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Non-unique singular solutions for KP and modified KP equations on $\mathbb T^2$ and $\mathbb R^2$

This paper establishes the non-uniqueness of weak singular solutions for third- and fifth-order Kadomtsev-Petviashvili (KP) and modified KP equations on both $\mathbb T^2$ and $\mathbb R^2$ by constructing infinitely many such solutions with zero initial data and compact time support, while also determining sharp regularity thresholds for their existence.

Original authors: Alexandru F. Radu

Published 2026-08-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexandru F. Radu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematical physics, certain equations serve as the fundamental blueprints for understanding how waves move and interact in fluids, plasmas, and other continuous media. Among these, the Kadomtsev-Petviashvili equations, often shortened to KP equations, are celebrated for describing how ripples on a surface behave when they travel in two directions at once. For decades, mathematicians have studied these equations to understand the rules of stability: if you start with a perfectly calm, flat surface, does it stay that way, or do tiny, invisible disturbances grow into something larger? The prevailing hope in this field has been that the answer is usually "stay that way," meaning that if you begin with nothing, you should end with nothing. This expectation of uniqueness is a cornerstone of predictability in science; it suggests that the future is determined solely by the present state.

However, the behavior of these waves becomes far more complex when the equations include nonlinear terms, where the wave's height affects its own speed and shape. In these scenarios, the mathematics can become incredibly delicate. For a long time, it was believed that even in these complex, nonlinear settings, starting with a flat surface would always result in a flat surface remaining flat. But this assumption relied on the solutions being "smooth," meaning they had no sudden jumps or infinite spikes. The question remained: what happens if we allow for solutions that are rougher, or "singular," where the mathematical description breaks down in specific ways? If we relax the requirement for perfect smoothness, does the rule of uniqueness still hold, or does the system allow for multiple different futures to emerge from the exact same calm beginning?

A new study by Alexandru F. Radu has now answered this question with a definitive and surprising result. The researcher has constructed an infinite number of distinct, weak solutions to these wave equations that all start from a state of absolute zero. Imagine a pond that is perfectly still at the beginning of an experiment. According to the standard rules of physics, it should remain still. Radu's work demonstrates that, under specific mathematical conditions involving rough, singular waves, the pond could suddenly develop a complex, moving pattern of ripples, even though it started with no motion at all. This is not a simulation or a guess; it is a rigorous mathematical proof that these equations do not have a single, unique outcome when the solutions are allowed to be singular.

The paper focuses on two main types of these wave equations: the standard quadratic versions and the modified cubic versions, which describe different kinds of wave interactions. These equations were studied on two different types of spaces: a flat, infinite plane and a torus, which is a shape like a donut where the edges wrap around to meet themselves. In both settings, the researcher built these strange solutions using a technique that involves layering many tiny, high-frequency oscillations together. These oscillations are carefully designed to cancel each other out in the initial state but to interact in a way that generates a new, non-zero wave pattern as time progresses. The solutions constructed are "weak" in the sense that they are not perfectly smooth; they possess singularities, or points where the mathematical description becomes rough, but they are still valid solutions within the broader framework of the equations.

One of the most striking findings is the sheer variety of these solutions. The study proves that for any given time interval, there are infinitely many different ways the water could start moving from a standstill. These solutions are not just theoretical curiosities; they are constructed to be extremely small, meaning they can be made to have a tiny amount of energy or height, yet they still break the rule of uniqueness. The researcher showed that these solutions belong to specific categories of mathematical smoothness, defined by how their energy is distributed across different frequencies. For the quadratic equations, the solutions are rougher, while for the cubic, modified equations, the solutions can be slightly smoother but still retain the property of non-uniqueness. The study also identified precise thresholds for how rough these solutions can be before they become smooth and unique again, effectively mapping the boundary between order and chaos in these systems.

The research also tackled the issue of stationary solutions, which are wave patterns that do not change over time. The paper proves that for the KP-I equation, which describes a specific type of wave behavior, there are infinitely many stationary solutions that are singular and rough. However, the study also establishes a sharp limit: if a stationary solution is smooth enough to have a finite amount of energy (specifically, if it belongs to the L2 class), then it must be perfectly smooth and unique. This finding is crucial because it pinpoints exactly where the breakdown of uniqueness occurs. It suggests that the "wild" behavior of these equations is confined to the rougher, more singular realm, while the smoother, more physical solutions remain predictable and unique.

In the context of the whole infinite plane, the researcher also addressed a technical constraint regarding the frequencies of the waves. The constructed solutions have a "gap" in their frequencies, meaning they do not contain certain very low-frequency components. This gap is necessary to make the mathematical definitions work correctly for these rough solutions. Despite this constraint, the results hold true: even with this gap, the equations allow for infinitely many different futures starting from zero. The study further showed that these solutions can be scaled and shifted in time, creating a vast family of non-unique behaviors that can be made arbitrarily small.

This work fundamentally changes our understanding of the KP equations. It does not suggest that the physical world is unpredictable in a chaotic, random sense, but rather that the mathematical models we use to describe it have a hidden layer of complexity. When we allow for solutions that are not perfectly smooth, the equations lose their ability to predict a single future from a single past. The paper provides a concrete, mathematical demonstration that the universe of solutions to these wave equations is much richer and more diverse than previously thought, containing an infinite number of paths that all begin at the same point of stillness. By proving that these singular solutions exist and are non-unique, the study closes a long-standing gap in the theory of dispersive waves, showing that the assumption of uniqueness is not a universal law but a property that depends on the smoothness of the solution.

