Introduction

Solitonic Solutions for the Coupled Burgers Equation in Space-Time Using Composite Derivatives

Nápoles Valdes

Juan Eduardo

UNNE, FaCENA, Ave. Libertad 5450, Corrientes 3400, Argentina; UTN-FRRE, French 414, Resistencia,Chaco 3500, Argentina.

In this article, we investigate the coupled Burgers equation in spacetime governed by the local composite derivative CN. By defining a generalized traveling variable transformation, we reduce the coupled system of partial differential equations to integrable ordinary differential equations. Using a hyperbolic ansatz, we derive soliton solutions and evaluate them for three representative kernel classes: potential, exponential, and Mittag-Leffler. We also show that the conformable derivative formulation represents only a restricted monomial subset of our generalized integral framework. Numerical simulations and contour visualizations illustrate how the choice of kernel distorts the wavefront geometry and phase propagation in nonhomogeneous media.

Coupled Burgers’ equationComposite CN-derivativeExact soliton solutionsIntegral

traveling wave variable.

1Introduction

Unlike other local operators proposed in recent literature, which are often criticized for simply rescaling the ordinary derivative using the chain rule, the composite derivative (CN-derivative) introduces a non-trivial deformation of the domain that formally extends the class of differentiable functions. By evaluating the local rate of change over an argument transformed by a kernel N(x, α), this operator allows us to analyze and differentiate functions that exhibit sharp points or vertical tangents in the classical sense, while preserving a manageable algebraic structure. This ability to re-metrize space and time makes the composite derivative a powerful analytical tool for modeling complex nonlinear phenomena in heterogeneous media and with non-uniform spatiotemporal response rates.

In [6] we define the composite derivative, as follows:

Definition 1. Given a function f : [0, +∞) → R. Then the CN-derivative of f of order α (the composite N derivative) is defined by

α f(N(x, α)) − f(N(a, α)) fCN (a) = lim , α ∈ (0, 1].

x→a x − a

If f is differentiable at N, and N is differentiable at a, then

fCN α (a) = fN ′ (a)N ′(a, α)

We must point out that, unlike other local operators, where the criticism focuses on the fact that they are a simple rescaling of the ordinary derivative, this composite derivative expands the class of differentiable functions.

1.1Examples of Functions Not Classically Differentiable but Differentiable in the CN Sense

The fundamental property of the CN operator is that the function f does not need to be differentiable at the original point a, but only at the transformed point N(a, α). Furthermore, the appropriate choice of kernel N(x, α) can "smooth" out sharp corners, peaks, or classical non-differentiable behavior.

Below are two examples of functions that are not differentiable in the classical sense but are differentiable in the composite sense.

Example 1. Consider the absolute value function f(t) = |t| at the origin t = 0.

Left-hand − |t|−0 = −1 limit: limt→0 t

|t|−0

Right-hand limit: limt→0+ t = 1

Since the left-hand and right-hand limits do not coincide, the classical derivative f ′(0) does not exist (it has a corner).

Analysis with Composite Derivatives (CN). Let’s choose a smooth shift kernel for the variable, for example:

N(t, α) = t + (1 − α)

for a fixed parameter α ∈ (0, 1).

Evaluating the kernel at t = 0:

N(0, α) = 1 − α > 0

Behavior of f at N(0, α): For any α ∈ (0, 1), the value N(0, α) is strictly positive. Since f(x) = |x| = x for all x > 0, the function f is completely differentiable in the neighborhood of the point N(0, α), with derivative f ′(N(0, α)) = 1.

Let’s corroborate this by the definition:

α |N(t, α)| − |N(0, α)| |t + 1 − α| − (1 − α) fCN (0) = lim = lim t→0 t − 0 t→0 t

Since 1 − α > 0, for t close enough to 0, we have that t + 1 − α > 0:

α (t + 1 − α) − (1 − α) t fCN (0) = lim = lim = 1 t→0 t t→0 t If we calculate using the kernel formula, given that N ′(t, α) = dt d (t + 1 − α) = 1:

fCN α (0) = f ′ (N(0, α)) · N ′(0, α) = f ′(1 − α) · 1 = 1 · 1 = 1

The function f(t) = |t| is not differentiable at t = 0 in the classical sense, but it is differentiable at t = 0 in the CN sense with fCN α (0) = 1.

