Solitary wave solutions, periodic and superposition solutions to the system of first-order (2+1)-dimensional Boussinesq's equations derived from the Euler equations for an ideal fluid model

This paper concludes the study of (2+1)-dimensional ideal fluid models by deriving a system of first-order Boussinesq equations and demonstrating the existence of various traveling wave solutions, including solitary, cnoidal, and superposition types, for the auxiliary velocity potential function.

The Gist

The Big Picture: Predicting the Shape of a Wave

Imagine you are standing by a calm lake. You throw a stone, and a wave ripples out. Physicists have spent decades trying to write the perfect "recipe" (a mathematical equation) that predicts exactly how that wave will look, how fast it will move, and how it will change as it travels.

For a long time, scientists mostly studied waves moving in a straight line (like a wave in a long, narrow canal). This is the "1D" world. But real oceans are wide and deep; waves move in all directions (forward, sideways, and diagonally). This is the "2D" world.

This paper is the final chapter in a series by two Polish physicists (Piotr Rozmej and Anna Karczewska) who wanted to build a perfect recipe for waves moving in a wide, open ocean, based on the fundamental laws of how water actually moves (the Euler equations).

The Problem: The "Square" vs. The "Rectangle"

In their previous work, the authors studied waves where the ocean was treated like a long, narrow rectangle. They stretched the math to fit a 2D ocean, but they had to assume the wave moved differently in the "length" direction than in the "width" direction. It was a bit like forcing a square peg into a round hole, but it worked well enough to find some cool wave shapes.

The New Challenge:
In this paper, they tackled the "square" problem. They asked: What if the wave spreads out equally in all directions? (Imagine a ripple from a stone dropped in a perfectly still pond).

They discovered something surprising: You cannot simply copy the famous "KdV equation" (the gold standard for 1D waves) and just add a "y" variable to it. If you try to force the math to work like the old 1D version, the equations break down. The water refuses to behave that way when it spreads out evenly.

The Solution: The "Shadow" Function

Since they couldn't solve for the wave height directly (let's call the wave height "The Surface"), they had to use a clever trick.

Imagine trying to figure out the shape of a shadow on a wall without looking at the object casting it. Instead, you look at the object itself.

• The Object: A hidden mathematical function they call $f$ (the velocity potential).
• The Shadow: The actual water surface you see, $\eta$ (the wave height).

The authors found that while they couldn't write a simple equation for the surface ($\eta$), they could write a complex equation for the hidden object ($f$). Once they solved for the hidden object, they could easily calculate what the surface wave looked like.

The Three Types of Waves They Found

Using this "Shadow" method, they discovered three distinct families of waves that can travel across a 2D ocean. Think of these as three different "costumes" the water can wear:

1. The Solitary Wave (The Lone Wolf)

• What it is: A single, perfect hump of water that travels forever without changing shape.
• The Analogy: Imagine a surfer riding a single, massive wave that never breaks and never fades. It's a "soliton."
• The Paper's Finding: They found that these waves exist in 2D, but they are made of two layers: a main "hump" and a smaller, sharper "spike" on top. Depending on how fast the wave moves, the shape changes. If it moves too slowly, the wave becomes so huge it would physically break (unrealistic). If it moves just right, it's a perfect, stable wave.

2. The Cnoidal Wave (The Rolling Hills)

• What it is: A series of waves that repeat over and over, like a train of ocean swells.
• The Analogy: Think of a rolling hillside. Some hills are sharp and pointy (like a sawtooth), while others are smooth and round (like a sine wave).
• The Paper's Finding: They found that these repeating waves can be "tuned." By changing a mathematical knob (called the "elliptic parameter"), you can turn a smooth, gentle swell into a sharp, jagged peak. They also had to ensure that the amount of water pushed up by the wave equals the amount of water pulled down in the trough, so the ocean doesn't magically gain or lose water volume.

3. The Superposition Wave (The "Table-Top" Wave)

• What it is: This is the most exotic discovery. It's a wave that looks like a flat plateau with steep sides.
• The Analogy: Imagine a wave that looks like a table or a mesa (a flat-topped mountain). Instead of a smooth curve, it has a flat top and drops off sharply.
• The Paper's Finding: These are "superposition" waves, meaning they are built by adding two different types of mathematical waves together. The authors call them "table-top" waves. They found that these can exist in 2D, moving in any direction, maintaining their flat, table-like shape as they travel.

