Travelling wave-like solutions for some nonlinear equations with cubic
nonlinearities
Abstract: we exercised the series-expansion method to extract solitary wave solutions for complex nonlinear evo-
lution equations (NLEEs) such as the coupled Higgs equation (CHE), the (3+1)-dimensional dynamical system,
the (3+1)-dimensional nonlinear Schrodinger(NLSE) equation which have many physical importance in different
branches. Varieties of periodic and solitonic wave-like solutions are extracted. Computational work has been
done and plots and counter graphs are plotted using Wolfram Mathematica 11.
Key-Words: Complex nonlinear evolution equations, coupled Higgs equation. PACS No:05.45.Yv,94.05.Fg.
1 Introduction
Importance of nonlinear evolution equations
(NLEEs) in different breaches such as physical
sciences, engineering, and biological sciences cannot
be ignored. Some well known phenomena in chem-
ical kinetics, nonlinear optics [1], flow mechanics
[2], nuclear physics [3], condense matter physics
[5], etc population dynamics etc [6], are described
by NLEEs. Once a problem is identified, then its
mathematical models can be traced with a suitable
equations, preferably in NLEEs [7]. Further we
generally wish to find exact algebraic and numerical
solutions of such nonlinear equations in order to
predict, control and quantify the underlying features
of the system under study. In last few decades,
numerous exercise have been made to extract the
exact algebraic and numerical solutions of NLEEs
and a many methods have been fabricated to cater
the explicit travelling wave like solutions of NLEEs
[8]. Recently, Wang et al. [2] developed a new
formulation called (F0
F)-expansion method for a
dependable prescription of nonlinear wave equations.
Consequently many more utilisation of this technique
have also been reported [9]. A summarized version of
(F0
F)-expansion formalism is also prescribed recently
[10]. In next section one, few equation [11] and [12]
are identified and solved by (F0
F)method. At last few
remarks are noted in summary section .
1.1 The Series-expansion method
In brief, here we prescribe the important steps of the
(F0
F)-expansion method [13]. Hypothesize that a non-
linear evolution differential equation (NLEEs) is of
the form polynomial R as
R(u, ut, ux, utt, uxt, uxx, ...) = 0,(1)
Step 1: The standard explication of (1) can be written
by a polynomial in (F0
F)i.e.
u(ξ) = αmF0
Fm
+αm−1F0
Fm−1
+....., (2)
where F=F(ξ)satisfies the second order linear
ODE of the form
F00 +τF 0+κF = 0,(3)
where αm,αm−1,....., α0,τand κare constants to
be of single-minded and can ce calulated later with
αm6= 0.
Step 2: Substituting (2) into (1) and using (3), collect-
ing all terms with the same order of (F0
F)together, and
then equating each coefficient of the resulting poly-
nomial to zero yields a set of algebraic equations for
αm,αm−1,....., α0, c, τand κ.Further depending on
the sign of the discriminant 4=τ2−4κpossess the
many general solutions as
F0
F=
√τ2−4κ
2Asinh(pτ2−4κ
2)χ+Bcosh(pτ2−4κ
2)χ
Acosh(pτ2−4κ
2)χ+Bsinh(pτ2−4κ
2)χ−κ
2,
√4κ−τ2
2−Asinh(p4κ−τ2
2)χ+Bcosh(p4κ−τ2
2)χ
Acosh(p4κ−τ2
2)χ+Bsinh(p4κ−τ2
2)χ−κ
2,
B
A+Bχ −κ
2
or more simplified version as
F0
F=
√τ2−4κ
2tanh √τ2−4κ
2χ+χ0−κ
2,
√τ2−4κ
2coth √τ2−4κ
2χ+χ0−κ
2,
√4κ−τ2
2cot √4κ−τ2
2χ+χ0−κ
2,
B
A+Bχ −κ
2
Well known general solutions of eq.(3) have been fa-
miliar for us, then simulating αm,αm−1,.....,α0with
eq.(3) into eq.(2) we exract more traveling wave-like
solutions of NLEEs eq.(1).
JASVINDERPAL SINGH VIRDI
VSS University of Technology (VSSUT)
Odisha-768018, INDIA.
Received: January 19, 2023. Revised: October 25, 2023. Accepted: November 26, 2023. Published: December 31, 2023.
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1.2 The Coupled Higgs Equation
Acknowledge the following form of coupled Higgs
equation (CHE) [14]
utt −uxx +|u|2u−2uv = 0,
vtt +vxx −(|u|2)xx = 0.(4)
Tajiri obtained N-solition solutions to the system (4)
in[14]. Zhao calculated more accustomed traveling
wave solutions of eq. (4) in [14].