Example 2. Consider the function f(t) = t1/3 at t = 0.

f ′(t) = 1 3 t−2/3 Evaluating at the origin, f ′(0) = limt→0 t1/3 t−0 = limt→0 t21 /3 = +∞

The function has a vertical tangent at t = 0, so the classical derivative f ′(0) does not exist in R.

We choose a kernel of the shifted quadratic power type:

N(t, α) = (t + 1)3/α − 1 or simply a potential kernel N(t, α) = (t + α)3

For greater algebraic simplicity, let’s take N(t, α) = (t + α)3 with α > 0.

Evaluating the kernel at t = 0:

N(0, α) = α3 > 0

Ordinary derivative of the kernel:

′ ∂ 3 2 ′ 2 N (t, α) = (t + α) = 3(t + α) =⇒ N (0, α) = 3α

Evaluating f at the kernel:

(︁ 3)︁1/3 f(N(t, α)) = (t + α) = t + α

Note that the composition f(N(t, α)) transforms the function with a vertical tangent into a smooth linear polynomial t + α.

Applying the definition by limit:

α f(N(t, α)) − f(N(0, α)) (t + α) − α t fCN (0) = lim = lim = lim = 1 t→0 t − 0 t→0 t t→0 t

Verification by core product rule:

′ ′ ′ 3 1 3 −2/3 1 fN (0) = f (N(0, α)) = f (α ) = (α ) = 2 3 3α

Multiplying by N ′(0, α) = 3α2:

(︃ )︃ α ′ ′ 1 (︁ 2)︁ fCN (0) = fN (0) · N (0, α) = 2 · 3α = 1

The function f(t) = t1/3 does not have a finite derivative at t = 0 in the classical sense, but its composite derivative at t = 0 exists and is a finite real value (fCN α (0) = 1).

The Burgers equation is one of the fundamental mathematical models in the physics of nonlinear phenomena, as it represents the simplest scalar formulation that balances nonlinear convection with diffusion or dissipation effects. Historically considered a one-dimensional simplification of the Navier-Stokes equations, its importance in current research transcends fluid mechanics, extending to hydrodynamics, nonlinear acoustics, traffic dynamics, cosmology (in the formation of large-scale structures in the universe), and shock wave propagation. In the context of contemporary science, the analysis of the Burgers equation using generalized differential operators represents a field of intense research, as it allows for the modeling of soliton and wavefront propagation in non-homogeneous, complex, or microstructured media, offering a key theoretical window for understanding realistic transport processes in systems far from equilibrium.

Initially, the basic reference to the Burger equation was introduced by J. M. Burger in 1974 ( [1]). Hopf and Cole analyzed the Burger equation in the context of gas dynamics (see [3, 7]). Subsequently, Lian Yang et al. (in [9]) studied the small-amplitude, long-wavelength limit of the gas dynamics system for the Burger equation. Under the effect of gravity, the coupled Burger equation (1 + 1) (see [4]) is a simple model of the evolution and sedimentation of volumetric deposition at scales of two types of elements in fluid suspensions or colloids.

Let N1(t, α) and N2(s, β) be the kernels associated with the time variables t and space variables s, respectively. Consider the coupled nonlinear partial system:

wCN α (t) − wCN 2β (s) + 2w(s, t) wCN β (s) + k [w(s, t)z(s, t)]β CN (s) = 0 α 2β β β (1) zCN (t) − zCN (s) + 2z(s, t) zCN (s) + l [w(s, t)z(s, t)]CN (s) = 0

(︂ )︂β where the second-order operator is equivalent to the iterated application wCN 2β (s) = wCN β (s) (s), s > 0,

t > 0, 0 < α, β ≥ 1. In the case of N is the identity we have the conventionally coupled Burger’s equation.