Why Does This Matter?

You might ask, "Who cares about flat-topped waves in a math paper?"

1. Realism: Most previous models of 2D waves were "made up" by mathematicians just to see if they could solve them. This paper is different. These equations were derived from the actual laws of physics (how water moves, gravity, and pressure). This means these waves are not just mathematical fantasies; they are what should happen in a real, ideal ocean.
2. Completing the Puzzle: The authors have now finished the job of generalizing the famous 1D wave equations to 2D. They have shown that whether the ocean is narrow or wide, whether the wave is a single hump or a flat table, the math holds up.
3. Future Tech: Understanding these exact shapes helps in designing better ships, predicting coastal erosion, and understanding how energy moves through the ocean.

The Takeaway

The authors successfully solved a difficult puzzle: How do waves behave when they spread out equally in all directions?

They found that you can't just copy the old rules. You have to look at the "hidden" math first. But once you do, you find that the ocean is full of beautiful, stable patterns: lone surfer waves, rolling hills, and even flat-topped "table" waves. They have closed the book on this specific chapter of fluid dynamics, proving that even in a complex 2D world, nature still follows elegant, predictable patterns.

Technical Summary

1. Problem Statement

The paper addresses the derivation and solution of nonlinear wave equations in two spatial dimensions and one temporal dimension ($(2+1)$-D) for an ideal, incompressible, irrotational fluid with a free surface.

• Context: While many studies propose $(2+1)$-dimensional wave equations (like the Kadomtsev-Petviashvili or KP equation) based on mathematical analogy or integrability, few are rigorously derived from the fundamental Euler equations of hydrodynamics.
• Specific Challenge: The authors focus on the case of uniform scaling of horizontal coordinates ($x$ and $y$), where the scaling factors are identical ($L_x = L_y$). In this scenario, the standard perturbation approach used to derive a single nonlinear wave equation for the surface elevation $\eta(x,y,t)$ (analogous to the KdV equation in 1D) fails. The resulting system consists of two coupled first-order Boussinesq equations that cannot be easily decoupled into a single equation for $\eta$ due to the presence of mixed spatial derivatives in the zeroth-order terms.
• Goal: To overcome this decoupling difficulty, derive a governing equation for the velocity potential's auxiliary function $f(x,y,t)$, and find exact traveling wave solutions (solitary, periodic cnoidal, and superposition types) for the surface profile $\eta$.

2. Methodology

The authors employ a rigorous perturbation approach based on the Euler equations for an ideal fluid.

• Scaling and Non-dimensionalization:
  • The fluid domain is scaled using small parameters: $\alpha = A/H$ (nonlinearity), $\beta = (H/L_x)^2$ (shallowness in $x$), and $\gamma = (H/L_y)^2$ (shallowness in $y$).
  • Uniform Scaling Assumption: The paper specifically investigates the case where $\alpha \approx \beta \approx \gamma \ll 1$. This implies $L_x \approx L_y$, representing waves with similar wavelengths in all horizontal directions.
• Derivation of the System:
  • Starting from the Laplace equation for the velocity potential $\phi$ and boundary conditions, the potential is expanded as a power series in the vertical coordinate $z$.
  • Substituting this expansion into the kinematic and dynamic boundary conditions yields a system of two first-order Boussinesq equations for the surface elevation $\eta$ and the auxiliary function $f$ (where $f = \phi|_{z=0}$).
  • Unlike previous works where $\gamma$ was of higher order ($O(\beta^2)$), here $\gamma$ is of the same order as $\beta$, preventing the elimination of $f$ to get a single equation for $\eta$.
• Transformation to Traveling Waves:
  • The authors assume traveling wave solutions of the form $\xi = kx + ly - \omega t$.
  • By substituting the traveling wave ansatz into the coupled system, they derive a single fourth-order Ordinary Differential Equation (ODE) for the derivative of the auxiliary function, $F(\xi) = f'(\xi)$.
  • The resulting ODE is of the form: $aF'' + 2bFF'' + cF^{(4)} = 0$, which integrates to a first-order equation resembling the energy equation of the KdV equation: $(F')^2 = P_3(F)$, where $P_3$ is a cubic polynomial.
• Solution Strategy:
  • The authors solve the integrated ODE using various ansatzes based on Jacobi elliptic functions ($cn, dn, sn$) and their combinations.
  • They determine integration constants and coefficients to satisfy the equation exactly.
  • Finally, they reconstruct the surface elevation $\eta(\xi)$ using the relation derived from the boundary conditions: $\eta = -[f_t + \frac{1}{2}\alpha(f_x^2 + f_y^2) - \dots]$.
  • Volume Conservation: A constant $A_{00}$ is added to the periodic solutions to ensure the conservation of fluid volume (mass) over one wavelength.