By using the wave variables
u=eıθU(ξ), v =V(ξ), θ =px +rt, ξ =x+ct, (5)
eq. (4) are carried to ODEs
(c2−1)U00 + (p2−r2)U+U3−2UV = 0,
(c2+ 1)V00 −(U2)00 = 0.(6)
Integrating the second equation of (6) twice and set-
ting constants of integration is zero, then we get
V=U2
c2+ 1.(7)
Substituting (7) into first equation of (6), we obtained
(c4−1)U00 + (c2+ 1)(p2−r2)U+ (c2−1)U3= 0.(8)
Now, balancing the terms u00 with u3we find, m=1.
Here, we suppose that
U(ξ) = α1F0
F+α0, α16= 0,(9)
where F=F(ξ)satisfies the second order linear
ODE
F00 +τF 0+κF = 0,(10)
where α1,α0,τand κare constants to be determined
later. Using (9) into (8) and rationalizing in power of
F0
Fsimultaneously, the LHS of (8) are converted into
the polynomials in F0
F. Segregating each coefficient
of the polynomials to zero, leads a set of concurrent
analytic equations as follows:
2(c4−1)α1+ (c2−1)α3
1= 0,3τ(c4−1)α1
+3(c2−1)α2
1α0= 0,
(2κ+τ2)(c4−1)α1+ (c2+ 1)(p2−r2)α1
+3(c2−1)α1α2
0= 0,
τκ(c4−1)α1+ (c2+ 1)(p2−r2)α0
+(c2−1)α3
0= 0.(11)
Further solving, we have
α0=±ιrc2+ 1
2τ, α1=±ιp2(c2+ 1),
c=±r2(p2−r2)
τ2−4κ+ 1,(12)
here τ,κ, p and r are arbitrary constant.
Again by taking (12) into (9), leads to
U(ξ) = ±ιp2(c2+ 1)F0
F±ιrc2+ 1
2τ. (13)
Exchanging the second order LODE (10) into the (13)
we got wide different category of solutions of (8) and
the coupled Higgs equation (CHE) as follows:
With τ2−4κ > 0,
U(ξ) = ±ιrc2+ 1
2pτ2−4κ
Asinh(1
2√τ2−4κ)ξ+Bcosh(1
2√τ2−4κ)ξ
Acosh(1
2√τ2−4κ)ξ+Bsinh(1
2√τ2−4κ)ξ,
(14)
V(ξ) = (4κ−τ2)
2
Asinh(1
2√τ2−4κ)ξ+Bcosh(1
2√τ2−4κ)ξ
Acosh(1
2√τ2−4κ)ξ+Bsinh(1
2√τ2−4κ)ξ2
,
(15)
u(ξ) = ±ιrc2+ 1
2pτ2−4κ
Asinh(1
2√τ2−4κ)ξ+Bcosh(1
2√τ2−4κ)ξ
Acosh(1
2√τ2−4κ)ξ+Bsinh(1
2√τ2−4κ)ξeιθ,
(16)
v(ξ) = (4κ−τ2)
2
Asinh(1
2√τ2−4κ)ξ+Bcosh(1
2√τ2−4κ)ξ
Acosh(1
2√τ2−4κ)ξ+Bsinh(1
2√τ2−4κ)ξ2
,
(17)
here θ=px +rt,ξ=x±q2(p2−r2)
τ2−4κ+ 1 t.
In Precisely, if B6= 0,A=0, τ > 0,κ= 0, then uand
vbecomes
u(ξ) = ±ιrc2+ 1
2τtanh(1
2τξ)eıθ,(18)
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v(ξ) = −τ2
2tanh2(1
2τξ).(19)
These are the possible periodic wave-like solution of
coupled Higgs equation (CHE).
With τ2−4κ < 0,
U(ξ) = ±ιrc2+ 1
2p4κ−τ2
−Asin(1
2√4κ−τ2)ξ+Bcos(1
2√4κ−τ2)ξ
Acos(1
2√4κ−τ2)ξ+Bsin(1
2√4κ−τ2)ξ,
(20)
V(ξ) = (τ2−4κ)
2
−Asin(1
2√4κ−τ2)ξ+Bcos(1
2√4κ−τ2)ξ
Acos(1
2√4κ−τ2)ξ+Bsin(1
2√4κ−τ2)ξ2
,
(21)
u(ξ) = ±ιrc2+ 1
2p4κ−τ2
−Asin(1
2√4κ−τ2)ξ+Bcos(1
2√4κ−τ2)ξ
Acos(1
2√4κ−τ2)ξ+Bsin(1
2√4κ−τ2)ξeιθ,
(22)
v(ξ) = (τ2−4κ)
2
−Asin(1
2√4κ−τ2)ξ+Bcos(1
2√4κ−τ2)ξ
Acos(1
2√4κ−τ2)ξ+Bsin(1
2√4κ−τ2)ξ2
,
(23)
where θ=px +rt,ξ=x±q2(p2−r2)
τ2−4κ+ 1 t.