In [12], the coupled system of fractional Burgers equations in space-time defined in the sense of the conformable derivative was investigated:

∂αw ∂2βw ∂βw ∂β(wz) α − 2β + 2w β + k β = 0 ∂t ∂s ∂s ∂s

∂αz ∂2βz ∂βz ∂β(wz) α − 2β + 2z β + l β = 0 ∂t ∂s ∂s ∂s

where t > 0, s > 0, 0 < α, β ≤ 1, and the derivatives ∂t ∂α α and ∂s ∂β β are the local operators of the introduced conformable derivative in [8].

The main objective of this work is to extend the framework of coupled nonlinear systems by studying the spacetime-coupled Burgers equation governed by the composite derivative CN. Employing a generalized traveling variable transformation defined by integrals over the upper bound, we reduce the system of partial differential equations to a system of integrable ordinary differential equations. Furthermore, using an ansatz of hyperbolic functions, we obtain exact soliton solutions and evaluate them for three distinct kernel classes: potential, exponential, and Mittag-Leffler. This approach demonstrates how the choice of kernel distorts the waveform profile and phase geometry, providing a more general framework than that of conventional and conformable local operators.

2Results

Theorem 1. Under the composite traveling variable transformation

∫︂ s ∫︂ t dτ dτ ξ(s, t) = m − n 0 N2(τ, β) 0 N1(τ, α) with w(s, t) = W(ξ) and z(s, t) = Z(ξ) (where m, n ∈ R \ {0}), the nonlinear system 1 reduces to the following system of integrated ordinary differential equations (fixing the constant of integration to zero):

−nW(ξ) − m2W ′(ξ) + mW 2(ξ) + kmW(ξ)Z(ξ) = 0 −nZ(ξ) − m2Z′(ξ) + mZ2(ξ) + lmW(ξ)Z(ξ) = 0

Proof : Calculating the derivative of the traveling variable ξ(s, t) we have:

(︃ ∫︂ s )︃ ∂ξ ∂ dτ m = m = ∂s ∂s 0 N2(τ, β) N2(s, β) (︃ ∫︂ t )︃ ∂ξ ∂ dτ −n = −n = ∂t ∂t 0 N1(τ, α) N1(t, α)

Applying Definition 1 to w(s, t) = W(ξ(s, t)) in the variable t:

[︃ ]︃ α ∂w wCN (t) = · N1(t, α) ∂t t→N1(t,α)

Using the chain rule for the ordinary partial derivative:

(︃ )︃ ∂w ′ ∂ξ ′ −n = W (ξ) = W (ξ) ∂t ∂t N1(t, α)

Substituting into the definition of wCN α (t):

(︃ )︃ α ′ −n ′ wCN (t) = W (ξ) · N1(t, α) = −nW (ξ) N1(t, α)

Identically for the spatial coordinate s:

[︃ ]︃ (︃ )︃ β ∂w ′ m ′ wCN (s) = · N2(s, β) = W (ξ) ′ · N2(s, β) = mW (ξ) ∂s s→N2(s,β) N2(s, β)

(︂ )︂β For the double composition wCN β (s), let us denote the intermediate function g(s) = wCN β (s) = mW ′(ξ(s, t)).

Applying the composite operator on g(s):

(︂ )︂ [︃ ]︃ β β β ∂g wCN (s) = gCN (s) = · N2(s, β) CN ∂s s→N2(s,β)

Let’s calculate the ordinary derivative of g(s):

(︃ )︃ ∂g ∂ ′ ′′ ∂ξ ′′ m = (mW (ξ)) = mW (ξ) = mW (ξ) ∂s ∂s ∂s N2(s, β)

Substituting in the composition double:

(︂ )︂ (︃ )︃ β β 2 ′′ 1 2 ′′ wCN (s) = m W (ξ) ′ · N2(s, β) = m W (ξ) CN N2(s, β)

Similarly, for the nonlinear term of the product and the components of z(s, t) = Z(ξ):

