# Laplace Decomposition Method to Study Solitary Wave Solutions of Coupled Nonlinear Partial Differential Equation

Authors: Arun Kumar; Ram Dayal Pankaj
Journal: ISRN Computational Mathematics
 (2012)

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## Abstract

Analytical and numerical solutions are obtained for coupled nonlinear partial differential equation by the well-known Laplace decomposition method. We combined Laplace transform and Adomain decomposition method and present a new approach for solving coupled Schrödinger-Korteweg-de Vries (Sch-KdV) equation. The method does not need linearization, weak nonlinearity assumptions, or perturbation theory. We compared the numerical solutions with corresponding analytical solutions.


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## Body

## 1. Introduction

Systems of partial differential equations have attracted much attention in a variety of applied sciences because of their wide applicability. These systems were formally derived to describe wave propagation to model the shallow water waves [1–5] and to examine some chemical reaction-diffusion model of Brusselator [4–6]. Some of the commonly used methods to solve these equations are the method of characteristics, the Riemann invariants, and Adomian decomposition method [6].In this work, we used Laplace decomposition method introduced by Khuri [7, 8] which is further used by Yusufoglu to solve Duffing equation [9] and Elgasery for Falkner-Skan equations [10]. This technique, modified by Hussain and Khan [11], illustrates how the Laplace transform may be used to approximate the solutions of the nonlinear partial differential equations by extending the decomposition method [12, 13].


## 2. Laplace Decomposition Method (LDM)

In this section, we outline the main steps of the method. We consider the nonlinear partial differential equations in an operator form:(1)Ltu+R1(u,v)+N1(u,v)=f1,Ltv+R2(u,v)+N2(u,v)=f2.
With initial data
(2)u(x,0)=g1(x),v(x,0)=g2(x),
where Lt=∂/∂t is a first-order partial differential operator, R1, R2 and N1, N2 are linear and nonlinear operators, respectively, and f1,f2 are the source terms. By applying the Laplace transform to both sides of (1) and using initial conditions (2), we have
(3)L(Ltu)+L(R1(u,v))+L(N1(u,v))=L(f1),L(Lt(v))+L(R2(u,v))+L(N2(u,v))=L(f2).
By the differentiation property of Laplace transform, we get
(4)L(u)=g1(x)p+L(f1)p-1p[L(R1(u,v))+L(N1(u,v))],L(v)=g2(x)p+L(f2)p-1p[L(R2(u,v))+L(N2(u,v))],
where “p”   is Laplace domain function. Solutions u(x,t) and v(x,t) in LDM are defined as
(5)u(x,t)=∑n=0∞un,v(x,t)=∑n=0∞vn.
The nonlinear terms N1,N2, represented by infinite series,
(6)N1(x,t)=∑n=0∞An,N2(x,t)=∑n=0∞Bn
are the Adomian polynomials [14], generated for all forms of nonlinearity and they are determined by the following relations:
(7)An=1n![dndλn(N1∑i=0∞λiui)]λ=0,Bn=1n![dndλn(N2∑i=0∞λivi)]λ=0.
Substituting (5) and (6) into (4), gives
(8)L(∑n=0∞un)=g1(x)p+L(f1)p-1p[L(R1[(∑n=0∞un),(∑n=0∞vn)])+L(∑n=0∞An)],L(∑n=0∞vn)=g2(x)p+L(f2)p-1p[L(R2[(∑n=0∞un),(∑n=0∞vn)])+L(∑n=0∞Bn)].
Applying the linearity of the Laplace transform, we deduct the following recursive relations:
(9)L(u0)=g1(x)p+L(f1)p,L(v0)=g2(x)p+L(f2)p,L(u1)=-1pL[(R1(u0,v0))]-1pL[A0],L(v1)=-1pL[(R2(u0,v0))]-1pL[B0].
In general, for k≥1, the recursive relations are given by
(10)L(uk+1)=-1pL[R1(uk,vk)]-1pL[Ak],L(vk+1)=-1pL[R2(uk,vk)]-1pL[Bk].
Applying the inverse Laplace transform, we can evaluate uk and vk  (k≥0). In some cases the exact solution in the closed form can be obtained.


