in the same waywhich successively makes the coefficients β₂ and β₃ equal to 0and implies that W ≃ W₁.Case No.2 Without loss of generality, here we take β₁ = 0 and β₂ = 1, the generator becomes

Now, act on the aforementioned W,Subcase No.2.1 If β₃ < 0, thenSubcase No.2.2 If β₃ > 0, thenCase No.3 For β₁ = β₂ = 0 and β₃ = 1. Thus, in the meanwhile we have W ≃ W₃.Case No.4 Let consider β₁ = 0 = β₃ and β₂ ≠ 0. In this case, the generator is W ≃ W₂.

The optimal system of one-dimensional subalgebras admitted by Eq. 3) is as follows:

2.3 Reductions and invariant solutions

2.3.1 Reduction by W₂

The invariants for corresponding characteristic equation are as follows:where a and b are arbitrary constants.

The invariant solution can be written in the form of b = f(a), implies thatsubstituting this value in Eq. 3), we obtain

The solution of this reduced equation for β = 1 is given in the form of solution set as.

2.3.2 Reduction by W₃

The corresponding characteristic equation to this generator is as follows:this gives two invariantswhere a₁ and b₁ are arbitrary constants. It impliesputting this in Eq. 3, we obtainwhich gives a trivial solution for u = f(x).

2.3.3 Reduction by W₁

The characteristic equation issolving this, we obtain corresponding invariants of the formfrom thiswhere we obtainsubstituting these derivatives into Eq. 3, we obtain

Thus, non-linear PDE (3) reduces to a non-linear ODE.

2.3.4 Reduction by W₂ + W₃

The invariants that we gain by solving characteristic equation are as follows:a₃ and b₃ are arbitrary constants. The invariant solution corresponding to them is u = f (a₃). Inserting this solution into Eq. 3 will give us a non-linear ODE of the form

2.3.5 Reduction by W₂− W₃

The invariants corresponding to characteristic equation for this case are a₄ = x + t and b₄ = u. Furthermore, its invariant solution is given as u = f (a₄). Therefore, the Eq. 3 will be converted into an ODE of the form

2.4 Determining lie symmetry of SPE for the case and k(u) = u_xx (n > 1)

The equation becomes

The one-parameter Lie group of transformations and the second prolongation with coefficients are given in Eqs 4, 6, respectively for Eq. 43. Let the generator be

Therefore, we havesimplification gives the following equation:which is solved for the values of ν^xx and ν^xt, will give us the equation involving derivatives of infinitesimals with respect to dependent and independent variables and also the derivatives of dependent variable w.r.to independent variables. Substituting Eq. 43 and comparing the values of coefficients on both sides gives an under-determined system of equations

To solve this system, we consider ν as:which satisfies the aforementioned equations and then by solving the aforementioned system, we obtainc₁ and c₂ are any arbitrary constants. The infinitesimal generators for the one-parameter of Lie groups of transformations admitted in Eq. 43) are given by

These symmetry generators give us the symmetry groups for i = 1, 2:If u = R (x,t) is a solution of Eq. 43), then uⁱ for i = 1, 2, and 3 and ϵ ≪ 1 also satisfies Eq. 43, Commutator Table:also,

Adjoint Table.

Proposition 5.1: The generators Z₁ = ∂_t and Z₂ = ∂_x form a two-dimensional abelian Lie symmetry algebra.

2.5 Optimal system, reductions and invariant solutions

Considering a generator Z = b₁Z₁ + b₂Z₂. This generator will established a set of optimal system comprising of Lie algebrawhere b₁, and b₂ are arbitrary constants. The reduction of PDE Eq. 39 by using the generator Z₁ leads to an invariant solution u = f (c₁). The reduced non-linear ODE will be

The reduction through Z₂ generates a trivial case for Eq. 39.

3 Conclusion

In this paper, we have carried out the LSA for computing the similarity solutions (symmetries) of the non-linear SPE for the cases when h(u) = e^u and k(u) = u_xx and and k(u) = u_xx in SPE (2). In addition, an optimal system of one-dimensional subalgebra has been set up. The reductions and invariant solutions for the generalized SPE are calculated corresponding to this optimal system as well. The graphs are formed by the maple for the functions obtained from the transformed points of one-parameter Lie groups.