[w(s, t)z(s, t)]CN (s) = m(W Z) (ξ)

(︂ )︂β zCN α (t) = −nZ′(ξ), zCN β (s) = mZ′(ξ), zCN β (s) = m2Z′′(ξ)

Substituting the simplified expressions in the first ODE:

−nW ′(ξ) − m2W ′′(ξ) + 2mW(ξ)W ′(ξ) + km(W Z)′(ξ) = 0

Since 2W W ′ = (W 2)′, we group under the ordinary derivative with respect to ξ:

d (︁ 2 ′ 2 )︁ −nW(ξ) − m W (ξ) + mW (ξ) + kmW(ξ)Z(ξ) = 0 dξ

Integrating with respect to ξ and setting the integration constant as 0:

−nW − m2W ′ + mW 2 + kmW Z = 0 (2)

Performing the exact same procedure for the second PDE:

d (︁ 2 ′ 2 )︁ −nZ(ξ) − m Z (ξ) + mZ (ξ) + lmW(ξ)Z(ξ) = 0 dξ

Integrating with a constant of zero:

−nZ − m2Z′ + mZ2 + lmWZ = 0 (3)

This completes the proof of the Theorem. □

Remark 1. It is worth emphasizing that the conformable derivative used in [12] does not constitute an independent or more general operator, but a particular subcase of the composite derivative CN. Indeed, by selecting the specific potential kernel

xα NC(x, α) = , α ∈ (0, 1],

whose derivative is NC ′ (x, α) = xα−1 = x11 −α , the composite derivative reduces directly to

(︃ α )︃ α ′ x 1 α 1−α df(x) fCN (x) = f · 1−α ⇐⇒ D f(x) = x .

α x dx

Therefore, under the choice of spatial and temporal kernels N2(s, β) = sβ β and N1(t, α) = tα α , our generalized traveling variable ∫︂ s ∫︂ t dτ dτ ξ(s, t) = m − n 0 N2(τ, β) 0 N1(τ, α) (︂ )︂ (︂ )︂ reduces exactly to the conformable traveling variable ξconf(s, t) = m 1− ββ s1−β − n 1− αα t1−α , where the scaling factors are absorbed within the wave speed parameters m and n. This analytical reduction demonstrates rigorously that the results obtained in said work, represent a particular case of our composite framework CN, which additionally accommodates exponential, Mittag-Leffler behaviors and non-differentiable functions in the classical sense.

Theorem 2. (Generalized Exact Soliton Solutions)

Under the hyperbolic transformation given by dγ dξ = sinh(ξ) (with sinh(γ) = −csch(ξ) and cosh(γ) = −coth(ξ)), the generalized soliton solutions for kl ̸= 1 are given by:

m(1 − k) w(s, t) = [1 + coth (ξ(s, t)) − csch (ξ(s, t))] 2(kl − 1)

m(1 − l) z(s, t) = [1 + coth (ξ(s, t)) − csch (ξ(s, t))] 2(kl − 1)

where the exact traveling variable in the composite sense is defined as before.

Proof : In the reduced ODE −nW −m2W ′ +mW 2 +kmW Z = 0, we balance the term with the highest derivative W ′ with the nonlinear term W 2:

Degree(W ′) = N + 1, Degree(W 2) = 2N =⇒ N + 1 = 2N =⇒ N = 1

We formulate the hyperbolic polynomial ansatz of degree 1:

W(ξ) = A0 + A1 cosh(γ) + B1 sinh(γ)

Z(ξ) = C0 + C1 cosh(γ) + D1 sinh(γ)

Using dγ dξ = sinh(ξ) and the relation sinh(ξ) = −csch(γ):