## 3.  Application

At the classical level, a set of coupled nonlinear wave equations describes the interaction between high-frequency and low-frequency waves [15], and the calculation of exact and numerical solutions of the equations, in particular, travelling wave solutions, plays an important role in wave-wave interaction and soliton theory [1, 16].We consider the Schrödinger-KdV (Sch-KdV) equation as a model for the interaction of long and short nonlinear waves:(11)iEt=Exx+Eη,ηt=-6Eηx-ηxxx+(|E|2)x.
With initial conditions
(12)E(x,0)=22k2(1-3tanh2(kx))eiαx,η(x,0)=8k2-α3-6k2tanh2(kx),
where α,k are arbitrary constant.With Laplace decomposition method on (11) and using the differentiation property of Laplace transform, initial conditions and the inverse Laplace transforms are(13)∑n=0∞En(x,t)=E(x,0)-iL-1[1pL(Enxx+∑n=0∞An(η,E))],∑n=0∞ηn(x,t)=η(x,0)+L-1[1pL(∑n=0∞Bn(E)-6∑n=0∞Cn(η,E)-ηnxxx)],
where (x,t)=∑n=0∞En,η(x,t)=∑n=0∞ηn, and ∑n=0∞An(η,E)=ηE   ∑n=0∞Bn(E)=(|E|2)x   ∑n=0∞Cn(η,E)=Eηx are adomian polynomials that represent nonlinear terms.So the recursive relation is deduced as(14)E(x,0)=E0=22k2(1-3tanh2(kx))eiαx,E1(x,t)=-iL-1[1pL(E0xx+∑n=0∞A0(η,E))],En+1(x,t)=-iL-1[1pL(Enxx+∑n=0∞An(η,E))],η(x,0)=η0=8k2-α3-6k2tanh2(kx),η1(x,t)=L-1[1pL(∑n=0∞B0(E)-6∑n=0∞C0(η,E)-η0xxx)],ηn+1(x,t)=L-1[1pL(∑n=0∞Bn(E)-6∑n=0∞Cn(η,E)-ηnxxx)],n≥1.
By this recursive relation we can find other components of the solution as
(15)E1(x,t)=-iL-1[1pL(E0xx+η0E0)],E1(x,t)=22k2t3×[(kx)(αx)(3α2+α+10k2)(1-3tanh(kx))×cos(αx)+36kαsech2(kx)×tanh(kx)cos(αx){(3α2+α+10k2)}(αx)(αx)(kx)]+i22k2t3×[(kx){αsech2(kx)tanh(kx)cos(αx)}(3α2+α+10k2)×(1-3tanh(kx))sin(αx)-36kαsech2(kx)tanh(kx)cos(αx)],η1(x,t)=L-1[1pL((|E0|2)x-6E0η0x-η0xxx)],η1(x,t)=-48k5tsech2(kx)(4-9tanh2(kx))+1442k5tsech2(kx)tanh(kx)×(1-3tanh2(kx))cos(αx)+144i2k5t×sech2(kx)tanh(kx)(1-3tanh2(kx))×sin(αx),E2(x,t)=-iL-1[1pL(E1xx+η1E0+η0E1)],E2(x,t)=-2k2t23×[1p6k2(15α2+α+10k2)sech2(kx)×(1-3tanh2(kx))+(3α2+α+10k2)×(1-3tanh2(kx))×(3α2+α-8k23-6k23tanh2(kx))]eαix+482k7t2isech2(kx)tanh(kx)×(4-9tanh2(kx))(1-3tanh2(kx))eαix-288k7t2isech2(kx)tanh(kx)×(1-3tanh2(kx))2e2αix-2k2t23×i[216k3αsech2(kx)tanh3(kx)-24αk(3α2+α+13k2)×sech2(kx)tanh(kx)]eαix,η2(x,t)=η1(x,t)-6tk4sech5(kx)×{k4tsech3(kx)×(1208-1191cosh(2kx)+120cosh(4kx)-cosh(6kx))+24e2iαx(iαcosh(kx)+2ksinh(kx))}.The other components of the decomposition series can be determined in a similar way; we can obtain the expression ofE(x,t) in a Taylor series, which gives the closed form solutions as:(16)E(x,t)=22k2(1-3tanh2(k(x+2αt)))×exp(i[αx+13(3α2+α-10k)t]),η(x,t)=8k2-α3-6k2tanh2(k(x+2αt)).


## 4. Numerical Description of the Solution

The Laplace decomposition method is used for finding the exact and approximate travelling-waves solutions of the Sch-KdV equation. Both the exact and approximate solutions obtained forn = 2 using LDM are plotted in Figure 1. It is evident that when compute more terms for the decomposition series the numerical results are getting much closer to the corresponding analytical solutions.The plots of results for solution of Sch-KdV equations with a fixed values ofα=k=0.05 and for different values of time. (a) Analytical solutions for E(x,t). (b) Numerical results for E2(x,t) by means of LDM. (c) Analytical solutions for η(x,t). (d) Numerical results for η2(x,t) by means of LDM.
(a)
(b)
(c)
(d)


## 5. Conclusion

The Laplace decomposition method is a powerful method which has provided an efficient potential for the solution of physical applications modeled by nonlinear differential equations. The algorithm can be used without any need to complex calculations except for simple and elementary operations.



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