With τ2−4κ= 0,
U(ξ) = ±ιp2(c2+ 1)B
A+Bξ ,(24)
V(ξ) = −2B
A+Bξ 2
,(25)
u(ξ) = ±ιp2(c2+ 1)B
A+Bξ eιθ,(26)
v(ξ) = −2B
A+Bξ 2
,(27)
where p=r, c = 1, θ =px +rt, ξ =x+ct. Solu-
tions for (CHE) are prepaired and curves are plotted
for all possible wave-like solutions are crafted by their
3D plots and further their analogous contour plots.
Figure 1: Traveling wave solution corresponding to
the C. Higgs equation
(a) Plot of u1(ζ), when 4>0, τ = 0.1, κ =
0.02A= 0.02, B = 0.03
(b) Plot of u2(ζ), when 4<0, τ = 0.2, κ =
0.2, A = 0.02, B = 0.03
(c) Plot of u3(ζ), when 4= 0, τ = 0.2, κ =
0.01, A = 0.02, B = 0.03
1.3 The (2+1)-dimensional NLS Equation
Consider the following (2+1)-dimensional NLS equa-
tion [15]
ιut+uxx +uyy +a|u|2u= 0,(28)
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We take the transformation u=eıθU(ξ),θ=αx +
βy+τ t,ξ=x+y+ct. The substitution of transformation
into (28) yields the ODE
2U00 −(α2+β2+τ)U+aU3= 0.(29)
under the condition c=-2(α+β).
Now, balancing the terms u00 with u3we find, m=1.
Here, we suppose that
U(ξ) = α1F0
F+α0, α16= 0,(30)
where F=F(ξ)satisfies the second order linear
ODE
F00 +τF 0+κF = 0,(31)
where α1,α0,τand κare well known to us.
Following the steps and equating each coefficient
leads to group of equations as follows:
4α+α3
1= 0, , 6τ α + 3aα2
1α0= 0,
2(2κ+τ2)α−(α2+β2+τ)α+ 3aαα2
0= 0,
2τκα −(α2+β2+τ)α0+aα3
0= 0.(32)
further, one can have
α0=±ιτ
√a, α =±2ι
√a, τ2−4κ=−(α2+β2+τ),(33)
where τ,κ,α,β,τare well known to us.
By using (33) in expression (30), we get
U(ξ) = ±2ι
√aF0
F±ιτ
√a.(34)
following the earlier steps (31) into the (34) we have
three types of travelling wave solutions of (2+1)-
dimensional NLS equation (28) as follows:
With τ2−4κ > 0,
U(ξ) = ±r(4κ−τ2)
a
Asinh(1
2√τ2−4κ)ξ+Bcosh(1
2√τ2−4κ)ξ
Acosh(1
2√τ2−4κ)ξ+Bsinh(1
2√τ2−4κ)ξ,
(35)
u(ξ) = ±r(4κ−τ2)
a
Asinh(1
2√τ2−4κ)ξ+Bcosh(1
2√τ2−4κ)ξ
Acosh(1
2√τ2−4κ)ξ+Bsinh(1
2√τ2−4κ)ξeιθ,
(36)
where θ=αx +βy +τ t,ξ=x+y-2(α+β)t.
Precisely„ if A6= 0,B= 0,τ > 0,κ= 0, then u and
v becomes
u(ξ) = ±ιτ
√atanh(τξ
2)eıθ,(37)
which are the periodic wave solution of (2+1)-
dimensional NLS equation.
With τ2−4κ < 0,
U(ξ) = ±r(τ2−4κ)
a
−Asin(1
2√4κ−τ2)ξ+Bcos(1
2√4κ−τ2)ξ
Acos(1
2√4κ−τ2)ξ+Bsin(1
2√4κ−τ2)ξ,
(38)
u(ξ) = ±r(τ2−4κ)
a
−Asin(1
2√4κ−τ2)ξ+Bcos(1
2√4κ−τ2)ξ
Acos(1
2√4κ−τ2)ξ+Bsin(1
2√4κ−τ2)ξeιθ,
(39)
With τ2−4κ= 0,
U(ξ) = ±2ι
√aB
A+Bξ ,(40)
u(ξ) = ±2ι
√aB
A+Bξ eιθ,(41)
where c=∓2√2αβ −τ,θ=αx +βy +τ t,ξ=
x+y∓2√2αβ −τt. Solutions of (FNE) are pre-
paired and curves are plotted for all possible wave-
like solutions are crafted by their 3D plots and further
their analogous contour plots.