References

• 1.

  OliveriF. Lie symmetries of differential equations: Classical results and recent contributions. Symmetry (2010) 2(2):658–706. 10.3390/sym2020658

  • CrossRef
  • Google Scholar

• 2.

  KnightJCBroengJBirksTARussellPSJ. Photonic band gap guidance in optical fibers. Science (1998) 282(5393):1476–8. 10.1126/science.282.5393.1476

  • CrossRef
  • Google Scholar

• 3.

  ClarksonPALudlowDKPriestleyTJ. The classical, direct, and nonclassical methods for symmetry reductions of nonlinear partial differential equations. Methods Appl Anal (1997) 4(2):173–95. 10.4310/maa.1997.v4.n2.a7

  • CrossRef
  • Google Scholar

• 4.

  LiuHZhangL. Symmetry reductions and exact solutions to the systems of nonlinear partial differential equations. Physica Scripta (2018) 94(1):015202. 10.1088/1402-4896/aaeeff

  • CrossRef
  • Google Scholar

• 5.

  CocliteGMdi RuvoL. Well-posedness results for the short pulse equation. Z für Angew Mathematik Physik (2015) 66(4):1529–57. 10.1007/s00033-014-0478-6

  • CrossRef
  • Google Scholar

• 6.

  SchäferTWayneCE. Propagation of ultra-short optical pulses in cubic nonlinear media. Physica D: Nonlinear Phenomena (2004) 196(1-2):90–105. 10.1016/j.physd.2004.04.007

  • CrossRef
  • Google Scholar

• 7.

  AgrawalGP. Nonlinear fiber optics. In: Nonlinear science at the dawn of the 21st century. Berlin, Heidelberg: Springer (2000). p. 195–211.

  • Google Scholar

• 8.

  KarasawaNNakamuraSNakagawaNShibataMMoritaRShigekawaHet alComparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber. IEEE J Quan Electron (2001) 37(3):398–404. 10.1109/3.910449

  • CrossRef
  • Google Scholar

• 9.

  MoloneyJVNewellAC. Nonlinear optics. Physica D: Nonlinear Phenomena (1990) 44(1-2):1–37. 10.1016/0167-2789(90)90045-q

  • CrossRef
  • Google Scholar

• 10.

  XiaoYMaywarDNAgrawalGP. New approach to pulse propagation in nonlinear dispersive optical media. JOSA B (2012) 29(10):2958–63. 10.1364/josab.29.002958

  • CrossRef
  • Google Scholar

• 11.

  LiuHLiJ. Lie symmetry analysis and exact solutions for the short pulse equation. Nonlinear Anal Theor Methods Appl (2009) 71(5-6):2126–33. 10.1016/j.na.2009.01.075

  • CrossRef
  • Google Scholar

• 12.

  FakharKWangGKaraAH. Symmetry reductions and conservation laws of the short pulse equation. Optik (2016) 127(21):10201–7. 10.1016/j.ijleo.2016.08.013

  • CrossRef
  • Google Scholar

• 13.

  MatsunoY. Parametric solutions of the generalized short pulse equations. J Phys A: Math Theor (2020) 53(10):105202. 10.1088/1751-8121/ab6f18

  • CrossRef
  • Google Scholar

• 14.

  AliAT. New generalized Jacobi elliptic function rational expansion method. J Comput Appl Math (2011) 235(14):4117–27. 10.1016/j.cam.2011.03.002

  • CrossRef
  • Google Scholar

• 15.

  El-SabbaghMFAliAT. New generalized Jacobi elliptic function expansion method. Commun Nonlinear Sci Numer Simulation (2008) 13(9):1758–66. 10.1016/j.cnsns.2007.04.014

  • CrossRef
  • Google Scholar

• 16.

  ZhaoWMunirMMBashirHAhmadDAtharM. Lie symmetry analysis for generalized short pulse equation. Open Phys (2022) 20(1):1185–93. 10.1515/phys-2022-0212

  • CrossRef
  • Google Scholar