Lie Symmetries, Conservation Laws, and
Invariant Solutions of Multidimensional
Modified Korteweg–de Vries Equations
a
Department of Mathematics, Riphah International University, Faisalabad
Campus, Faisalabad 38000, Pakistan.
b
Department of Mathematics, Riphah International University, Faisalabad
Campus, Faisalabad 38000, Pakistan.
Abstract
This paper presents a classical Lie symmetry analysis, conservation laws, and
invariant solutions for a generalized family of modified Korteweg–de Vries (mKdV)
equations in (1+1), (1+2), and (1+3) dimensions. Using the multiplier method and
double reduction approach, several new conservation laws and exact analytical
solutions are obtained. The results extend existing studies on KdV-type equations
to higher dimensions and demonstrate the power of symmetry-based techniques in
understanding nonlinear partial differential equations with physical applications
in fluid dynamics and wave propagation.
Keywords: Lie symmetries, conservation laws, double reduction theory, op-
timal system and invariant solutions.
1 Introduction
Partial differential equations (PDEs) play a fundamental role in modeling a wide
range of natural and physical phenomena, including fluid dynamics, plasma phys-
ics, nonlinear optics, and wave propagation. The search for exact solutions to
nonlinear PDEs remains a central topic in applied mathematics, as such solutions
provide important insight into the structural and qualitative behavior of complex
systems. Analytical methods that exploit the intrinsic symmetries of differential
equations have proven especially powerful in reducing, classifying, and solving
nonlinear problems.
Among these analytical methods, the Lie symmetry approach, developed by
Sophus Lie, has established itself as a systematic and elegant framework for study-
ing continuous symmetry transformations. Lie symmetries enable the reduction of
PDEs to simpler forms, often transforming them into ordinary differential equa-
tions (ODEs) that are more manageable to solve. Over the years, numerous re-
searchers have extended and refined Lie’s classical method to handle increasingly
complex and multidimensional systems. This approach has become a cornerstone
in the qualitative analysis of nonlinear differential equations.
An equally important contribution to the theory of symmetries and conserved
quantities was made by Emmy Noether, whose theorem established a deep and
fundamental link between variational symmetries and conservation laws. Conserva-
tion laws are essential for understanding the stability, energy balance, and physical
meaning of differential systems. They also provide useful checks for the correctness
of analytical and numerical solutions. The concept of a conservation law is closely
associated with physical principles such as the conservation of mass, momentum,
and energy. Various researchers, including Bluman and Kumei, Hereman, and Naz,
1
,have developed systematic frameworks for constructing conservation laws of non-
linear PDEs.
One of the most effective modern techniques for deriving conservation laws
is the multiplier approach, proposed and developed by several authors, including
Filho and Figueiredo. This method does not require the existence of a Lagrangian
and can therefore be applied to a wide class of non-variational PDEs. By de-
termining appropriate multipliers, one can construct conserved vectors associated
with the governing equations. These conserved quantities, when combined with Lie
symmetries, play a crucial role in obtaining exact solutions through the double re-
duction theory, which allows for the simultaneous reduction of both independent
and dependent variables.
The present study focuses on the Lie symmetry analysis, conservation laws, and
invariant solutions of a family of generalized modified Korteweg–de Vries (mKdV)
equations in (1+1), (1+2), and (1+3) dimensions. These equations arise in many
areas of applied science, including plasma waves, nonlinear lattice theory, and fluid
mechanics. By employing the classical Lie group method together with the mul-
tiplier technique and double reduction approach, new classes of conservation laws
and exact analytical solutions are derived. The obtained results extend previously
known one-dimensional models to higher dimensions, revealing deeper structural
properties and providing new insights into the integrability of nonlinear evolution
equations.
2 Fundamental Operators
This section provides the mathematical framework and operators used in the sym-
metry and conservation law analysis. The key definitions of infinitesimal generat-
ors, conserved vectors, and Euler–Lagrange operators are summarized below for
completeness.
Assume a system of partial differential equations (PDEs) that can be written
in the nth order with r independent variables z = (z 1 , z 2 , z 3 , ..., z r ) and t dependent
variables d = (d1 , d2 , d3 , ..., dt ) and µ = 1, 2, 3, ..., t,
Fµ = (z, d, d1 , d2 , d3 , ..., dn ), (2.1)
here d1 , d2 , d3 , ..., dn are the nth order partial derivatives.
Definition 1. The infinitesimal generator Z’s prolongation of order i is defined
as
∂ ∂
Z [i] = Z + τx + ... + τx1 ...xi , (2.2)
∂dx ∂dx1 ...xi
′
here τ s are given by
τxs = Dx (η s ) − dsl Dx (ξ l ),
..
.
τxs1 ...xi = Dxi (ηxs 1 ...xi−1 ) − dskx1 ...xi−1 Dxi (ξ k ).