Technical Summary: Non-Unique Singular Solutions for KP and Modified KP Equations

Problem Statement
The paper addresses the issue of uniqueness for the Kadomtsev–Petviashvili (KP) equations and their modified counterparts on both the torus $\mathbb{T}^2$ and the whole plane $\mathbb{R}^2$ . Specifically, it investigates the third-order ( $d=3$ ) and fifth-order ( $d=5$ ) equations with quadratic ( $r=2$ ) and cubic ( $r=3$ ) nonlinearities. The central problem is to determine whether the initial value problem for these equations admits unique solutions within specific weak regularity classes. The author focuses on constructing weak singular solutions that possess zero initial data ( $u(0)=0$ ) yet are non-trivial and possess compact time support, thereby demonstrating non-uniqueness.

Methodology
The construction relies on the convex integration method, adapted to the dispersive setting of the KP equations. The core strategy involves an iterative scheme where a sequence of smooth approximate solutions $(u_q, E_q)$ is constructed. At each step, a perturbation $w_q$ is added to the current solution $u_q$ to reduce the error $E_q$ while maintaining control over the regularity and support of the solution.

Key technical components include:

Intermittent Profiles: The perturbations are built using highly oscillatory "intermittent" profiles. For quadratic nonlinearities, these are trigonometric polynomials (periodic) or localized functions (whole-space) with separated Fourier support. For cubic nonlinearities, the author constructs two distinct families of profiles:
- Isotropic profiles: Concentrated in both spatial variables with frequencies scaling as $\lambda$ .
- Parabolic profiles: Concentrated with $x$ -frequencies scaling as $\lambda$ and $y$ -frequencies scaling as $\lambda^2$ , exploiting the anisotropic dispersion of the KP equations.
Fourier-Based Nonlinear Definitions: To handle the singular nature of the solutions, the nonlinear terms (e.g., $u^r$ ) are defined via absolutely convergent Fourier convolutions. This definition ensures the nonlinearity is well-defined as a distribution and is independent of Fourier cutoffs, a crucial requirement for the weak solution definition.
Error Decomposition: The error introduced by adding the perturbation is decomposed into oscillation, Nash, dispersion, and temporal errors. The parameters of the profiles (frequency $\lambda$ , amplitude, and spacing) are tuned so that these errors decay rapidly in specific weighted Wiener norms ( $A^{-S}$ ) as the iteration proceeds.
Frequency Gaps: On $\mathbb{R}^2$ , the construction enforces a fixed gap in the $x$ -frequency ( $|\xi| \ge c_0$ ) to ensure the nonlocal operator $\partial_x^{-1}$ is well-defined and to control the transverse dispersion term $\partial_x^{-1}\partial_y^2$ .

Key Contributions and Results

Non-Uniqueness for Zero Initial Data: The paper proves the existence of infinitely many weak singular solutions with zero initial data and compact time support for:
- KP-I and KP-II ( $\kappa = \pm 1$ ) with $d \in \{3, 5\}$ and $r \in \{2, 3\}$ .
- Modified KP equations ( $r=3$ ).
- Both periodic ( $\mathbb{T}^2$ ) and whole-space ( $\mathbb{R}^2$ ) domains.
Sharpness of Regularity Thresholds: The constructed solutions belong to specific function spaces, and the author demonstrates that the regularity exponents are sharp at the $L^3$ product threshold:
- Quadratic Solutions ( $r=2$ ): Belong to $C_t L^p$ for $p < 2$ and $C_t (H^{-\sigma, 0} \cap H^{-\sigma})$ for $\sigma > 0$ .
- Cubic Solutions ( $r=3$ ):
  - One family lies in $C_t H^\alpha$ for $\alpha < 1/3$ (isotropic).
  - Another family (parabolic support) lies in $C_t H^{s, 0}$ for $s < 1/2$ and $C_t H^\alpha$ for $\alpha < 1/4$ .
- The exponents $1/3$ and $1/2$ are identified as sharp. For the fifth-order quadratic KP on $\mathbb{R}^2$ , the non-uniqueness range is shown to be "almost sharp," as unconditional uniqueness is known for $s > 0$ (Patterson [22]), leaving only the endpoint $s=0$ open.
Stationary Solutions: The paper constructs infinitely many periodic stationary weak singular solutions for KP-I and KP-II. It further proves that $L^2$ is the sharp regularity threshold for stationary KP-I solutions: any stationary solution in $L^2(\mathbb{T}^2)$ satisfying the equation in the distributional sense must be smooth ( $C^\infty$ ).
Periodic vs. Whole-Space: The results hold for both domains. On $\mathbb{R}^2$ , the solutions possess a fixed $x$ -frequency gap, and the non-uniqueness persists even with this constraint.

Significance
The paper establishes that the Cauchy problem for KP and modified KP equations fails to be well-posed in the sense of uniqueness within low-regularity weak solution classes. By constructing solutions with zero initial data that become non-trivial immediately, the work highlights the limitations of standard well-posedness theories when the solution regularity drops below certain critical thresholds (specifically related to the $L^3$ product and Sobolev embeddings).

The results refine the understanding of the "resonant obstruction" and the behavior of the flow map near the origin, showing that uniqueness can fail even when the nonlinearity is defined via absolutely convergent Fourier series. The construction of stationary solutions and the proof of the $L^2$ sharpness threshold for KP-I provide a complete picture of the regularity requirements for uniqueness in the stationary case, contrasting with the non-uniqueness found in the time-dependent singular class.

The work relies on and extends previous theories regarding well-posedness (e.g., Ionescu–Kenig–Tataru, Bourgain, Hadac) and non-uniqueness in other dispersive and fluid equations (e.g., Navier–Stokes), demonstrating that similar convex integration techniques can be successfully adapted to the anisotropic and nonlocal structure of the KP equations.

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