′ dW dγ W (ξ) = = (A1 sinh(γ) + B1 cosh(γ)) sinh(ξ) = −A1 − B1 coth(γ) dγ dξ

Substituting W(ξ), Z(ξ) and W ′(ξ) into the reduced ODEs of Theorem 1 and setting each coefficient of the hyperbolic basis {1, cosh(γ), sinh(γ), . . . } equal to zero, we solve the nonlinear algebraic system:

m(1 − k) m(1 − k) m(1 − k) A0 = , A1 = − , B1 = 2(kl − 1) 2(kl − 1) 2(kl − 1)

m(1 − l) m(1 − l) m(1 − l) C0 = , C1 = − , D1 = 2(kl − 1) 2(kl − 1) 2(kl − 1)

Substituting the coefficients obtained in W(ξ):

m(1 − k) W(ξ) = [1 − cosh(γ) + sinh(γ)] 2(kl − 1)

Replacing cosh(γ) = − coth(ξ) and sinh(γ) = −csch(ξ):

m(1 − k) m(1 − k) W(ξ) = [1 − (− coth(ξ)) + (−csch(ξ))] = [1 + coth(ξ) − csch(ξ)] 2(kl − 1) 2(kl − 1)

Analogously for Z(ξ):

m(1 − l) Z(ξ) = [1 + coth(ξ) − csch(ξ)] 2(kl − 1) Substituting the defined traveling variable, and given that the solutions in continuous space are given by w(s, t) = W(ξ(s, t)) and z(s, t) = Z(ξ(s, t)), we evaluate ξ in the hyperbolic expressions:

For the first component w(s, t):

[︃ (︃ ∫︂ s ∫︂ t )︃ (︃ ∫︂ s ∫︂ t )︃]︃ m(1 − k) dτ dτ dτ dτ w(s, t) = 1 + coth m ′ − n ′ − csch m ′ − n ′ 2(kl − 1) 0 N2(τ, β) 0 N1(τ, α) 0 N2(τ, β) 0 N1(τ, α)

For the second component z(s, t):

[︃ (︃ ∫︂ s ∫︂ t )︃ (︃ ∫︂ s ∫︂ t )︃]︃ m(1 − l) dτ dτ dτ dτ z(s, t) = 1 + coth m ′ − n ′ − csch m ′ − n ′ 2(kl − 1) 0 N2(τ, β) 0 N1(τ, α) 0 N2(τ, β) 0 N1(τ, α)

Remark 2. To ensure the rigor that these solutions satisfy the original PDEs, let’s check the cancellation of the kernels when deriving w(s, t):

Derivative with respect to t:

(︃ ∫︂ t )︃ ∂ξ ∂ dτ 1 = −n ′ = −n ′ ∂t ∂t 0 N1(τ, α) N1(t, α)

Applying the composite derivative:

[︃ ]︃ (︃ )︃ α ∂w ′ ′ −n ′ ′ wCN (t) = · N1(t, α) = W (ξ) · ′ N1(t, α) = −nW (ξ) ∂t t→N1(t,α) N1(t, α)

Derivative with respect to s:

(︃ ∫︂ s )︃ ∂ξ ∂ dτ 1 = m ′ = m ′ ∂s ∂s 0 N2(τ, β) N2(s, β)

Applying the composite derivative and its double composition:

(︃ )︃ β ′ m ′ ′ wCN (s) = W (ξ) · ′ N2(s, β) = mW (ξ) N2(s, β)

(︂ )︂ (︃ )︃ β β ′′ m ′ 2 ′′ wCN (s) = mW (ξ) · ′ N2(s, β) = m W (ξ) CN N2(s, β)

Substituting these derivatives into the original coupled equations, the differential identity 0 = 0 is trivially satisfied for all (s, t) in the domain of definition of the integrals.

3Numerical Examples

To simplify the final numerical expressions and keep the focus on the action of the nuclei, we fix the following algebraic and system constants:

Burgers system parameters: k = 2, l = 3. Note that kl − 1 = (2)(3) − 1 = 5 = 0 ̸ .

Amplitude coefficients:

m(1 − k) m(1 − 2) m = = − 2(kl − 1) 2(5) 10 m(1 − l) m(1 − 3) 2m m = = − = − 2(kl − 1) 2(5) 10 5

Wave velocities/weights: m = 2, n = 1. Derivative orders α = 1 2 , β = 2 1 .