3. Key Contributions

1. Derivation of the $(2+1)$-D Boussinesq System: The paper rigorously derives the first-order Boussinesq system for the case of uniform horizontal scaling, a scenario where previous attempts to derive a single KdV-type equation failed.
2. Reduction to a Solvable ODE: The authors successfully transformed the coupled system of PDEs into a single ODE for the auxiliary function $f$, allowing for the analytical determination of wave profiles.
3. Discovery of New Solution Families: The paper presents three distinct families of exact traveling wave solutions for the surface elevation $\eta$:
   • Solitary Waves: Solutions involving $\text{sech}^2$ and $\text{sech}^4$ terms.
   • Periodic Cnoidal Waves: Solutions involving $\text{cn}^2$ and $\text{cn}^4$ terms.
   • Superposition Solutions: A novel class of solutions involving linear combinations of Jacobi elliptic functions (specifically $\text{dn}^2 \pm \sqrt{m}\text{cn}\cdot\text{dn}$), which were previously identified in 1D but are now extended to this $(2+1)$-D model.
4. Physical Interpretation: The paper provides detailed analysis of the wave profiles, including the decomposition into components (e.g., $\eta_2$ and $\eta_4$) and the conditions under which these solutions are real and physically meaningful (e.g., constraints on wave velocity $v$ and elliptic modulus $m$).

4. Results

• Solitary Waves: The surface profile $\eta$ is found to be a sum of $\text{sech}^2$ and $\text{sech}^4$ functions. The amplitude grows rapidly with wave velocity. Two branches of solutions exist: one with velocity $v > 1$ (physically relevant) and one with $v < 1/\sqrt{3}$ (unphysical due to large amplitudes).
• Cnoidal Waves: The solutions are expressed as $\eta = A_{00} + A_2 \text{cn}^2(B\xi, m) + A_4 \text{cn}^4(B\xi, m)$.
  • The paper identifies two branches of admissible solutions based on the elliptic parameter $m$: $m \in (0.5, 1)$ corresponding to $v > 1$, and $m \in (0, 0.5)$ corresponding to $1/\sqrt{3} < v < 1$.
  • As $m \to 1$, the periodic waves converge to solitary waves.
• Superposition Solutions: The paper demonstrates the existence of "table-top" periodic waves formed by the superposition of $\text{dn}^2$ and $\text{cn}\cdot\text{dn}$ terms. These solutions exhibit flat tops and are distinct from standard cnoidal waves.
• 3D Profiles: The solutions exhibit translational symmetry along the direction perpendicular to the wave propagation vector, confirming their nature as $(2+1)$-dimensional traveling waves.

5. Significance

• Closing the Theoretical Gap: This work completes the generalization of KdV-type equations to $(2+1)$ dimensions derived directly from the Euler equations for an ideal fluid. It covers both non-uniform scaling (previous work) and uniform scaling (this work).
• Validation of 1D Analogy: The results confirm that for small-amplitude waves, the solutions to the $(2+1)$-dimensional equations derived from fundamental hydrodynamics are structurally analogous to the well-known $(1+1)$-dimensional solutions (solitons, cnoidal waves, superposition waves).
• Physical Relevance: By deriving these equations from the Euler equations rather than constructing them by analogy, the authors provide a stronger theoretical basis for applying these models to real-world shallow water phenomena, particularly in scenarios where wave propagation is isotropic (uniform scaling).
• New Mathematical Structures: The identification of superposition solutions in this specific hydrodynamic context expands the known landscape of exact solutions to nonlinear wave equations, offering new tools for modeling complex wave interactions in 2D.

In conclusion, Rozmej and Karczewska successfully navigated the mathematical difficulties of uniform scaling in $(2+1)$-D fluid dynamics to provide a comprehensive set of exact analytical solutions, bridging the gap between fundamental fluid mechanics and nonlinear wave theory.