Definition 2. A conserved vector is a j-tuple Y = (Y 1 , Y 2 , Y 3 , ..., Y (j ) ),i.e
Dn Y n = 0, n = 1, 2, 3, ..., j (2.3)
2
, holds for all solutions of partial differential equations (2.1).
Definition 3. The operator for each α, that is given by
δ ∂ X ∂
Eu = = α+ (−1)s Dx1 ...Dxi τxα1 ...xi α , α = 1, 2, ..., t, (2.4)
δu ∂d ∂dx1 ...xi
i≥1
is known as the Euler-Lagrange operator or simply an Euler operator.
Definition 4. Let Λµ be the multiplier which is equal to Λµ (z, d, d1 , d2 , d3 , ..., dn̂ )
for the conserved vectors of the system (2.1), which satisfies
Λµ (Fµ ) = Dn Yn . (2.5)
Definition 5. Suppose that the generalized symmetry operator denoted by Z which
is associated the conserved quantities denoted by Ý and given (2.1) [13]. If Z and
Y satisfy
Z (Y i ) − Y j Dj (ξ i ) + Y i Dj (ξ j ) = 0, (2.6)
then Z is related to Y .
The conservation laws satisfied the divergence properties that is Dr Y r +Ds Y s =
0, where Y r and Y s in terms of (t1 , x1 ) are
Y t1 Dt1 (r) + Y x1 Dx1 (r)
Yr = , (2.7)
Dt1 (r)Dx1 (s) − Dt1 (s)Dx1 (r)
Y t1 Dt1 (s) + Y x1 Dx1 (s)
Ys = . (2.8)
Dt1 (r)Dx1 (s) − Dt1 (s)Dx1 (r)
Theorem 1. Suppose that Y i is conservation laws and their divergence properties
represents as Di Y i = 0. The mathematical formulations defined as the following
Y1 Y1 Y1 Ỹ 1
Y2 Y2 Y2 Ỹ 2
. .
, J . = MT
−1 T
.
. = J (M ) . , (2.9)
. .
. . . .
Yn Yn Yn Ỹ n
where M , M −1 and J can be obtained from
D1 t1 D1 t2 . . . D1 tn D1 t1 D1 t2 . . . D1 tn
D2 t1 D2 t2 . . . D2 tn D2 t1 D2 t2 . . . D2 tn
. . ... . , M −1 = . . ... .
M =
.
, (2.10)
. ... .
.
. ... .
. . ... . . . ... .
Dn t1 Dn t2 . . . Dn tn Dn t1 Dn t2 . . . Dn tn ,
and J = det(M ).
3
Invariant Solutions of Multidimensional
Modified Korteweg–de Vries Equations
a
Department of Mathematics, Riphah International University, Faisalabad
Campus, Faisalabad 38000, Pakistan.
b
Department of Mathematics, Riphah International University, Faisalabad
Campus, Faisalabad 38000, Pakistan.
Abstract
This paper presents a classical Lie symmetry analysis, conservation laws, and
invariant solutions for a generalized family of modified Korteweg–de Vries (mKdV)
equations in (1+1), (1+2), and (1+3) dimensions. Using the multiplier method and
double reduction approach, several new conservation laws and exact analytical
solutions are obtained. The results extend existing studies on KdV-type equations
to higher dimensions and demonstrate the power of symmetry-based techniques in
understanding nonlinear partial differential equations with physical applications
in fluid dynamics and wave propagation.
Keywords: Lie symmetries, conservation laws, double reduction theory, op-
timal system and invariant solutions.
1 Introduction
Partial differential equations (PDEs) play a fundamental role in modeling a wide
range of natural and physical phenomena, including fluid dynamics, plasma phys-
ics, nonlinear optics, and wave propagation. The search for exact solutions to
nonlinear PDEs remains a central topic in applied mathematics, as such solutions
provide important insight into the structural and qualitative behavior of complex
systems. Analytical methods that exploit the intrinsic symmetries of differential
equations have proven especially powerful in reducing, classifying, and solving
nonlinear problems.
Among these analytical methods, the Lie symmetry approach, developed by
Sophus Lie, has established itself as a systematic and elegant framework for study-
ing continuous symmetry transformations. Lie symmetries enable the reduction of
PDEs to simpler forms, often transforming them into ordinary differential equa-
tions (ODEs) that are more manageable to solve. Over the years, numerous re-
searchers have extended and refined Lie’s classical method to handle increasingly
complex and multidimensional systems. This approach has become a cornerstone
in the qualitative analysis of nonlinear differential equations.
An equally important contribution to the theory of symmetries and conserved
quantities was made by Emmy Noether, whose theorem established a deep and
fundamental link between variational symmetries and conservation laws. Conserva-
tion laws are essential for understanding the stability, energy balance, and physical
meaning of differential systems. They also provide useful checks for the correctness
of analytical and numerical solutions. The concept of a conservation law is closely
associated with physical principles such as the conservation of mass, momentum,
and energy. Various researchers, including Bluman and Kumei, Hereman, and Naz,
1
,have developed systematic frameworks for constructing conservation laws of non-
linear PDEs.