With these numerical parameters fixed, the solutions reduce to:

w(s, t) = − [1 + coth (ξ(s, t)) − csch (ξ(s, t))]

z(s, t) = − [1 + coth (ξ(s, t)) − csch (ξ(s, t))]

where the traveling variable takes the explicit form:

∫︂ s ∫︂ t dτ dτ ξ(s, t) = 2 1 − n 1 0 N2(τ, 2 ) 0 N1(τ, 2 )

Let us consider three particular kernels to illustrate Theorem 3.

Example 3. Potential Kernel

N1(t, 1/2) = 2√t, N2(s, 1/2) = 2√s

Space integral:

∫︂ s dτ = √s 0 2√τ Time integral:

∫︂ t dτ = √t 0 2√τ Traveling variable and Solutions

ξP otential(s, t) = 2√s − √t

[︂ (︂ )︂ (︂ )︂]︂ w1(s, t) = − 1 + coth 2√s − √t − csch 2√s − √t

[︂ (︂ )︂ (︂ )︂]︂ z1(s, t) = − 1 + coth 2√s − √t − csch 2√s − √t

Example 4. Exponential Kernel

N1(t, 1/2) = et/2, N2(s, 1/2) = es/2

Space integral:

∫︂ s ∫︂ s (︂ )︂ dτ −τ /2 −s/2 τ /2 = e dτ = 2 1 − e 0 e 0 Time integral:

∫︂ t (︂ )︂ dτ −t/2 τ /2 = 2 1 − e

Traveling variable and Solutions

[︂ ]︂ [︂ ]︂ ξExponential(s, t) = 2 2(1 − e−s/2) − 1 2(1 − e−t/2) = 2 − 4e−s/2 + 2e−t/2

[︂ (︂ )︂ (︂ )︂]︂ 1 −s/2 −t/2 −s/2 −t/2 w2(s, t) = − 1 + coth 2 − 4e + 2e − csch 2 − 4e + 2e

[︂ (︂ )︂ (︂ )︂]︂ 2 −s/2 −t/2 −s/2 −t/2 z2(s, t) = − 1 + coth 2 − 4e + 2e − csch 2 − 4e + 2e

Example 5. Mittag-Leffler Type Kernel (Regularized Sum) Let’s consider the linearized kernel associated with Mittag-Leffler:

N1(t, 1/2) = 1 + √t, N2(s, 1/2) = 1 + √s

Space integral:

∫︂ s dτ 0 1 + √τ

Setting u = √τ =⇒ dτ = 2u du:

∫︂ √s ∫︂ √s (︃ )︃ 2u du 1 = 2 1 − du = 2√s − 2 ln(1 + √s) 0 1 + u 0 1 + u

Time integral:

∫︂ t dτ = 2√t − 2 ln(1 + √t) 0 1 + √τ

Traveling variable and Solutions

[︁ ]︁ [︂ ]︂ ξM L(s, t) = 2 2√s − 2 ln(1 + √s) − 1 2√t − 2 ln(1 + √t)

ξM L(s, t) = 4√s − 2√t − 4 ln(1 + √s) + 2 ln(1 + √t)

w3(s, t) = − [1 + coth (ξM L(s, t)) − csch (ξM L(s, t))]

z3(s, t) = − [1 + coth (ξM L(s, t)) − csch (ξM L(s, t))]

A heat map (or contour plot) for w1(s, t) visually represents how the amplitude of the soliton wave w(s, t) varies simultaneously in space (s) and time (t) under the influence of a given nucleus. In other words, the heat map represents the spatiotemporal evolution of the soliton solution w(s, t) for the composite-coupled Burgers equation. The color at each point (s, t) denotes the local value of the wave profile. Unlike conventional traveling waves that move at a constant speed along straight lines in the (s, t) plane, the colored bands in Wpotential exhibit a parabolic curvature (with potential kernel). This is because the composite traveling variable ξ = 2√s − √t imposes a nonlinear relationship between position s and time t.

Figure 1