2 Conclusion
We can expand this work and rterritory applications
of the (F0
F)-expansion method. During this work we
tried to obtain new solutions of some NLEES with
complex NLEEs well known the coupled Higgs equa-
tion, the (3+1)-dimensional nonlinear Schrodinger
equation(NLSE). The result we got are purely in trav-
eling wave nature and can further leads to solitary
wave or periodic solutions under wide parametric re-
strictions. Further interesting to claim here the from
of the generic results, we can easily extract the solu-
tions which are accomplished from different methods.
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Figure 2: Traveling wave solution corresponding to
the NLS Equation
(a) Plot of u1(χ), when 4>0, τ = 0.1, ρ =
0.02, A = 0.02, B = 0.03
(b) Plot of u2(χ), when 4= 0, τ = 0.2, ρ =
0.01, A = 0.01, B = 0.02
References:
[1] J S Virdi, On different approaches for inte-
grals of physical dynamical systems, Disconti-
nuity, Nonlinearity, and Complexity 9(2) (2000)
, pp.299-307.,
[2] Wang M.: Solitary Wave Solutions for Vari-
ant Boussinesq Equations, Phys. Lett. A. 199,
(1995) 169.
[3] Swain S.N, Virdi J.S., (2019) Lecture Notes in
Mechanical Engineering, doi.org 10.1007-978-
981-15-0287-3-15, 187-200.
[4] J.S. Virdi, Some New Solutions of Non Lin-
ear Evolution Equations With Mutable Co-
efficients. Front. Appl. Math. Stat. (2021)
https://doi.org/10.3389/fams.2021.631052.
[5] S Behera, J S Virdi, Some More Solitary
Traveling Wave Solutions of Nonlinear Evolu-
tion Equations, Discontinuity, Nonlinearity, and
Complexity 12(1) (2023) 75-85.
[6] Virdi J.S, Some new solutions of nonlinear evo-
lution equations with variable coefficients, AIP
Proceedings 1728, (2016), 020039.
[7] S.Behera, N.H.Aljahdaly, J.S.Virdi, On
the modified (G0
G2)-expansion method for
finding some analytical solutions of the
traveling waves, J. Ocean Eng. Sci. (2022)
https://doi.org/10.1016/j.joes.2021.08.013
[8] S. Behera and J.S. Virdi, (2022) Generalized
soliton solutions to Davey-Stewartson equation,
Nonlinear Optics, Quantum Optics, 57, 325
(2023).
[9] Behera, S. and Virdi, J. P. S. (2021). Travelling
wave solutions of some nonlinear systems
by Sine-cosine Approach. Proceedings of
the Seventh International Conference on
Mathematics and Computing, Advances in
Intelligent Systems and Computing 1412,
https://doi.org/10.1007/978-981-16-6890-6_66.
[10] S Behera, J S Virdi, Some More Solitary
Traveling Wave Solutions of Nonlinear Evolu-
tion Equations, Discontinuity, Nonlinearity, and
Complexity 11(4) (2022).
[11] W Yao, S Behera, Mustafa, H Rezazadeh,
J S Virdi, W. Mahmoud, Omar Abu Ar-
qub, M.S. Osman, Analytical solutions of con-
formable Drinfel’d–Sokolov–Wilson and Boiti
Leon Pempinelli equations via sine–cosine
method, Results in Physics 42, 105990 (2022)
[12] S. Behera and J.S. Virdi, Generalized soliton so-
lutions to Davey-Stewartson equation, Nonlin-
ear Optics, Quantum Optics, vol. 56 5, (2023).
[13] Behera, S. and Virdi, J. S. (2021). Travelling
wave solutions of some nonlinear systems
by Sine-cosine Approach. Proceedings of
the Seventh International Conference on
Mathematics and Computing, Advances
in Intelligent Systems and Computing
1412,https://doi.org/10.1007/978-981-16-
6890-6 66.
[14] S Behera, J S Virdi, Some More Solitary
Traveling Wave Solutions of Nonlinear Evolu-
tion Equations, Discontinuity, Nonlinearity, and
Complexity 12(3) 75-85, (2023).
[15] J.S.Virdi, Some Study On Solitary Traveling
Wave Solutions For Nonlinear Evolution Equa-
tions, Journal of Tianjin University Science and
Technology, Vol:55 03, (2022)
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The authors equally contributed in the present
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problem to the final findings and solution.
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Scientific Article or Scientific Article Itself
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Conflict of Interest
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