One of the most effective modern techniques for deriving conservation laws
is the multiplier approach, proposed and developed by several authors, including
Filho and Figueiredo. This method does not require the existence of a Lagrangian
and can therefore be applied to a wide class of non-variational PDEs. By de-
termining appropriate multipliers, one can construct conserved vectors associated
with the governing equations. These conserved quantities, when combined with Lie
symmetries, play a crucial role in obtaining exact solutions through the double re-
duction theory, which allows for the simultaneous reduction of both independent
and dependent variables.
The present study focuses on the Lie symmetry analysis, conservation laws, and
invariant solutions of a family of generalized modified Korteweg–de Vries (mKdV)
equations in (1+1), (1+2), and (1+3) dimensions. These equations arise in many
areas of applied science, including plasma waves, nonlinear lattice theory, and fluid
mechanics. By employing the classical Lie group method together with the mul-
tiplier technique and double reduction approach, new classes of conservation laws
and exact analytical solutions are derived. The obtained results extend previously
known one-dimensional models to higher dimensions, revealing deeper structural
properties and providing new insights into the integrability of nonlinear evolution
equations.
2 Fundamental Operators
This section provides the mathematical framework and operators used in the sym-
metry and conservation law analysis. The key definitions of infinitesimal generat-
ors, conserved vectors, and Euler–Lagrange operators are summarized below for
completeness.
Assume a system of partial differential equations (PDEs) that can be written
in the nth order with r independent variables z = (z 1 , z 2 , z 3 , ..., z r ) and t dependent
variables d = (d1 , d2 , d3 , ..., dt ) and µ = 1, 2, 3, ..., t,
Fµ = (z, d, d1 , d2 , d3 , ..., dn ), (2.1)
here d1 , d2 , d3 , ..., dn are the nth order partial derivatives.
Definition 1. The infinitesimal generator Z’s prolongation of order i is defined
as
∂ ∂
Z [i] = Z + τx + ... + τx1 ...xi , (2.2)
∂dx ∂dx1 ...xi
′
here τ s are given by
τxs = Dx (η s ) − dsl Dx (ξ l ),
..
.
τxs1 ...xi = Dxi (ηxs 1 ...xi−1 ) − dskx1 ...xi−1 Dxi (ξ k ).
Definition 2. A conserved vector is a j-tuple Y = (Y 1 , Y 2 , Y 3 , ..., Y (j ) ),i.e
Dn Y n = 0, n = 1, 2, 3, ..., j (2.3)
2
, holds for all solutions of partial differential equations (2.1).
Definition 3. The operator for each α, that is given by
δ ∂ X ∂
Eu = = α+ (−1)s Dx1 ...Dxi τxα1 ...xi α , α = 1, 2, ..., t, (2.4)
δu ∂d ∂dx1 ...xi
i≥1
is known as the Euler-Lagrange operator or simply an Euler operator.
Definition 4. Let Λµ be the multiplier which is equal to Λµ (z, d, d1 , d2 , d3 , ..., dn̂ )
for the conserved vectors of the system (2.1), which satisfies
Λµ (Fµ ) = Dn Yn . (2.5)
Definition 5. Suppose that the generalized symmetry operator denoted by Z which
is associated the conserved quantities denoted by Ý and given (2.1) [13]. If Z and
Y satisfy
Z (Y i ) − Y j Dj (ξ i ) + Y i Dj (ξ j ) = 0, (2.6)
then Z is related to Y .
The conservation laws satisfied the divergence properties that is Dr Y r +Ds Y s =
0, where Y r and Y s in terms of (t1 , x1 ) are
Y t1 Dt1 (r) + Y x1 Dx1 (r)
Yr = , (2.7)
Dt1 (r)Dx1 (s) − Dt1 (s)Dx1 (r)
Y t1 Dt1 (s) + Y x1 Dx1 (s)
Ys = . (2.8)
Dt1 (r)Dx1 (s) − Dt1 (s)Dx1 (r)
Theorem 1. Suppose that Y i is conservation laws and their divergence properties
represents as Di Y i = 0. The mathematical formulations defined as the following
Y1 Y1 Y1 Ỹ 1
Y2 Y2 Y2 Ỹ 2
. .
, J . = MT
−1 T
.
. = J (M ) . , (2.9)
. .
. . . .
Yn Yn Yn Ỹ n
where M , M −1 and J can be obtained from
D1 t1 D1 t2 . . . D1 tn D1 t1 D1 t2 . . . D1 tn
D2 t1 D2 t2 . . . D2 tn D2 t1 D2 t2 . . . D2 tn
. . ... . , M −1 = . . ... .
M =
.
, (2.10)
. ... .
.
. ... .
. . ... . . . ... .
Dn t1 Dn t2 . . . Dn tn Dn t1 Dn t2 . . . Dn tn ,
and J = det(M ).
3
